Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments
aa r X i v : . [ m a t h . A P ] J a n Asymptotic spreading of KPP reactive fronts inheterogeneous shifting environments ∗ King-Yeung Lam
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
Xiao Yu † School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Abstract
We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts inheterogeneous shifting habitats, with any number of shifting speeds, by further developingthe method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Ourframework addresses both reaction-diffusion equation and integro-differential equations with adistributed time-delay. The latter leads to a class of limiting equations of Hamilton-Jacobi-typedepending on the variable x/t and in which the time and space derivatives are coupled together.We will first establish uniqueness results for these Hamilton-Jacobi equations using elementaryarguments, and then characterize the spreading speed in terms of a reduced equation on a one-dimensional domain in the variable s = x/t . In terms of the standard Fisher-KPP equation,our results leads to a new class of “asymptotically homogeneous” environments which sharethe same spreading speed with the corresponding homogeneous environments. In this paper, we consider the spreading property of positive solutions u ( t, x ) to a class of functionaldifferential equations with diffusion on R : ∂ t u = ∂ xx u + f ( t, x, u )+ R τ R R Γ( τ, y ) f ( t − τ, x − y, u ( t − τ, x − y )) dydτ, in (0 , ∞ ) × R ,u ( t, x ) = φ ( t, x ) , on [ − τ , × R , (1.1) ∗ Keywords: Asymptotic speed of spread, shifting environment, Hamilton-Jacobi equation, spatio-temporal de-lays.2010 Mathematics Subject Classification. Primary: 35B40, 35K57, 35R10; Secondary: 35D40. † Corresponding Author: [email protected] < τ < ∞ . The existence and uniqueness of solutions can be established as in [48]. When f ( t, x, u ) = ( r ( t, x ) − u ) u and f ≡
0, we give new results concerning the classical Fisher-KPPequation with heterogeneous coefficients. When f is non-trivival and f <
0, the model (1.1)was motivated by the study of structured populations with distributed maturation delay, in whichjuveniles and adults have different movement patterns and f , − f are regarded as birth and deathfunctions of adult population, respectively; see [22, 23, 41, 43, 48] and the references therein.We will treat the following classes of initial data φ . Definition 1.1.
We say that the initial data φ satisfies (IC µ ) for some µ ∈ (0 , ∞ ) provided φ ∈ L ∞ ([ − τ , × R ) is non-negative and there is 0 < a < b such that ae − ( µ + o (1)) x ≤ φ ( t, x ) ≤ be − ( µ + o (1)) x for x ≫ , ∀ t ∈ [ − τ , . We say that the initial data φ satisfies (IC ∞ ) provided φ ∈ L ∞ ([ − τ , × R ) is non-negative, andlim sup x →∞ " e µx sup t ∈ [ − τ , φ ( t, x ) = 0 for every µ > . In particular, φ satisfies ( IC µ ) for some µ ∈ (0 , ∞ ) if there are positive constants a and b such that ae − µx ≤ φ ( t, x ) ≤ be − µx ; whereas φ satisfies ( IC ∞ ) if it is compactly supported in [ − τ , × R .The estimation of the asymptotic speeds of spread, or spreading speeds, is central in the studyof biological invasions. The concept, originally introduced by Aronson and Weinberger [1], saysthat a species residing in a one-dimensional domain R with population density u ( t, x ) has spreadingspeed c ∗ > η > t →∞ " sup x ≥ ( c ∗ + η ) t u ( t, x ) = 0 and lim inf t →∞ (cid:20) inf ≤ x ≤ ( c ∗ − η ) t u ( t, x ) (cid:21) > . For the Fisher-KPP equation, ∂ t u = ∂ xx u + u ( r ( t, x ) − u ) in (0 , ∞ ) × R , (1.2)it is well-known [19, 29, 1] that when r ( t, x ) ≡ r for some positive constant r , then the singlespecies has spreading speed c ∗ = 2 √ r . In addition, the spreading speed c ∗ coincides with theminimal speed of the traveling wave solutions to (1.2).For spatially periodic environment, Weinberger [45] introduced an elaborate method and showedthe existence of spreading speeds for discrete-time recursions. Subsequently, the general theory onthe existence of spreading speeds and its coincident with the minimal wave speed was developed in[33] for monotone dynamical systems and [18] for the time-space periodic monotone systems withweak compactness.Using the super- and sub-solutions method and the principal eigenvalue of time-space periodicparabolic problems, spreading properties in time-space periodic and more general environmentsare investigated in [8], as well as in [42]. 2ore recently, by combining the Hamilton-Jacobi approach [16] and homogenization ideas,Berestycki and Nadin [9, 10] showed the existence of spreading speeds for spatially almost periodic,random stationary ergodic, and very general environments. Their spreading speed is expressed asa minimax formula in terms of suitable notions of generalized principal eigenvalues in unboundeddomains. See also [34] for the relative result on the nonlocal KPP models.In the following, we outline the basic ideas of the Hamilton-Jacobi approach. The Hamilton-Jacobi approach was introduced by Freidlin [20], who employed probabilistic ar-guments to study the asymptotic behavior of solution to the Fisher-KPP equation modeling thepopulation of a single species. Subsequently, the result was generalized by Evans and Souganidis[16] using PDE arguments; see also [47]. The method can be briefly outlined as follows:1. The WKB-Ansatz: u ǫ ( t, x ) = u (cid:0) tǫ , xǫ (cid:1) and w ǫ ( t, x ) = − ǫ log u ǫ ( t, x ).2. Observe that the half-relaxed limits w ∗ ( t, x ) and w ∗ ( t, x ), given by w ∗ ( t, x ) = lim sup ǫ → t ′ ,x ′ ) → ( t,x ) w ǫ ( t ′ , x ′ ) , w ∗ ( t, x ) = lim inf ǫ → t ′ ,x ′ ) → ( t,x ) w ǫ ( t ′ , x ′ ) , (1.3)are respectively viscosity sub- and super-solutions of a given Hamilton-Jacobi equation [5].3. Show that w ∗ ( t, x ) ≤ w ∗ ( t, x ) by establishing a strong comparison result (SCR).4. Since w ∗ ≥ w ∗ by construction, we have w ∗ = w ∗ and thus w ǫ converges locally uniformly tothe unique vicosity solution ˆ w ( t, x ) of the Hamilton-Jacobi equation.5. Since ˆ w ( t, x ) ≥ t, x ), the spreading speed c ∗ can be characterized by the freeboundary separating the regions Ω := Int { ( t, x ) : ˆ w ( t, x ) = 0 } and Ω := { ( t, x ) : ˆ w ( t, x ) > } .For example, when r ( t, x ) ≡ r in the Fisher-KPP equation (1.2) with Heaviside-like initialdata, the limiting Hamilton-Jacobi equation ismin { w, w t + | w x | + r } = 0 for ( t, x ) ∈ (0 , ∞ ) × R , (1.4)with initial condition w (0 , x ) = 0 for x ≤ , and w (0 , x ) = + ∞ for x > . (1.5)By a duality correspondence of the viscosity solution of (1.4) with the value function of certain zerosum, two player differential game with stopping times, and the dynamic programming principle,it can be shown [16] that (1.4) with initial data (1.5) has a unique viscosity solutionˆ w ( t, x ) = t max (cid:26) (cid:12)(cid:12)(cid:12) xt (cid:12)(cid:12)(cid:12) − r , (cid:27) . c ∗ = 2 √ r in this case. (Note that L ( v ) = v − r is the Legendre transform of H ( p ) = p + r .)The above formula for spreading speed holds also for environments which are compact perturbationsof homogeneous environment, i.e. r ( t, x ) = r for ( t, x ) outside a compact set [8, 30]. Remark 1.2.
As we shall see, the above holds for a more general class of “asympoptoticallyhomogeneous” environments, namely, r ( t, x ) satisfying lim inf M →∞ inf [ M, ∞ ) × [ M, ∞ ) r > , and lim sup ǫ → t ′ ,x ′ ) → ( t,x ) r (cid:18) t ′ ǫ , x ′ ǫ (cid:19) = r everywhere, and lim inf ǫ → t ′ ,x ′ ) → ( t,x ) r (cid:18) t ′ ǫ , x ′ ǫ (cid:19) = r almost everywhere . See Theorem 5. An example is r ( t, x ) = r − r ( x − c t ) for some positive constants r , c and acompactly supported, non-negative function r ; see Figure (1.1a) . When r ( t, x ) = r ( x ) and is 1-periodic, it is shown in [9] that the limiting H-J equation ismin { w, w t + ˆ H ( w x ) } = 0 for ( t, x ) ∈ (0 , ∞ ) × R , where ˆ H ( p ) = k perp is characterized as the principal eigenvalue of the elliptic eigenvalue problem φ ′′ − pφ ′ + [ r ( x ) + p ] φ = kφ and φ ( x ) = φ ( x + 1) in R . Then ˆ w ( t, x ) = t max n ˆ L (cid:0) xt (cid:1) , o , where ˆ L ( v ) is the Legendre transform of ˆ H ( p ), given byˆ L ( v ) := sup p> n vp − ˆ H ( p ) o . Since w ( t, c ∗ x ) = 0, we need to solve ˆ L ( c ∗ ) = 0, i.e. sup p> n c ∗ p − ˆ H ( p ) o = 0 . Hence, c ∗ = inf p> ˆ H ( p ) p = inf p> k perp p , which gives an alternative derivation of the results in [8, 20, 46]. This framework is substantiallygeneralized recently by Berestycki and Nadin [9, 10] to almost periodic, random stationary ergodic,and more general environments, via the homogenization point of view using suitable notions ofprincipal eigenvalues in unbounded domains of the form { ( t, x ) : t > M, x > M } .On the other hand, the application of the Hamilton-Jacobi framework has largely been limitedto reaction-diffusion or nonlocal diffusion equation [12], and has not been extended to models withtime-delay such as (1.1). Yet another type of spatio-temporal heterogeneity is introduced by the recent work of Potapov andLewis [39] and Berestycki et al. [6] to model the effect of shifting of isotherms. Such heterogeneities,4ncorporating the variable x − ct in the coefficients, are not considered in the aforementioned results.By assuming that the moving source patch for a focal species is finite and is being surrounded bysink patches, [6, 39] investigated the critical patch size for species persistence. In [31], Li et al.proposed to study the Fisher-Kpp equation with a shifting habitat r ( t, x ) = r ( x − c t ): u t = u xx + u ( r ( x − c t ) − u ) in (0 , ∞ ) × R , (1.6)which describes the situation that the favorable environment is shrinking in the sense that c > r ∈ C ( R ) is increasing and satisfies r ( −∞ ) < < r (+ ∞ ). It is proved in [31] (see also [26])that if the species persists, then the species spreads at the speed 2 p r (+ ∞ ). We refer to [7, 17]for the existence of forced waves, and to [51, 50] for related results for two-competing species. Wealso mention [11] for habitats with two-shifts.More recently, the general theory on the propagation dynamics without spatial translationalinvariance was established by Yi and Zhao [49] for monotone evolution systems. A key assumptionin [49] is that the given monotone system is sandwiched by two limiting homogeneous systemsin certain translation sense, and that one of the limiting homogeneous system is unsuitable forspecies persistence while the other one has KPP structure. It was shown that the spreading speedcoincides with the spreading speed in the limiting homogeneous systems with KPP structure. Inparticular, [49] generalizes [31] the context of (1.6).An interesting case arises when both of the limiting systems has KPP structures, but withdifferent spreading speeds, e.g. c − = 2 p r ( −∞ ) , c + = 2 p r (+ ∞ ) for (1.6). The spreading behaviorwhen 0 < r ( −∞ ) < r (+ ∞ ) is especially subtle. In [25], it is proved that c ∗ = c + if c ≪ c ∗ = c − if c ≫
1. But the general case remains open. By the maximum principle, it is notdifficult to see that the actual spreading speed of the species must be no slower than c − andno faster than c + . However, there is a fundamental difference between the homogeneous andheterogeneous cases as far as the spreading speed is concerned. As discussed earlier, the spreadingspeed can be computed locally when the environment is homogeneous or periodic. However, whenthe environment is heterogeneous and shifting, it is not always possible to calculate the spreadingspeed using local considerations [38]. By the Hamilton-Jacobi approach, we can gain a more”global” point of view and show that the spreading speed of (1.6) can be subject to nonlocalpulling effect [24, 21], and is influenced by the speed c of the shifting environment: c ∗ = c + = 2 √ r if c ≤ √ r , c − √ r − r + r c −√ r − r if 2 √ r < c < √ r − r + √ r ) ,c − = 2 √ r if c ≥ √ r − r + √ r ) , where r = r (+ ∞ ), r = r ( −∞ ) and r > r >
0. See Theorem 6(iv) for details. We point out thatit is possible to derive this particular result as a consequence of [24], which relies on the changeof coordinates x ′ = x − c t to transform (1.6) into a problem with spatially heterogeneous, but5emporally constant coefficients. But what about environments with more than one speed of shift,such as r ( t, x ) = r ( x − c t ) + r ( x − c t )? In this paper, we will further develop the method based on Hamilton-Jacobi equations to determinethe spreading speed of a species in a heterogeneous shifting habitat, with multiple (or indeedinfinitely many) shifting speeds, which leads to a new class of Hamilton-Jacobi equations. Ourapproach will provide a unified framework to address both reaction-diffusion equation, and integro-differential equation with exponentially decaying or compactly supported initial data; see Definition1.1. The spreading speed will be characterized in terms of a reduced Hamilton-Jacobi equation in asingle variable s = x/t . We will also provide a new proof of uniqueness for the underlying Hamilton-Jacobi equations with elementary arguments. As a by-product of our approach, we obtain a newclass of “asymptotically homogeneous” environments which share the same spreading speed withthe corresponding homogeneous environment; see Theorem 5.For i = 1 ,
2, let R i , R i ∈ L ∞ ( R ) be given by (note that R i , R i only depend on x/t below) R i ( x/t ) := lim sup ǫ → t ′ ,x ′ ) → ( t,x ) ∂ u f i ( t ′ /ǫ, x ′ /ǫ,
0) for ( t, x ) ∈ (0 , ∞ ) × R R i ( x/t ) := lim inf ǫ → t ′ ,x ′ ) → ( t,x ) ∂ u f i ( t ′ /ǫ, x ′ /ǫ,
0) for ( t, x ) ∈ (0 , ∞ ) × R . (1.7)We recall the concept of local monotonicity from [13]. Definition 1.3.
We say that h : R → R is locally monotone if, for each s , eitherlim δ → inf | s i − s | <δs f ∈ C ( R + × R × R + ) ∩ C ( R + × R × [0 , δ ∗ ]) and f ∈ C ([ − τ , ∞ ) × R × R + ) ∩ C ([ − τ , ∞ ) × R × [0 , δ ∗ )) satisfy f i ( t, x,
0) = 0 and f i ( t, x, u ) ≤ u∂ u f i ( t, x,
0) for all ( t, x, u ) , i = 1 , , sup R + × R × [0 ,M ] | ∂ u f i ( t, x, s ) | < ∞ for each M > i = 1 , , Furthermore, for any η ′ >
0, there exists δ ∗ > t, x ) such that f i ( t, x, u ) ≥ u ( ∂ u f i ( t, x, − η ′ ) , if 0 ≤ u ≤ δ ∗ , i = 1 , . (1.8)(H2) There exists L > G ( t, x, L, L ) ≤ t, x ) ∈ (0 , ∞ ) × R and L ∈ [ L , ∞ ), where G ( t, x, u, v ) := f ( t, x, u ) + Z τ Z R Γ( τ, y ) f ( t − τ, x − y, v ) dydτ. f ( t, x, v ) ≡
0, or f ( t, x, v ) > − τ , ∞ ) × R × (0 , ∞ ) and Γ ∈ L ([0 , τ ] × R ) is non-negativeand satisfies R τ R R Γ( τ, y ) dydτ = 1 and R τ R R Γ( τ, y ) e py + qτ dydτ < ∞ for all ( p, q ) ∈ R .(H4) The functions R ( s ) and R ( s ), given by (1.7), satisfy R i ( s ) = R i ( s ) a.e. in (0 , ∞ ) , and R ( s ) + R ( s ) > s > . (1.9)and one of the following holds:(i) R and R are both non-increasing, or both non-decreasing;(ii) R is continuous, and R is monotone;(iii) R is piecewise constant, and R and R + R are both locally monotone.(H5) For any φ ∈ L ∞ ([ − τ , × R ) with φ ≥6≡ s > u of(1.1) satisfies lim inf t → + ∞ inf ≤ x ≤ st u ( t, x, φ ) > . The hypothesis (H1) says that the nonlinearity is sublinear. In case f is nontrivial, we only assume f ( · , · , u ) is monotone close to 0, in other words, the full system might not admit the comparisonprinciple; (H2) is a self-limitation assumption; (H3) says that Γ has finite moments to ensure afinite spreading speed; some sufficient conditions to guarantee uniqueness in the underlying HJequation are given in (H4); (H5) means the population spreads successfully to the right. Remark 1.4.
Hypothesis (H5) can be guaranteed if lim inf t →∞ (cid:20) inf x ∈ [0 ,st ] ∂ u f ( t, x, (cid:21) > . (See [8].)This is equivalent to R ( s ) > s ∈ [0 , s ]. More generally, if there exist s, c > c ( c ) /
4, one can apply a change of coordinates x ′ = x − c t , which introducesa drift term, then (H5) holds, so that the arguments of this paper can also be applied. This isconnected with the results in [25, 26, 31] when R ( s ) < −∞ , c ] and R ( s ) > c , ∞ ). Remark 1.5.
For example, take Γ to be any probability kernel on [0 , τ ] × R with finite moments,and f ( t, x, u ) = ( r ( x − c t ) − u ) u and f ( t, x, v ) = r ( x − c t ) ve − v . where r i are monotone functions such that inf r + inf r >
0. If r , r are both increasing or bothdecreasing, then (H1)-(H5) are satisfied with R i ( s ) = r i ( −∞ ) for s < c i r i ( −∞ ) ∨ r i (+ ∞ ) for s = c i r i (+ ∞ ) for s > c i and R i ( s ) = r i ( −∞ ) for s < c i r i ( −∞ ) ∧ r i (+ ∞ ) for s = c i r i (+ ∞ ) for s > c i . Remark 1.6.
When f ≡
0, and f ( t, x, u ) = ( r ( t, x ) − u ) u , then (H1)-(H5) reduces to7 ˜H) R ( s ) is locally monotone, R ( s ) > s ≥ R = R a.e.where R ( s ) = lim sup t →∞ s ′→ s r ( t, s ′ t ) and R ( s ) = lim inf t →∞ s ′→ s r ( t, s ′ t ).As observed in [36], the exact spreading speed can be characterized in terms of the followingHamilton-Jacobi equation on a one-dimensional domain:min (cid:26) ρ, ρ − sρ ′ + | ρ ′ | + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e ( ρ − sρ ′ ) τ + ρ ′ y dydτ (cid:27) = 0 in (0 , ∞ ) . (1.10)To tackle the classical issue of uniqueness for the Hamilton-Jacobi equation (1.10), we need one ofthe following to hold for the function R , and sometimes R , as s → + ∞ .(H6) R ( s ) is identically zero, or non-increasing in [ s , ∞ ) for some s ,(H6 ′ ) lim s → + ∞ R ( s ) exists and is positive, and R ( s ) ≥ sup ( s, ∞ ) R ! − o (cid:18) s (cid:19) for s ≫ ′′ ) One of lim s → + ∞ R i ( s ) exists, lim sup s →∞ R ( s ) >
0, and Γ( τ, y ) = 0 in [0 , τ ] × ( −∞ , , for some τ ∈ (0 , τ ].Next, we specify the spreading speeds ˆ s µ as a free boundary point of the solution of (1.10),which depends implicitly on R ( s ), R ( s ), Γ( τ, y ) and µ ∈ (0 , ∞ ]. For the the definition of viscositysuper- and sub-solutions, see Section 2. Proposition 1.7.
Let R ∈ L ∞ (0 , ∞ ) be locally monotone, R ∈ L ∞ (0 , ∞ ) non-negative, andeither monotone or piecewise constant. Suppose either (a) µ ∈ (0 , ∞ ) , or (b) µ = ∞ and one of (H6), (H6 ′ ) or (H6 ′′ ) holds,then there exists a unique viscosity solution ˆ ρ µ ∈ C ([0 , ∞ )) of (1.10) such that ˆ ρ µ (0) = 0 , and lim s → + ∞ ˆ ρ µ ( s ) s = µ. (1.11) Furthermore, s ˆ ρ µ ( s ) is non-decreasing in [0 , ∞ ) , so that the free boundary point ˆ s µ := sup { s : ˆ ρ µ ( s ) = 0 } (1.12) is well-defined. Theorem 1.
Assume (H1)-(H5) . Let u be a solution to (1.1) with initial data satisfying (IC µ ) forsome µ ∈ (0 , ∞ ) , then lim t →∞ sup x ≥ (ˆ s µ )+ η ) t u ( t, x ) = 0 for each η > , lim inf t →∞ inf ≤ x ≤ (ˆ s µ − η ) t u ( t, x ) > for each < η < ˆ s µ , (1.13) where ˆ s µ is given in (1.12) . heorem 2. Assume (H1)-(H5) . Let u be a solution to (1.1) with initial data satisfying (IC ∞ ).If one of the conditions (H6) , (H6 ′ ) or (H6 ′′ ) holds, then lim t →∞ sup x ≥ (ˆ s ∞ )+ η ) t u ( t, x ) = 0 for each η > , lim inf t →∞ inf ≤ x ≤ (ˆ s ∞ − η ) t u ( t, x ) > for each < η < ˆ s ∞ , (1.14) where ˆ s ∞ is given in (1.12) with µ = ∞ . Remark 1.8.
Since we do not impose assumptions on f i ( t, x, u ) for large u , except that it eventu-ally becomes negative in (H2), the convergence of u to a homogeneous equilibrium does not holdin general. However, if we strengthen (H2) to(H2 ′ ) For i = 1 , f i ( t, x, ≡
0, and ( u − f i ( t, x, u ) < t, x, u ) such that u = 1.Then one can argue as in [48] that u ( t, x ) → x < ˆ s µ t and t ≫ Remark 1.9.
To compare our approach with that of Berestycki and Nadin, we only homogenizealong the ray x/t = s for each s here, while in [9, 10], the information in { ( t, ye ) : t ≫ , x ≫ } foreach direction e is homogenized via the notion of principal eigenvalues of the parabolic problems.In [10], it is demonstrated that in higher dimensions, sometimes the spreading speed in direction e does not depend only on what happens in the e direction. The same can be observed in a shiftinghabitat on R , where the spreading speed is nonlocally determined; see Theorems 4 and 6. The following theorem concerns the spreading in “asymptotically homogeneous” environments, andgeneralizes the spreading results of [22, 48]. For simplicity, we assume that Γ( τ, y ) is symmetric inthe variable y for each τ ∈ [0 , τ ] in the next theorems. Theorem 3.
Let u be a solution to (1.1) with initial data satisfying (IC µ ) for some µ ∈ (0 , ∞ ) .Assume (H1)-(H5) and, in addition, for i = 1 , , there are positive constants r , r + r such that R i ( s ) = r i for every s ∈ R , R i ( s ) = r i for almost every (a.e.) s ∈ R . (1.15) Then (1.13) holds, and the spreading speed ˆ s µ is given by ˆ s µ = inf p> λ ( p ) p if µ ∈ [ µ ∗ , ∞ ) , λ ( µ ) µ if µ ∈ (0 , µ ∗ ) , (1.16) where λ ( p ) : R → (0 , ∞ ) is uniquely defined by the implicit formula ∆( λ, p ) := − λ + p + r + r Z τ Z R Γ( τ, y ) e py − λτ dydτ = 0 . (1.17) and µ ∗ > such that λ ( µ ∗ ) µ ∗ = inf p> λ ( p ) p > . emark 1.10. A sufficient condition of (H4)-(H5) and (1.15) is when f i are independent of t, x such that f ′ i (0) = r i . Moreover, the homogeneous coefficients case could be extended to a largeclass of space-time heterogeneous problems, e.g., P i =1 inf [0 , ∞ ) × R ∂ u f i ( t, x, >
0, and r − m X i =1 k i ( x − c i t α i ) ≤ ∂ u f ( t, x, ≤ r and 0 < r − m X i =1 ˜ k i ( x − c i t α i ) ≤ ∂ u f ( t, x, ≤ r , where m ∈ N , α i , c i , r , r + r are positive constants, and k i ( · ) and ˜ k i ( · ) are non-negative functionsthat are compactly supported on R .Next, we turn our attention to environments with one shift. Let R i, − , R i, + be fixed constants.For p ∈ R , define λ − ( p ) and λ + ( p ) implicitly by0 = ∆ ± ( λ, p ) = − λ + p + R , ± + R , ± Z τ Z R Γ( t, y ) e py − λτ dydτ, Define ( c ∗− , µ ∗− ) and ( c ∗ + , µ ∗ + ) by 0 < c ∗± = inf p> λ ± ( p ) p = λ ( µ ∗± ) µ ∗± . Theorem 4.
Assume (H1)-(H5) and, in addition, for i = 1 , , there are constants R i, − , R i, + and c > such that R i ( s ) = R i, + if s ≥ c R i, − if s < c for every s ∈ R , R i ( s ) = R i, + if s > c R i, − if s ≤ c for a.e. s ∈ R , (1.18) and R i, − ≤ R i, + and R , + + R , + > R , − + R , − > . Then (1.13) holds, and the rightward spreading speed ˆ s µ can be given by (see Figure 1.2)(i) µ ∈ (0 , µ ∗ + ] , then ˆ s µ ( c ) = λ + ( µ ) /µ if c ≤ λ + ( µ ) /µ,λ − ( p ( c , µ )) /p ( c , µ ) if c > λ + ( µ ) /µ and p ( c , µ ) < µ ∗− c ∗− , otherwise (1.19) where p ( c , µ ) is the smallest root of c p − λ − ( p ) = c µ − λ + ( µ ) . (1.20)10 ii) µ ∈ ( µ ∗ + , ∞ ) , then ˆ s µ ( c ) = c ∗ + if c ≤ c ∗ + ,λ − (¯ p ( c )) / ¯ p ( c ) if λ ′ + ( µ ∗ + ) < c ≤ λ ′ + ( µ ) and ¯ p ( c ) < µ ∗− ,λ − ( p ) /p if c > λ ′ + ( µ ) and p ( c , µ ) < µ ∗− ,c ∗− , otherwise . (1.21) where p ( c , µ ) is the smallest root of (1.20) and ¯ p ( c ) is the smallest root of c p − λ − ( p ) = c Ψ + ( c ) − λ + (Ψ + ( c )) (1.22) In particular, if µ = ∞ , then ˆ s ∞ ( c ) = c ∗ + if c ≤ c ∗ + ,λ − (¯ p ( c )) / ¯ p ( c ) if c ∈ ( c ∗ + , ¯ c ) ,c ∗− if c ≥ ¯ c . (1.23) where ¯ c is the unique positive number such that ¯ p (¯ c ) = µ ∗− . Remark 1.11.
A sufficient condition of (H4)-(H5) and (1.18) is when there exists c > f i ( t, x, u ) = f i ( x − c t, u ) satisfying R i, − = ∂ u f i ( −∞ , ≤ ∂ u f i ( y, ≤ ∂ u f i (+ ∞ ,
0) = R i, + holds for all y ∈ R ,R , + + R , + > R , − + R , − > . -50 -40 -30 -20 -10 0 10 20 30 40 500.20.30.40.50.60.70.80.91 (a) -50 -40 -30 -20 -10 0 10 20 30 40 500.20.30.40.50.60.70.80.91 (b) Figure 1.1: In the left panel, r ( t, x ) = r − r ( x − c t ) for a compactly supported, non-negativeand bounded function r with r , c > r ( t, x ) = ˜ r ( x − c t ) for a non-decreasing function ˜ r with 1 / r ( −∞ ) < ˜ r (+ ∞ ) = 1where Theorem 6 applies; red curves represents r ( t, x ) = α ˜ r ( x − c t ) + (1 − α )˜ r ( x − c t ) for c > c > α ∈ (0 , / Figure 1.2: Parameter regions in ( µ, c )-plane corresponding to various cases of Theorem 4. In this subsection, we state our new results for the Fisher-KPP equation (1.2), which could beeasily derived from the results in last section or the arguments in Section 3. Throughout thissubsection, we impose the following assumption on r .(F) r ∈ L ∞ ( R × R ) and lim inf M →∞ (cid:20) inf [ M, ∞ ) × [ M, ∞ ) r ( t, x ) (cid:21) > r > t →∞ , s ′ → s r ( t, s ′ t ) = r for every s ∈ (0 , ∞ ) , lim inf t →∞ , s ′ → s r ( t, s ′ t ) = r for a.e. s ∈ (0 , ∞ ) . (1.24)An example is r ( t, x ) = r − R ∞ r ( x − st ) dµ ( s ), where r is a non-negative function with compactsupport, and µ is σ -finite measure on [0 , ∞ ). See Figure (1.1a) for the prototypical example r ( t, x ) = r − r ( x − c t ). Theorem 5.
Consider the Cauchy problem (1.2) , with initial data u ( x ) satisfying (IC µ ) for some µ ∈ (0 , + ∞ ] , respectively. If r ( t, x ) satisfies (F) , and also (1.24) for some constant r > , thenthe spreading speed ˆ s µ is given by (1.12) . Furthermore, it can be given explicitly as follows ˆ s µ = µ + r µ if µ ∈ (0 , √ r ) , and ˆ s µ = 2 √ r if µ ∈ [ √ r , ∞ ] . (1.25) Remark 1.12.
In (1.24), the convergence of limit superior “everywhere” cannot be relaxed to“almost everywhere”, because it is possible for locked waves to form, i.e. c ∗ = max { √ r , c } [24].On the other hand, if the condition “almost everywhere” in the convergence of limit inferior isstrengthened to “everywhere”, the spreading result is proved in [10, Proposition 3.1].12ore generally, we observe that (H1)-(H6) are satisfied for the classical KPP equation (1.2)when r ( t, x ) = m X i =1 r i ( x − c t )such that c i ∈ R are distinct, and for each i , r i ∈ C ( R ) is strictly positive and monotone; seeRemark 1.6. Hence, the spreading speed can be characterized by the free-boundary problem(1.12). Below, we completely work out the spreading speed when there is only one shifting speed. Theorem 6.
Consider the Cauchy problem (1.2) , with initial data u ( x ) satisfying (IC µ ) for some µ ∈ (0 , ∞ ] , respectively. If r ( t, x ) satisfies (F) , and there are constants r > r > such that lim sup ǫ → t ′ ,x ′ ) → ( t,x ) r (cid:18) t ′ ǫ , x ′ ǫ (cid:19) = r if x ≥ c t,r if x < c t. for all ( t, x ) ∈ (0 , ∞ ) × R , and lim inf ǫ → t ′ ,x ′ ) → ( t,x ) r (cid:18) t ′ ǫ , x ′ ǫ (cid:19) = r if x > c t,r if x ≤ c t. for a.e. ( t, x ) ∈ (0 , ∞ ) × R . (e.g. r ( t, x ) = ˜ r ( x − c t ) for some increasing function ˜ r such that ˜ r ( −∞ ) = r and ˜ r ( ∞ ) = r ,see figure (1.1b) .) Then the spreading speed ˆ s µ is given by (1.12) . Furthermore, it can be givenexplicitly as follows.(i) µ ∈ (0 , √ r ]ˆ s µ ( c ) = µ + r µ if c ∈ ( −∞ , µ + r µ ] , c − √ ( c − µ ) +4( r − r )2 + r c − √ ( c − µ ) +4( r − r ) if c ∈ ( µ + r µ , + ∞ ) , (1.26) (ii) µ ∈ ( √ r , √ r )ˆ s µ ( c ) = µ + r µ if c ∈ ( −∞ , µ + r µ ] , c − √ ( c − µ ) +4( r − r )2 + r c − √ ( c − µ ) +4( r − r ) if c ∈ ( µ + r µ , µ + r − r µ −√ r ) , √ r if c ∈ [ µ + r − r µ −√ r . + ∞ ) , (1.27) (iii) µ ∈ [ √ r , √ r + √ r − r )ˆ s µ ( c ) = √ r if c ∈ ( −∞ , √ r ] , c − √ r − r + r c −√ r − r if c ∈ (2 √ r , µ ] , c − √ ( c − µ ) +4( r − r )2 + r c − √ ( c − µ ) +4( r − r ) if c ∈ (2 µ, µ + r − r µ −√ r ) , √ r if c ∈ [ µ + r − r µ −√ r . + ∞ ) , (1.28)13 iv) µ ∈ [ √ r + √ r − r , ∞ ]ˆ s µ ( c ) = √ r if c ∈ ( −∞ , √ r ] , c − √ r − r + r c −√ r − r if c ∈ (2 √ r , √ r + 2 √ r − r ) , √ r if c ∈ [2 √ r + 2 √ r − r , ∞ ) . (1.29)Before we close, we also state the spreading speed when there are two shifts in the environment,which makes use of a previous result regarding a special case of (1.10) when R ≡ R ispiecewise constant. (For the derivation, see Theorem C in https://arxiv.org/pdf/1910.04217.pdf.) Theorem 7.
Consider (1.2) , with initial data u ( x ) satisfying (IC ∞ ) . Suppose r ( t, x ) satisfies (F) , and there are positive constants > r > r > and c > c > such that lim sup ǫ → t ′ ,x ′ ) → ( t,x ) r (cid:18) t ′ ǫ , x ′ ǫ (cid:19) = if x ≥ c t,r if c t ≤ x < c t,r if x < c t. for every ( t, x ) ∈ (0 , ∞ ) × R , and lim inf ǫ → t ′ ,x ′ ) → ( t,x ) r (cid:18) t ′ ǫ , x ′ ǫ (cid:19) = if x > c t,r if c t < x ≤ c t,r if x ≤ c t. for a.e. ( t, x ) ∈ (0 , ∞ ) × R . (e.g. r ( t, x ) = − r − r ˜ r ( x − c t )+ r − r − r ˜ r ( x − c t ) for some increasing function ˜ r such that ˜ r ( −∞ ) = r and ˜ r ( ∞ ) = 1 , see Figure (1.1b) .) Then the spreading speed ˆ s µ is given by (1.12) . Furthermore,it can be given explicitly as follows. ˆ s ∞ ( c , c ) = for c ≤ µ + r µ for c > , µ < √ r and c ≤ µ + r µ √ r for µ ≥ √ r and c ≤ √ r ¯ p + r ¯ p for c > , µ < √ r , ¯ p < √ r and c > µ + r µ , for µ ≥ √ r , c < µ, ¯ p < √ r and c > √ r p + r p for µ ≥ √ r , c ≥ µ, p < √ r and c > √ r , √ r otherwise, (1.30) where µ = c − √ − r , ¯ p = c − r ( c − µ ) + r − r , p = c − √ r − r . (1.31)14 Proof of Theorems 1 and 2
Let u ( t, x ) be the unique solution of (1.1) with initial data φ satisfying (IC µ ) for some µ ∈ (0 , ∞ ].To analyze the spreading behavior of u ( t, x ), we introduce the large time and large space scalingparameter ǫ u ǫ ( t, x ) = u (cid:18) tǫ , xǫ (cid:19) , (2.1)and relate the limit of u ǫ as ǫ → { w, H ( x/t, ∂ t w, ∂ x w ) } = 0 for ( t, x ) ∈ (0 , ∞ ) × (0 , ∞ ) , (2.2)and the second one is time-independent:min { ρ, H ( s, ρ − sρ ′ , ρ ′ ) } = 0 for s ∈ (0 , ∞ ) , (2.3)where H ( s, q, p ) = q + p + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e qτ + py dydτ. Note that H ( s, q, p ) is u.s.c., since the functions R ( s ) and R ( s ) are u.s.c by (H4). Remark 2.1.
We choose to work with (2.2) since it is more explicitly connected with (1.1) and(2.3). It is possible to rewrite (2.2) into a more standard form: min { w, ∂ t w + ˜ H ( x/t, ∂ x w ) } = 0,where ˜ H ( s, p ) is defined implicitly by H ( s, q, p ) = 0 if and only if − q = ˜ H ( s, p ) , by exploiting the monotonicity of H in q . See Appendix A for details.Indeed, under the scaling of (2.1), the problem (1.1) can be rewritten as ∂ t u ǫ = ǫ∂ xx u ǫ + ǫ [ F ǫ ( t, x, u ǫ ) + f ,ǫ ( t, x, u ǫ )] , t > , x ∈ R ,u ǫ ( θ, x ) = φ ( θ/ǫ, x/ǫ ) , ( θ, x ) ∈ [ − ǫτ, × R , (2.4)where F ǫ ( t, x, u ǫ ) = Z τ Z R Γ( τ, y ) f ,ǫ ( t − ǫτ, x − ǫy, u ǫ ( t − ǫτ, x − ǫy )) dydτ with f ,ǫ ( t, x, v ) := f ( t/ǫ, x/ǫ, v ) and f ,ǫ ( t, x, u ) := f ( t/ǫ, x/ǫ, u ).The proof of Theorem 1 relies on a WKB approach in front propagation, which was introducedin [4, 16, 40]. It is based on the real phase defined by the Hopf-Cole transform w ǫ ( t, x ) = − ǫ log u ǫ ( t, x ) . w ǫ is ∂ t w ǫ − ǫ∂ xx w ǫ + | ∂ x w ǫ | + F ǫ ( t,x,u ǫ )+ f ,ǫ ( t,x,u ǫ ) u ǫ = 0 , t > , x > ,w ǫ ( θ, x ) = − ǫ log φ ( θ/ǫ, x/ǫ ) , x ≥ , θ ∈ [ − ǫτ, ,w ǫ ( t,
0) = − ǫ log u ǫ ( t, , t ≥ − ǫτ. (2.5)In the following, we apply the half-relaxed limit method, due to Barles and Perthame [5], topass to the (upper and lower) limits of w ǫ . More precisely, for each ( t, x ) ∈ (0 , ∞ ) × R , we set w ∗ ( t, x ) = lim sup ǫ → t ′ ,x ′ ) → ( t,x ) w ǫ ( t ′ , x ′ ) , w ∗ ( t, x ) = lim inf ǫ → t ′ ,x ′ ) → ( t,x ) w ǫ ( t ′ , x ′ ) . (2.6)We will show that w ∗ and w ∗ are respectively viscosity sub- and super-solution of the time-dependent problem (2.2) (see Proposition 2.7). To pass to the (local uniform) limit, it is necessaryto establish the uniqueness of viscosity solution for such problems so that w ∗ = w ∗ and hence w ǫ converges uniformly as ǫ →
0. If the initial data of u has compact support, then the initial data of w ∗ , w ∗ can be infinite and the uniqueness does not follow from standard PDE proofs. Previously,this mathematical issue was tackled in [16] by way of a correspondence with the value functionof a zero sum, two player differential game with stopping times and the dynamics programmingprinciple; see also [15] for a method based on semigroup method. Our main novelty here is toprovide an elementary proof of the uniqueness based on the 1-homogeneity of w ∗ (resp. w ∗ ): w ∗ ( t, x ) = lim sup ǫ → t ′ ,x ′ ) → ( t,x ) − ǫ log u (cid:18) t ′ ǫ , x ′ ǫ (cid:19) = t lim sup ǫ → t ′′ ,x ′′ ) → (1 ,x/t ) − ( ǫ/t ) log u (cid:18) t ′′ ǫ/t , x ′′ ǫ/t (cid:19) = tw ∗ (1 , x/t ) . (2.7)Hence, there exists ρ ∗ ( s ) and ρ ∗ ( s ) such that w ∗ ( t, x ) = tρ ∗ ( x/t ) and w ∗ ( t, x ) = tρ ∗ ( x/t ) . (2.8)We will show that ρ ∗ ( s ) and ρ ∗ ( s ) are respectively the sub- and super-solution of the problem (2.3)in a one-dimensional domain (see Lemma 2.6). By showing a novel comparison result for (2.3), wehave ρ ∗ ( s ) = ρ ∗ ( s ) (see Proposition 2.11), so that they can be identified with the unique solutionˆ ρ µ of (1.10), which satisfies (see Proposition 1.7)ˆ ρ µ ( s ) = 0 in [0 , ˆ s µ ] , and ˆ ρ µ ( s ) > s µ , ∞ ) . Hence, w ∗ ( t, x ) = w ∗ ( t, x ) = t ˆ ρ µ ( x/t ), and we see that w ǫ ( t, x ) converges in C loc ((0 , ∞ ) × R ), andlim ǫ → w ǫ ( t, x ) = 0 for 0 ≤ x < ˆ s µ t, and lim ǫ → w ǫ ( t, x ) > x > ˆ s µ t. From this, the asymptotic behavior of u ǫ ( t, x ) can then be inferred.Let w ∗ and w ∗ be the half-relaxed limits as given by (2.6). The following lemma indicates that w ∗ and w ∗ are well-defined and finite-valued everywhere.16 roposition 2.2. Let φ satisfy ( IC µ ) for some µ ∈ (0 , ∞ ] and w ǫ be the solution of (2.5) , then ≤ w ∗ ( t, x ) ≤ w ∗ ( t, x ) < ∞ for each ( t, x ) ∈ (0 , ∞ ) × R , (2.9) and w ∗ ( t,
0) = w ∗ ( t,
0) = 0 for t > , w ∗ (0 , x ) = w ∗ (0 , x ) = µx if µ ∈ (0 , ∞ ) , ∞ if µ = ∞ . for x > . (2.10) Proof.
First, we claim that there exists
L > w ǫ ( t, x ) ≥ − ǫ log L in [0 , ∞ ) × R .Let L be given in (H2). It suffices to choose L ∈ [ L , ∞ ) such that 0 ≤ u ( t, x, φ ) ≤ L in[ − τ , × R . Then (H2) and the maximum principle yield u ( t, x, φ ) ≤ L for ( t, x ) ∈ [0 , ∞ ) × R . This proves that w ∗ ( t, x ) ≥ t, x ).The first part of (2.10) follows from hypothesis (H5). The second part of (2.10) follows fromLemma B.1 in case µ ∈ (0 , ∞ ), or Lemma B.3 in case µ = ∞ .It remains to show the upper bound of (2.9). By noting that ∂ t w ǫ − ǫ∂ xx w ǫ + | ∂ x w ǫ | ≤ C for ( t, x ) ∈ (0 , ∞ ) × (0 , ∞ ) , this follows from the proof of [35, Lemma 3.2].We recall the classical definition of discontinuous viscosity super- and sub-solutions to (2.3)following [3]. See Definition 2.5 for the corresponding definition for (2.2). Definition 2.3.
We say that a lower semicontinuous function ˆ ρ is a viscosity super-solution of(2.3) if ˆ ρ ≥
0, and for all test functions φ ∈ C , if s is a strict local minimum of ˆ ρ − φ , thenˆ ρ ( s ) − s φ ′ ( s ) + | φ ′ ( s ) | + R ∗ ( s ) + R ∗ ( s ) Z τ Z R Γ( τ, y ) e (ˆ ρ ( s ) − s φ ′ ( s )) τ + φ ′ ( s ) y dydτ ≥ . We say that an upper semicontinuous function ˆ ρ is a viscosity sub-solution of (2.3) if for all testfunctions φ ∈ C , if s is a strict local minimum of ˆ ρ − φ and ˆ ρ ( s ) >
0, thenˆ ρ ( s ) − s φ ′ ( s ) + | φ ′ ( s ) | + R , ∗ ( s ) + R , ∗ ( s ) Z τ Z R Γ( τ, y ) e (ˆ ρ ( s ) − s φ ′ ( s )) τ + φ ′ ( s ) y dydτ ≤ . Finally, ˆ ρ is a viscosity solution of (2.3) if and only if ˆ ρ is a viscosity super- and sub-solution.The functions R ∗ i and R i, ∗ appeared above denote respectively the upper semicontinuous (u.s.c)and lower semcontinuous (l.s.c) envelope of R i , that is, R ∗ i ( s ) = lim sup s ′ → s R i ( s ′ ) and R i, ∗ ( s ) = lim inf s ′ → s R i ( s ′ ) . Remark 2.4. If R i ( i = 1 ,
2) satisfies (H4), then they are upper semi-continuous (u.s.c.) so wehave R i ( s ) ≡ R ∗ i ( s ) everywhere in [0 , ∞ ). 17 efinition 2.5. We say that a lower semicontinuous function ˆ w is a viscosity super-solution of(2.2) if ˆ w ≥
0, and for all test functions φ ∈ C , if ( t , x ) is a strict local minimum of ˆ w − φ , then H ( x /t , ∂ t φ ( t , x ) , ∂ x φ ( t , x ) ≥ . We say that an upper semicontinuous function ˆ w is a viscosity sub-solution of (2.2) if for all testfunctions φ ∈ C , if ( t , x ) is a strict local minimum of ˆ w − φ and ˆ w ( t , x ) >
0, then H ∗ ( x /t , ∂ t φ ( t , x ) , ∂ x φ ( t , x ) ≤ . Finally, ˆ w is a viscosity solution of (2.2) if and only if ˆ w is a viscosity super- and sub-solution.In the above definition, H ∗ ( s, q, p ) = q + p + ( R ) ∗ ( s ) + ( R ) ∗ ( s ) R τ R R Γ( τ, y ) e qτ + py dydτ . Wehave also used the fact that H is u.s.c..Below, we relate the notion of viscosity super- and sub-solutions of (2.2) and (2.3). Lemma 2.6.
Suppose w ( t, x ) and ρ ( s ) are two functions such that w ( t, x ) = tρ ( x/t ) in Ω := { ( t, x ) : 0 < x < c b t } for some c b ∈ (0 , ∞ ] . Then ρ ( s ) is a viscosity sub-solution (resp. super-solution) of (2.3) in the interval (0 , c b ) if andonly if w ( t, x ) is a viscosity sub-solution (resp. super-solution) of (2.2) in Ω .Proof. The proof follows from a minor modification of that in [36, Lemma 2.3]. Below we onlyshow the equivalence of viscosity sub-solutions.Let ρ ( s ) be a viscosity sub-solution of (2.3) in (0 , c b ). We must verify that w ( t, x ) = tρ (cid:0) xt (cid:1) isa viscosity sub-solution of (2.2). For any test function ϕ ∈ C , suppose that w − ϕ attains a strictlocal maximum at point ( t ∗ , x ∗ ) ∈ Ω such that w ( t ∗ , x ∗ ) >
0. Since w ( t, x ) = tρ (cid:0) xt (cid:1) and t ∗ > ρ (cid:16) x ∗ t ∗ (cid:17) > f ( y ) := yρ (cid:16) x ∗ t ∗ (cid:17) − ϕ ( yt ∗ ,yx ∗ ) t ∗ admits a strict local maximum at y = 1,so that letting s ∗ = x ∗ /t ∗ , we have ρ ( s ∗ ) − ∂ t ϕ ( t ∗ , x ∗ ) − s ∗ ∂ x ϕ ( t ∗ , x ∗ ) = 0 . (2.11)Next, set φ ( s ) := ϕ ( t ∗ , st ∗ ) /t ∗ . We observe that ρ ( s ) − φ ( s ) takes a strict local maximum point s = s ∗ and ρ ( s ∗ ) >
0. Note that φ ′ ( s ∗ ) = ∂ x ϕ ( t ∗ , x ∗ ), it follows from (2.11) that ∂ t ϕ ( t ∗ , x ∗ ) = ρ ( s ∗ ) − s ∗ φ ′ ( s ∗ ) . Hence at the point ( t ∗ , x ∗ ), we have ∂ t ϕ + | ∂ x ϕ | + R , ∗ ( x ∗ /t ∗ ) + R , ∗ ( x ∗ /t ∗ ) Z τ Z R Γ( τ, y ) e τ∂ t ϕ + y∂ x ϕ dydτ = ρ ( s ∗ ) − s ∗ φ ′ ( s ∗ )+ | φ ′ ( s ∗ ) | + R , ∗ ( s ∗ ) + R , ∗ ( s ∗ ) Z τ Z R Γ( τ, y ) e τ ( ρ ( s ∗ ) − s ∗ φ ′ ( s ∗ ))+ yφ ′ ( s ∗ ) dydτ ≤ , where the last inequality holds since ρ is a viscosity sub-solution of (2.3) with φ ( s ) being the testfunction. Therefore, w is a viscosity sub-solution of (2.2).18onversely, let w ( t, x ) = tρ (cid:0) xt (cid:1) be a viscosity sub-solution of (2.2) in Ω. Choose any testfunction φ ∈ C such that ρ ( s ) − φ ( s ) attains a strict local maximum at s ∗ and ρ ( s ∗ ) >
0. Withoutloss of generality, we might assume ρ ( s ∗ ) = φ ( s ∗ ). Set ϕ ( t, x ) = tφ (cid:0) xt (cid:1) + ( t − . It then followsthat w ( t, x ) − ϕ ( t, x ) = t ( ρ ( x/t ) − φ ( x/t )) − ( t − attain a strict local maximum at (1 , s ∗ ). Hence,by the definition of w ( t, x ) being a sub-solution and the fact that ∂ t ϕ (1 , s ∗ ) = φ ( s ∗ ) − s ∗ φ ′ ( s ∗ ) and ∂ x ϕ (1 , s ∗ ) = φ ′ ( s ∗ ), we infer that ρ ( s ∗ ) − s ∗ φ ′ ( s ∗ ) + | φ ′ ( s ∗ ) | + R , ∗ ( s ∗ ) + R , ∗ ( s ∗ ) Z τ Z R Γ( τ, y ) e τ ( ρ ( s ∗ ) − s ∗ φ ′ ( s ∗ ))+ yφ ′ ( s ∗ ) dydτ = ∂ t ϕ (1 , s ∗ ) + | ∂ x ϕ (1 , s ∗ ) | + R , ∗ ( s ∗ ) + R , ∗ ( s ∗ ) Z τ Z R Γ( τ, y ) e τ∂ t ϕ + y∂ x ϕ dydτ ≤ , which implies ρ is a sub-solution of (2.3).In the following, we observe that the limit functions w ∗ , w ∗ satisfies an equation without non-local term, even though the original problem (1.1) has a nonlocal space/time delay. Proposition 2.7.
Assume u ( t, x ) is a solution of the nonlocal model (1.1) with initial data satisfy-ing (IC µ ) for some µ ∈ (0 , ∞ ] . The functions w ∗ , w ∗ , as given in Proposition 2.2, are respectivelyviscosity supersolution and subsolution of (2.2) in (0 , ∞ ) × (0 , ∞ ) .Proof. The proof essentially follows from a slight variation of [4, Propositions 3.1 and 3.2], weinclude it here only for the sake of completeness. First, we verify that w ∗ is a viscosity super-solutions of (2.2). By (2.9), w ∗ ( t, x ) ≥ , ∞ ) × (0 , ∞ ).Fix a smooth test function φ , without loss of generality, assume w ∗ − φ has a strict globalminimum at some point ( t , x ) ∈ (0 , ∞ ) × (0 , ∞ ). (We only need to check the strict globalminima here, due to [3, Prop. 3.1].) It then suffices to show that H ( x/t, ∂ t φ, ∂ x φ ) ≥ t , x ).(Here we used the fact that H ( s, q, p ) is u.s.c. (Remark 2.1), so it coincides with its upper envelope H ∗ ( s, q, p ).)Clearly, there exists a sequence ǫ n → t n , x n ) ∈ ( ǫ n τ, ∞ ) × (0 , ∞ )such that w ǫ n ( t, x ) − φ has a global minimum at ( t n , x n ) and that (see, e.g. [3, Lemma 6.1]) w ǫ n ( t n , x n ) → w ∗ ( t , x ) as n → ∞ and ( t n , x n ) → ( t , x ) as n → ∞ . (2.12)By definition of ( t n , x n ) being the global minimum, w ǫ n ( t n , x n ) − φ ( t n , x n ) ≤ w ǫ n ( t, x ) − φ ( t, x ) for ( t, x ) ∈ (0 , ∞ ) × R , (2.13)19t then follows from the maximum principle and (H1) that at the point ( t n , x n ) ≤ ∂ t φ ( t n , x n ) − ǫ n ∂ xx φ ( t n , x n ) + | ∂ x φ ( t n , x n ) | + f ,ǫ n ( t n , x n , u ǫ n ( t n , x n )) u ǫ n ( t n , x n )+ Z τ Z R Γ( τ, y ) f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, u ǫ n ( t n − ǫ n τ, x n − ǫ n y )) u ǫ n ( t n , x n ) dydτ ≤ ∂ t φ − ǫ n ∂ xx φ + | ∂ x φ | + ∂ u f ,ǫ n ( t n , x n , Z τ Z R Γ( τ, y ) ∂ u f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, u ǫ n ( t n − ǫ n τ, x n − ǫ n y ) u ǫ n ( t n , x n ) dydτ ≤ ∂ t φ − ǫ n ∂ xx φ + | ∂ x φ | + ∂ u f ,ǫ n ( t n , x n , Z τ Z R Γ( τ, y ) ∂ u f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, e wǫn ( tn,xn ) − wǫn ( t − ǫnτ,x − ǫny ) ǫn dydτ ≤ ∂ t φ − ǫ n ∂ xx φ + | ∂ x φ | + ∂ u f ,ǫ n ( t n , x n , Z τ Z R Γ( τ, y ) ∂ u f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, e φ ( tn,xn ) − φ ( tn − ǫnτ,xn − ǫny ) ǫn dydτ ≤ ∂ t φ − ǫ n ∂ xx φ + | ∂ x φ | + ∂ u f ,ǫ n ( t n , x n , Z τ Z R Γ( τ, y ) ∂ u f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, e ∂ t φ ( ξ n ,η n ) s + ∂ x φ ( ξ n ,η n ) y dydτ, where ξ n ∈ ( t n − ǫ n τ, t n ) and η n ∈ ( x n − ǫ n y, x n ). We used (2.13) in the fourth inequality. Letting n → ∞ , by Lebsegue dominated convergence theorem, we get0 ≤ ∂ t φ ( t , x ) + | ∂ x φ ( t , x ) | + R ( x /t ) + R ( x /t ) Z τ Z R Γ( τ, y ) e ∂ t φ ( t ,x ) τ + ∂ x φ ( t ,x ) y dydτ, where we use the first two parts of (1.7). This shows that w ∗ is a viscosity super-solution of (2.2).Next, we verify that w ∗ is a viscosity sub-solutions of (2.2). We argue that it is enough toverify that w ∗ is a viscosity subsolution ofmin { w, H ( x/t, ∂ t w, ∂ x w ) } = 0 in (0 , ∞ ) × (0 , ∞ ) , (2.14)which is obtained from (2.2) by replacing the Hamiltonian H ( s, q, p ) therein by H ( s, q, p ) = q + p + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e qτ + py dydτ, where R i ( s ) are given in (1.7). Indeed, suppose this is the case, then by (2.8) we have w ∗ ( t, x ) = tρ ∗ ( x/t ) for some u.s.c. function ρ ∗ ( s ). By arguing similarly as in Lemma 2.6 (with R i in place of R i ) it follows that ρ ∗ ( s ) satisfies, in viscosity sense,min { ρ, H ( s, ρ − sρ ′ , ρ ′ ) } ≤ , ∞ ) . (2.15)Since the Hamiltonian in (2.15) is convex in ρ ′ , a direct application of [28, Proposition 1.14] (seealso [2, Chap. II, Prop. 4.1]) yields that ρ ∗ ∈ Lip loc ([0 , ∞ )). It then follows from Rademacher’s20heorem that ρ ∗ is differentiable a.e. in (0 , ∞ ), so that it satisfies (2.15) a.e. in (0 , ∞ ). Since R i ( s ) = R i ( s ) a.e. (by (H4)), the following differential inequality holds a.e. in (0 , ∞ )min { ρ ∗ , H ( s, ρ ∗ − s ( ρ ∗ ) ′ , ( ρ ∗ ) ′ ) } ≤ . (2.16)However, by the convexity of the Hamiltonian, we can again apply [2, Chap. I, Prop. 5.1] toconclude that it in fact satisfies (2.16) in (0 , ∞ ) in viscosity sense, i.e., ρ ∗ is a viscosity sub-solution of (2.3). By Lemma 2.6 with c b = ∞ , we see that w ∗ ( t, x ) = tρ ∗ ( x/t ) is a viscositysub-solution of (2.2).Therefore, it remains to show that w ∗ is a viscosity sub-solution of (2.14). Fix a smooth testfunction φ and assume w ∗ − φ has a strict global maximum at some point ( t , x ) ∈ (0 , ∞ ) × (0 , ∞ )and w ∗ ( t , x ) >
0. We claim that H ( x /t , ∂ t φ ( t , x ) , ∂ x φ ( t , x )) ≤
0. By the definition of w ∗ in (2.6), there exist a sequence ǫ n → t n , x n ) ∈ (0 , ∞ ) × (0 , ∞ ) suchthat w ǫ n ( t, x ) − φ has a global maximum at ( t n , x n ), and satisfy w ǫ n ( t n , x n ) → w ∗ ( t , x ) > t n , x n ) → ( t , x ) as n → ∞ . (2.17)Next, we claim thatlim inf n →∞ f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, u ǫ n ( t n − ǫ n τ, x n − ǫ n y ) u ǫ n ( t n , x n ) ≥ R ( x /t ) e ∂ t φ ( t ,x ) τ + ∂ x φ ( t ,x ) y (2.18)for a.e. ( τ, y ) ∈ [0 , τ ] × R .For any given ( τ, y ) ∈ [0 , τ ] × R , by passing to a subsequence, we may divide into two cases:(i) u ǫ n ( t n − ǫ n τ, x n − ǫ n y ) → , as n → ∞ , or (ii) inf n u ǫ n ( t n − ǫ n τ, x n − ǫ n y ) > . In case (i), we use (H1) to obtain f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, u ǫ n ( t n − ǫ n τ, x n − ǫ n y ) u ǫ n ( t n , x n ) ≥ ( ∂ v f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, − η ′ ) exp (cid:18) w ǫ n ( t n , x n ) − w ǫ n ( t n − ǫ n τ, x n − ǫ n yǫ n (cid:19) ≥ ( ∂ v f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, − η ′ ) exp (cid:18) φ ( t n , x n ) − φ ( t n − ǫ n τ, x n − ǫ n yǫ n (cid:19) , where the first inequality follows from (H1), and then the second inequality from ( w ǫ n − φ )( t n , x n ) =max( w ǫ n − φ ). Letting n → ∞ , we deducelim inf n →∞ f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, u ǫ n ( t n − ǫ n τ, x n − ǫ n y ) u ǫ n ( t n , x n ) ≥ ( R ( x /t ) − η ′ ) e ∂ t φ ( t ,x ) τ + ∂ x φ ( t ,x ) y . Since η ′ > f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, u ǫ n ( t n − ǫ n τ, x n − ǫ n y ) is bounded from belowby a positive number, and that u ǫ n ( t n , x n ) = exp( − w ǫ n ( t n , x n ) /ǫ n ) → + (using (2.17) and that w ∗ ( t , x ) > n →∞ Z τ Z R Γ( τ, y ) f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, u ǫ n ( t n − ǫ n τ, x n − ǫ n y ) u ǫ n ( t n , x n ) dydτ (2.19) ≥ R ( x /t ) Z τ Z R Γ( τ, y ) e ∂ t φ ( t ,x ) τ + ∂ x φ ( t ,x ) y dydτ. Moreover, since u ǫ n ( t n , x n ) → n → ∞ , we use (H2) again to get for any η ′ >
0, there exists n such that f ,ǫ n ( t n , x n , u ǫ n ) u ǫ n ( t n , x n ) ≥ ∂ u f ,ǫ n ( t n , x n , − η ′ , for all n ≥ n . Letting n → ∞ and then η ′ →
0, we use (H4) to getlim inf n →∞ f ,ǫ n ( t n , x n , u ǫ n ) u ǫ n ( t n , x n ) ≥ R ( x /t ) . (2.20)Now we are ready to verify H ( x /t , ∂ t φ ( t , x ) , ∂ x φ ( t , x )) ≤ . Indeed, at the point ( t n , x n ), ∂ t φ − ǫ n ∂ xx φ + | ∂ x φ | + f ,ǫ n ( t n , x n , u ǫ n ) u ǫ n ≤ − Z τ Z R Γ( τ, y ) f ,ǫ n ( t n − ǫ n τ, x n − ǫ n y, u ǫ n ( t n − ǫ n τ, x n − ǫ n y ) u ǫ n ( t n , x n ) dydτ. Letting n → ∞ , while using (2.19) and (2.20), we obtain ∂ t φ + | ∂ x φ | + R ( x /t ) ≤ − R ( x /t ) Z τ Z R Γ( τ, y ) e ∂ t φ ( t ,x ) τ + ∂ x φ ( t ,x ) y dydτ. This concludes the proof.
Corollary 2.8.
Let w ∗ and w ∗ be given by (2.6) . There exists a u.s.c. function ρ ∗ ( s ) ( resp. l.s.c.function ρ ∗ ( s )) such that w ∗ ( t, x ) = tρ ∗ ( x/t ) , ( resp. w ∗ ( t, x ) = tρ ∗ ( x/t )) in (0 , ∞ ) × [0 , ∞ ) . (2.21) Moreover, ρ ∗ ( resp. ρ ∗ ) is a viscosity supersolution ( resp. subsolution ) of (2.3) .Proof. By Proposition 2.7, w ∗ and w ∗ are respectively sub- and super-solution of (2.2). Theexistence of ρ ∗ ( s ) and ρ ∗ ( s ) and (2.21) are proved in (2.7) and (2.8). The rest follows from Lemma2.6.The following lemma indicates that each sub-solution ρ ( s ) is strictly increasing in the interval { s ∈ (0 , ∞ ) : ρ ( s ) > } . Lemma 2.9.
Let ρ be a nonnegative viscosity subsolution to (2.3) , such that ρ (0) = 0 and ρ ( s ) →∞ as s → ∞ , then (i) s ρ ( s ) is non-decreasing; (ii) s ρ ( s ) s has no positive local maximumpoints in (0 , ∞ ) ; (iii) lim s →∞ ρ ( s ) s exists in [0 , ∞ ] ; (iv) if lim s →∞ ρ ( s ) s < ∞ , then sup s> ρ ( s ) s < ∞ . roof. By R ( s ) ≥ ρ ≥
0, it follows that ρ satisfies − sρ ′ + | ρ ′ | + R ( s ) ≤ | ρ ′ | ≤ s + 2 k R k ∞ in (0 , ∞ )in the viscosity sense. It follows from [28, Proposition 1.14] that ρ ∈ Lip loc ([0 , ∞ )). We first showassertion (i). Since ρ (0) = 0 and ρ ( s ) → ∞ as s → ∞ , it suffices to show that there does notexist s > ρ ( s ) > s ρ ( s ) has a local maximum points at s . Assume to thecontrary, then by definition of viscosity solution (using φ ≡ R ( s ) + R ( s ) < ρ ( s ) + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e τρ ( s ) dτ dy ≤ , which is a contradiction to (H4). This proves assertion (i).The assertion (ii) can be proved similarly by considering σ ( s ) = ρ ( s ) /s , which is a viscositysubsolution ofmin { σ, − s σ ′ + | σ + sσ ′ | + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e − s σ ′ τ +( σ + sσ ′ ) y dτ dy } = 0 . Next, we show (iii). Suppose there exists s > σ attains a local maximum at some s > σ ( s ) >
0. This implies R ( s ) + R ( s ) < | σ ( s ) | + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e σ ( s ) y dτ dy ≤ , which is a contradiction. This proves (ii). (iii) follows directly from (ii).Next, we assume lim s →∞ ρ ( s ) s < ∞ and prove (iv). Recall that ρ ∈ Lip loc ([0 , ∞ )) and ρ (0) = 0, sothat lim s → + ρ ( s ) /s < ∞ . By (ii), we have sup s> ρ ( s ) /s ≤ max (cid:26) lim s → + ρ ( s ) /s, lim s →∞ ρ ( s ) /s (cid:27) < ∞ .By Lemma 2.9(iii), lim s →∞ ρ ( s ) s exists. By dividing in to the following two cases,(i) 0 < lim s →∞ ρ ( s ) s < ∞ , (i ′ ) lim s →∞ ρ ( s ) s = ∞ , we will give several novel conditions so that (2.3) admits a comparison principle. Proposition 2.10.
Suppose R i ( s ) , i = 1 , be given by (H4) . Let ρ and ρ be non-negative super-and sub-solutions of (2.3) in (0 , ∞ ) , such that ρ (0) ≤ ρ (0) , and lim s →∞ ρ ( s ) s ≤ lim inf s →∞ ρ ( s ) s . (2.22) If lim s →∞ ρ ( s ) = ∞ and lim s →∞ ρ ( s ) s < ∞ , then ρ ( s ) ≤ ρ ( s ) in [0 , ∞ ) .Proof. See Appendix A. 23 roposition 2.11.
Let R i ( s ) , i = 1 , be given by (H4) . Let ρ and ρ be non-negative super- andsub-solutions of (2.3) in (0 , ∞ ) , such that ρ (0) ≤ ρ (0) , and lim s →∞ ρ ( s ) s ≤ lim inf s →∞ ρ ( s ) s . (2.23) If lim s →∞ ρ ( s ) s = ∞ and one of the following holds: (i) s R ( s ) is non-increasing in [ s , ∞ ) for some s , (ii) lim s → + ∞ R ( s ) exists and is positive, and R ( s ) ≥ sup ( s, ∞ ) R ! − o (cid:18) s (cid:19) for s ≫ , (iii) One of lim s → + ∞ R i ( s ) ( i = 1 , exists, and Γ( τ, y ) = 0 in [0 , τ ] × ( −∞ , , for some τ ∈ (0 , τ ] ,and lim sup s → + ∞ R ( s ) > ,then ρ ( s ) ≤ ρ ( s ) in [0 , ∞ ) .Proof. We first prove case (ii), recall from the proof of Lemma 2.9 that ρ is locally Lipchitzcontinuous, so that it is differentiable in S , where [0 , ∞ ) \ S has zero measure. Since ρ is aviscosity subsolution, it must satisfymin { ρ, ρ − sρ ′ + | ρ ′ | + Z τ Z R Γ( τ, y ) e τ ( ρ − sρ ′ )+ yρ ′ dydτ } ≤ s ∈ S . (2.24)Since ρ ( ∞ ) = ∞ and that R is upper semi-continuous and locally monotone (by (H4)) and that R (+ ∞ ) >
0, we can choose a sequence { s k } ⊂ S such that s k → ∞ , ρ ( s k ) > , R ( s k ) → lim sup s →∞ R ( s ) and inf k R ( s k ) > . (2.25)For each k , denote a k = ρ ( s k ), b k = ρ ′ ( s k ), then specializing (2.24) at s = s k gives a k − s k b k + | b k | + R ( s k ) + R ( s k ) Z τ Z R Γ( τ, y ) e ( a k − s k b k ) τ + b k y dydτ ≤ . (2.26)Using a k ≥ R ( s k ) > | b k | ≤ s k b k + | R | ∞ and hence | b k | ≤ O ( s k ). Using thelatter, along with inf k R ( s k ) > Z τ Z R Γ( τ, y ) e ( a k − s k b k ) τ + b k y dydτ ≤ O ( | s k | ) . (2.27)Next, define ν k = " sup [ s k , ∞ ) R − R ( s k ) + max ( , sup [ s k , ∞ ) R − R ( s k ) ) Z τ Z R Γ( τ, y ) e ( a k − s k b k ) τ + b k y dydτ, ν k →
0. Indeed, h sup [ s k , ∞ ) R − R ( s k ) i → s k , and " sup [ s k , ∞ ) R − R ( s k ) τ Z R Γ( τ, y ) e ( a k − s k b k ) τ + b k y dydτ ≤ o (cid:18) | s k | (cid:19) O ( | s k | ) = o (1) , where we used the assumption R ( s k ) ≥ sup ( s k , ∞ ) R ! − o (cid:18) s k (cid:19) and (2.27).Define ρ k ∈ Lip loc ([0 , ∞ )) by ρ k ( s ) := ρ ( s ) − ν k for s ∈ [0 , s k ) ,ρ ( s k ) + ( s − s k ) ρ ′ ( s k ) − ν k for s ∈ [ s k , ∞ ) . We claim that ρ k ( s ) satisfies ρ k ( s ) − sρ ′ k ( s ) + | ρ ′ k ( s ) | + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e ( ρ k ( s ) − sρ ′ k ( s )) τ + ρ ′ k y dydτ ≤ , ∞ ). Indeed, it is easy to see that ρ k ( s ) remains a viscosity subsolutionto (2.28) in [0 , s k ). It remains to show that it is a classical solution to (2.28) in [ s k , ∞ ). Indeed, ifwe denote a k = ρ ( s k ) and b k = ρ ′ ( s k )), then for s ≥ s k , ρ k ( s ) − sρ ′ k ( s ) + | ρ ′ k ( s ) | + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e ( ρ k ( s ) − sρ ′ k ( s )) τ + ρ ′ y dydτ ≤ a k − s k b k + | b k | − ν k + sup [ s k , ∞ ) R ! + sup [ s k , ∞ ) R ! Z τ Z R Γ( τ, y ) e ( a k − s k b k ) τ + b k y dydτ ≤ a k − s k b k + | b k | + R ( s k ) + R ( s k ) Z τ Z R Γ( τ, y ) e ( a k − s k b k ) τ + b k y dy ≤ . where we used ρ k ( s ) − sρ ′ k ( s ) = ρ ( s k ) − s k ρ ′ ( s k ) − ν k = a k − s k b k − ν k for s ≥ s k for the first inequality, the definition of ν k for the second inequality, and (2.26) for the last inequal-ity. This proves that ρ k is a viscosity subsolution of (2.3) in (0 , ∞ ).Now, note that ρ and ρ k form a pair of viscosity super- and subsolution of (2.3) that satisfiesthe setting of Proposition 2.10. By comparison, it follows in particular that ρ ( s ) − ν k ≤ ρ ( s ) for s ∈ [0 , s k ] . Letting k → ∞ , then s k → ∞ and ν k →
0, and the desired conclusion follows. This proves case(ii). Case (i) can be proven exactly as case (ii) (but the assumption R (+ ∞ ) > s k ∈ S such that R ( s k ) → lim sup s →∞ R ( s k ) and R ( s k ) → lim sup s →∞ R ( s k ) . (2.29)25his is possible in view of the assumption of case (iii), and local monotonicity of s R i ( s ). Denote a k = ρ ( s k ), and b k = ρ ′ ( s k ). From (2.26), we observe that a k − s k b k ≤ k R k ∞ for all k ≥ . (2.30)Fix an arbitrary ν >
0, and choose τ ′ ∈ (0 , τ ) and k ′ ∈ N such that for all k ≥ k ′ ,max ( sup [ s k , ∞ ) R − R ( s k ) , k R k ∞ Z τ ′ Z R Γ( τ, y ) e τ k R k ∞ dydτ ) < ν , which is possible in view of (2.29). Define function ρ k ( s ) by ρ k ( s ) := ρ ( s ) − ν for s ∈ [0 , s k ] ,ρ ( s k ) + ( s − s k ) ρ ′ ( s k ) − ν for s ∈ ( s k , ∞ ) . We claim that ρ k is a viscosity subsolution to (2.3) for k ≫
1. Again, it suffices to show, for k ≫ ρ k is a classical solution of (2.28) in [ s k , ∞ ).To this end, observe the assumption of case (iii) implies for each τ ′ ∈ (0 , τ ], Z τ Z R Γ( τ, y ) e ( a k − sb k ) τ − b k y dydτ = Z τ ′ Z ∞ Γ( τ, y ) e ( a k − sb k ) τ − b k y dydτ + Z τ τ ′ Z R Γ( τ, y ) e ( a k − sb k ) τ − b k y dydτ ≤ Z τ ′ Z ∞ Γ( τ, y ) e τ k R k ∞ dydτ + Z τ τ ′ Z R Γ( τ, y ) e ( a k − sb k ) τ − b k y dydτ, (2.31)where a k = ρ ( s k ), and b k = ρ ′ ( s k ), and we used (2.30) and b k ≥ ρ ′ ( s ) ≥ s ).Observe that for k ≫ s ∈ [ s k , ∞ ), ρ k ( s ) − sρ ′ k ( s ) + | ρ ′ k ( s ) | + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e ( ρ k ( s ) − sρ ′ k ( s )) τ − ρ ′ k ( s ) y dydτ = a k − ν − s k b k + | b k | + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e ( a k − ν − s k b k ) τ − b k y dydτ ≤ a k − s k b k + | b k | + R ( s k ) + R ( s ) Z τ τ ′ Z R Γ( τ, y ) e ( a k − ν − s k b k ) τ − b k y dydτ ≤ a k − s k b k + | b k | + R ( s k ) + e − ντ ′ ( sup [ s k , ∞ ) R ( s )) Z τ τ ′ Z R Γ( τ, y ) e ( a k − s k b k ) τ − b k y dydτ ≤ a k − s k b k + | b k | + R ( s k ) + R ( s k ) Z τ τ ′ Z R Γ( τ, y ) e ( a k − s k b k ) τ − b k y dydτ ≤ , where we used our choice of ν , τ ′ and (2.31) for the first inequality, and that sup [ sk, ∞ ) R R ( s k ) → R ( s k ) → lim sup s →∞ R >
0) in the third inequality. Hence, (2.28) holds for k ≫ s ∈ [ s k , ∞ ).26ow, ρ and ρ k defines a pair of viscosity super- and sub-solution of (2.3) that satisfies the settingof Proposition 2.10. Comparison implies that, for each ν > ρ ( s ) − ν ≤ ρ ( s ) for s ∈ [0 , s k ] and k ≫ . Letting k → ∞ and ν ց
0, we get ρ ( s ) ≤ ρ ( s ) in [0 , ∞ ). Proof of Propositions 1.7.
Suppose (H1)-(H5) hold, and either (i) µ ∈ (0 , ∞ ) or (ii) µ = ∞ andone of (H6), (H6 ′ ) or (H6 ′′ ) holds. By Proposition 2.11, there is at most one solution to (1.10)subject to the boundary conditions ρ (0) = 0 and lim s →∞ ρ ( s ) s = µ ∈ (0 , ∞ ].To show existence, let w ∗ ( t, x ) and w ∗ ( t, x ) be given by (2.6), and let ρ ∗ ( s ) and ρ ∗ ( s ) berespectively the super- and sub-solution of (1.10) that are given in Corollary 2.8, i.e. w ∗ ( t, x ) = tρ ∗ (cid:16) xt (cid:17) and w ∗ ( t, x ) = tρ ∗ (cid:16) xt (cid:17) for ( t, x ) ∈ (0 , ∞ ) × [0 , ∞ ) . By construction in (2.6), w ∗ ≤ w ∗ in (0 , ∞ ) × [0 , ∞ ), and hence ρ ∗ ≤ ρ ∗ .We claim that ρ ∗ (0) = ρ ∗ (0) = 0 and lim s →∞ ρ ∗ ( s ) s = lim s →∞ ρ ∗ ( s ) s = µ ∈ (0 , ∞ ] . (2.32)The assertions follow from Proposition 2.2, since ρ ∗ (0) = w ∗ (1 ,
0) = 0 and ρ ∗ (0) = w ∗ (1 ,
0) = 0 , (2.33)where we used the first part of (2.10). Also,lim inf s →∞ ρ ∗ ( s ) s = lim inf s →∞ w ∗ (cid:18) s , (cid:19) ≥ w ∗ (0 ,
1) = µ ∈ (0 , ∞ ] , (2.34)where we used the second part of (2.10), and that w ∗ is l.s.c.. Similarly, we havelim sup s →∞ ρ ∗ ( s ) s = lim sup s →∞ w ∗ (cid:18) s , (cid:19) ≤ w ∗ (0 ,
1) = µ ∈ (0 , ∞ ] . (2.35)We can combine (2.33)-(2.35) to obtain (2.32). This, and the fact that ρ ∗ and ρ ∗ are the super- andsub-solution to (1.10) enables the application of the comparison result (Proposition 2.11), whichimplies that ρ ∗ ≤ ρ ∗ in (0 , ∞ ) × [0 , ∞ ). Recalling that ρ ∗ ≥ ρ ∗ in (0 , ∞ ) × [0 , ∞ ) by construction,we conclude ρ ∗ ≡ ρ ∗ . This provides the existence of a viscosity solution ˆ ρ to (1.10) satisfyingˆ ρ (0) = 0 and lim s →∞ ˆ ρ ( s ) s = µ ∈ (0 , ∞ ].Finally, Lemma 2.9(i) says that ˆ ρ ( s ) is non-decreasing, and so s nlp ∈ [0 , ∞ ) is well-defined. Proof of Theorem 1.
Suppose (H1)-(H5) hold, and let u ( t, x ) be a solution of (1.1) with initialdata satisfying (IC µ ) for some µ ∈ (0 , ∞ ). Then Proposition 2.11 is applicable.Let w ∗ ( t, x ) and w ∗ ( t, x ) be given by (2.6). From the proof of Proposition 1.7, we have w ∗ ( t, x ) = t ˆ ρ ( x/t ) = w ∗ ( t, x ) for ( t, x ) ∈ (0 , ∞ ) × [0 , ∞ ) , ρ ( s ) is the unique viscosity solution to (1.10) subject to the boundary conditions ρ (0) = 0and lim s →∞ ρ ( s ) s = µ . Since ˆ ρ ( s ) = 0 in [0 , ˆ s µ ] and ˆ ρ ( s ) > s µ , ∞ ), we deduce that w ǫ ( t, x ) → { ( t, x ) : 0 ≤ x/t < ˆ s µ } (2.36)and that lim inf ǫ → inf K w ǫ ( t, x ) > K ⊂⊂ { ( t, x ) : x > ˆ s µ t } . (2.37)We show the first part of (1.13). First, by Remark B.2, there exists some s > t →∞ sup x ≥ st u ( t, x ) = 0 . (2.38)Now for given η >
0, take K = { (1 , x ) : ˆ s µ + η ≤ x ≤ s } in (2.37), thenlim t →∞ sup ( s nlp + η ) t ≤ x ≤ st u ( t, x ) = lim ǫ → sup s nlp + η ≤ x ′ ≤ s u (cid:18) ǫ , x ′ ǫ (cid:19) = lim ǫ → sup s nlp + η ≤ x ′ ≤ s exp (cid:18) − w ǫ (1 , x ′ ) ǫ (cid:19) = 0 . This proves the first part of (1.13).To show the second part of (1.13), fix ( t , x ) such that x /t < ˆ s µ and suppose to contrarythat there a sequence ǫ = ǫ k → t ǫ , x ǫ ) → ( t , x ) such that0 < x t < ˆ s µ and u ǫ ( t ǫ , x ǫ ) → . Now, consider the test function φ ǫ ( t, x ) := | t − t ǫ | + | x − x ǫ | . Since w ǫ ( t, x ) → B r ( x , t ), by taking ǫ so small that ( t ǫ , x ǫ ) ∈ B r ( x , t ), we see that w ǫ − φ ǫ hasan interior maximum point ( t ′ ǫ , x ′ ǫ ) ∈ B r ( x , t ). Observe that( t ′ ǫ , x ′ ǫ ) → ( t , x ) and ( t ǫ , x ǫ ) → ( t , x ) as ǫ → . (2.39)And, by construction, w ǫ ( t ′ ǫ , x ′ ǫ ) ≥ ( w ǫ − φ ǫ )( t ′ ǫ , x ′ ǫ ) ≥ ( w ǫ − φ ǫ )( t ǫ , x ǫ ) = w ǫ ( t ǫ , x ǫ ) (2.40)which, in view of u ǫ ( t, x ) = exp( − ǫ w ǫ ( t, x )), implies that 0 ≤ u ǫ ( t ′ ǫ , x ′ ǫ ) ≤ u ǫ ( t ǫ , x ǫ ). Since u ǫ ( t ǫ , x ǫ ) → u ǫ ( t ′ ǫ , x ′ ǫ ) → ǫ → . (2.41)Next, fix 0 < η ′ < (cid:20) − η ′ + R (cid:18) x t (cid:19)(cid:21) (1 − η ′ ) + R (cid:18) x t (cid:19) > η ′ . (2.42)Note that the above holds when η ′ = 0 (by (1.9)), so it also holds for small η ′ > M > Z τ Z M − M Γ( τ, y ) dydτ ≥ (1 − η ′ ) . (2.43)Then by the definition of φ ǫ and (2.39), and passing to a subsequence if necessary, we see that ∂ t φ ǫ ( t ′ ǫ , x ′ ǫ ) → , ∂ x φ ǫ ( t ′ ǫ , x ′ ǫ ) → , sup B r | D φ ǫ | ≤ , R τ R M − M Γ( τ, y ) ∂ u f ,ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy, e φǫ ( t ′ ǫ,x ′ ǫ ) − φǫ ( t ′ ǫ − ǫτ,x ′ ǫ − ǫy ) ǫ dydτ ≥ R (cid:16) x t (cid:17) R τ R M − M Γ( τ, y ) dydτ + o (1) , (2.44)where B r = B r ( t ′ ǫ , x ′ ǫ ) in the supremum, and R (cid:16) x t (cid:17) is given in (1.7) and we used (1.9).Having chosen M , we claim that (2.41) can be strengthened tosup ≤ τ ≤ τ | y |≤ M u ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy ) = sup ≤ τ ≤ τ | y |≤ M u (cid:18) t ′ ǫ ǫ − τ, x ′ ǫ ǫ − y (cid:19) → ǫ → . (2.45)Indeed, we can rewrite (1.1) as ∂ t u − ∂ xx u ≥ − C u in (0 , ∞ ) × R , (2.46)since 0 ≤ u ǫ ≤ M for some M >
0. Passing to a sequence, we may assume that˜ u ǫ ( t, x ) = u (cid:18) t ′ ǫ ǫ + t, x ′ ǫ ǫ + x (cid:19) → ˜ u ( t, x ) in C loc ( R × R ) . Moreover, the limit function ˜ u is a non-negative weak solution of (2.46) such that ˜ u (0 ,
0) = 0.By the strong maximum principle, we deduce that ˜ u ( t, x ) ≡ t ≤ x ∈ R . This showsthat ˜ u ǫ → −∞ , × R , which implies (2.45).In view of (1.8) and (2.45), we may consider ǫ small enough such that Z τ Z M − M Γ( τ, y ) [ − η ′ + ∂ u f ,ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy, u ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy ) dydτ ≤ Z τ Z M − M Γ( τ, y ) f ,ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy, u ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy )) dydτ, (2.47)and similarly, (1 − η ′ ) ∂ u f ,ǫ ( t ′ ǫ , x ′ ǫ , u ( t ′ ǫ , x ′ ǫ ) ≤ f ,ǫ ( t ′ ǫ , x ′ ǫ , u ( t ′ ǫ , x ′ ǫ )) . (2.48)29ence, (below w ǫ , φ ǫ and their derivatives are evaluated at ( t ′ ǫ , x ′ ǫ ), unless otherwise stated) (cid:20) − η ′ + R (cid:18) x t (cid:19)(cid:21) (1 − η ′ ) + o (1) ≤ (cid:20) − η ′ + R (cid:18) x t (cid:19)(cid:21) Z τ Z M − M Γ( τ, y ) dydτ + o (1)= ∂ t φ ǫ − ǫ∂ x φ ǫ − | ∂ x φ ǫ | + Z τ Z M − M Γ( τ, y )[ ∂ u f ,ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy, − η ′ ] e φǫ ( t ′ ǫ,x ′ ǫ ) − φǫ ( t ′ ǫ − ǫτ,x ′ ǫ − ǫy ) ǫ dydτ ≤ ∂ t w ǫ − ǫ∂ x w ǫ − | ∂ x w ǫ | + Z τ Z M − M Γ( τ, y )[ ∂ u f ,ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy, − η ′ ] e wǫ ( t ′ ǫ,x ′ ǫ ) − wǫ ( t ′ ǫ − ǫτ,x ′ ǫ − ǫy ) ǫ dydτ = ∂ t w ǫ − ǫ∂ x w ǫ − | ∂ x w ǫ | + Z τ Z M − M Γ( τ, y ) [ ∂ u f ,ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy, − η ′ ] u ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy ) u ǫ ( t ′ ǫ , x ′ ǫ ) dydτ ≤ ∂ t w ǫ − ǫ∂ x w ǫ − | ∂ x w ǫ | + Z τ Z M − M Γ( τ, y ) f ,ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy, u ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy )) u ǫ ( t ′ ǫ , x ′ ǫ ) dydτ = − f ,ǫ ( t ′ ǫ , x ′ ǫ , u ǫ ( t ′ ǫ , x ′ ǫ )) u ǫ ( t ′ ǫ , x ′ ǫ ) − Z τ Z | y | >M Γ( τ, y ) f ,ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy, u ǫ ( t ′ ǫ − ǫτ, x ′ ǫ − ǫy )) u ǫ ( t ′ ǫ , x ′ ǫ ) dydτ ≤ η ′ − ∂ u f ,ǫ ( t ′ ǫ , x ′ ǫ , , where we used (2.43) for the first inequality, (2.44) for the next equality, the fact that w ǫ − φ ǫ has local max at ( t ′ ǫ , x ′ ǫ ) for the next inequality, (2.47) for the third inequality, (2.5) for the nextequality, and (2.48) for the final inequality.Since ∂ u f ,ǫ ( t ′ ǫ , x ′ ǫ ,
0) = ∂ u f (cid:16) t ′ ǫ ǫ , x ′ ǫ ǫ , (cid:17) , the above chain of inequalities implies (cid:20) − η ′ + R (cid:18) x t (cid:19)(cid:21) (1 − η ′ ) + ∂ u f (cid:18) t ′ ǫ ǫ , x ′ ǫ ǫ , (cid:19) − η ′ ≤ o (1) . Letting ǫ →
0, we have (cid:20) − η ′ + R (cid:18) x t (cid:19)(cid:21) (1 − η ′ ) + R (cid:18) x t (cid:19) − η ′ ≤ , which is in contradiction with the choice of η ′ in (2.42). This completes the proof of the secondpart of (1.13). Proof of Theorem 2.
Let u ( t, x ) be a solution of (1.1) with initial data satisfying (IC ∞ ). Supposethat (H1)-(H5), and one of (H6), (H6 ′ ) or (H6 ′′ ) hold. Then Proposition 2.11 is again applicable.One can then repeat the proof of Theorem 1. Let u be a solution to (1.1) with initial data φ satisfying ( IC µ ) for some µ ∈ (0 , ∞ ]. By Theorem 1or Theorem 2, the (rightward) spreading speed of problem (1.1) is given by the number ˆ s µ , whichis characterized in (1.12) as a free-boundary point of certain first order Hamilton-Jacobi equation.In the following, we give the explicit formula for ˆ s µ in two class of environments: the first one being30he asymptotically homogeneous environments (see (1.15)), the second one being the environmentswith a single shifting speed (see (1.18)). We derive the exact spreading speed for asymptotically homogeneous environments, that is, thehypotheses of Theorem 3 and, in particular, the assumption (1.15) are enforced. When µ = ∞ and f i are independent of t and x , the problem was considered in [48]. Proof of Theorem 3.
Recall that ∆( λ, p ) is given in (1.17) and that λ ( p ) is the function that isimplicitly defined by ∆( λ, p ) = 0 . First, we observe that λ ( p ) is well-defined, since for each fixed p , we have ∆( λ, p ) → ∓∞ as λ → ±∞ , and that ∂ λ ∆( λ, p ) ≤ − λ ∆( λ, p ) is strictlydecreasing).Second, observe that p λ ( p ) is even, and strictly convex, i.e. λ ′′ ( p ) >
0. Indeed, λ ( p ) is even,since p ∆( λ, p ) is even. Furthermore, differentiating the relation ∆( λ ( p ) , p ) = 0 gives − ∂ λ ∆( λ, p ) · λ ′′ = ∂ λλ ∆( λ, p ) | λ ′ | + ∂ pp ∆( λ, p ) . Since ∂ λ ∆ < ∂ λλ ∆ ≥ ∂ pp ∆ >
0, we deduce λ ′′ ( p ) > λ ( p ) /p is unbounded as p → ∞ . Indeed, using ∆( λ ( p ) , p ) ≡ λ ( p ) p = λ ( p ) + ∆( λ ( p ) , p ) p ≥ p + r p → + ∞ as p → ∞ . By evenness, λ ( p ) /p → −∞ as p → −∞ . Recalling that λ ′′ >
0, we see that λ ′ : R → R is ahomeomorphism. We denote the inverse function of λ ′ to be Ψ : R → R .Next, observe that there exists a unique positive number µ ∗ such that p λ ( p ) p is decreasing in [0 , µ ∗ ) and increasing in ( µ ∗ , ∞ ) . (3.1)In particular λ ( p ) p ≥ λ ( µ ∗ ) µ ∗ for all p ≥ , and equality holds iff p = µ ∗ . (3.2)Indeed, let h ( p ) = λ ( p ) /p , then since λ ′′ >
0, it is not difficult to show that h ′ ( p ) = 0 for some p implies h ′′ ( p ) > . Observing also that h (0+) = −∞ and h (+ ∞ ) = ∞ (since λ ′ (0) = 0 < λ (0)), we deduce that p h ( p ) attains its global minimum at a positive number µ ∗ , and that h ′ ( p ) < , µ ∗ ) , h ′ ( µ ∗ ) = 0 and h ′ ( p ) > µ ∗ , ∞ ) . (3.3)This proves (3.1). 31ext, we considering the following three cases separately:(i) µ ∈ (0 , µ ∗ ] , (ii) µ ∈ ( µ ∗ , ∞ ) , and (iii) µ = ∞ . Case (i). Let µ ∈ (0 , µ ∗ ] be given, and define the function ρ µ ( s ) := max { µs − λ ( µ ) , } for s ∈ R + . (3.4)Then ρ µ satisfies (1.11) and is a classical solution ofmin (cid:26) ρ, ρ − sρ ′ + | ρ ′ | + r + r Z τ Z R Γ( τ, y ) e ( ρ − sρ ′ ) τ + ρ ′ y dydτ (cid:27) = 0 (3.5)in [0 , ∞ ) \ { λ ( µ ) /µ } . Since ρ ′ µ (( λ ( µ ) /µ ) − ) = 0 < ρ ′ µ (( λ ( µ ) /µ )+) = µ , we see that ρ µ is automat-ically a viscosity sub-solution of (3.5) in (0 , ∞ ) (since ρ µ − φ can never attain a strict maximumat s = λ ( µ ) /µ ). To show that it is a viscosity super-solution, suppose ρ µ − φ attains a strict localminimum at s ∈ (0 , ∞ ). If s = λ ( µ ) /µ , then ρ µ is differentiable and φ ′ ( s ) = ρ ′ µ ( s ) and there isnothing to prove. If s = λ ( µ ) /µ , then, denoting φ ′ = φ ′ ( s ), we have 0 ≤ φ ′ ≤ µ , and that − ( λ ( µ ) /µ ) φ ′ + | φ ′ | − g ′ (0) + f ′ (0) Z τ Z R Γ( τ, y ) e − ( λ ( µ ) /µ ) φ ′ τ + φ ′ y dydτ = ∆ (cid:18) λ ( µ ) µ φ ′ , φ ′ (cid:19) − ∆( λ ( φ ′ ) , φ ′ ) ≥ , where we used the fact that λ ∆( λ, p ) is decreasing in λ , and that λ ( µ ) µ φ ′ ≤ λ ( φ ′ ) (which is dueto 0 ≤ φ ≤ µ ≤ µ ∗ , and (3.1)). Hence ρ µ is a viscosity solution of (3.5) such that (1.11) holds. Bythe uniqueness result of Proposition 1.7, we deduce that ˆ ρ µ ( s ) = ρ µ ( s ), and hence ˆ s µ = λ ( µ ) /µ when µ ∈ (0 , µ ∗ ].Case (ii). Let µ ∈ [ µ ∗ , ∞ ) be given, define ρ µ ( s ) := µs − λ ( µ ) for s ∈ [ λ ′ ( µ ) , ∞ ) ,s Ψ( s ) − λ (Ψ( s )) for s ∈ [ λ ′ ( µ ∗ ) , λ ′ ( µ )) , s ∈ [0 , λ ′ ( µ ∗ )) . (3.6)It is straightforward to check that ρ µ is a classical solution of (3.5) in R \ { λ ′ ( µ ∗ ) } . Using the factthat Ψ = ( λ ′ ) − , it is straightforward to observe that ρ µ is continuous at λ ′ ( µ ∗ ) and is differentiableand a classical solution of (3.5) in [0 , ∞ ) \ { λ ′ ( µ ∗ ) } .Similar as before, ρ µ is a viscosity sub-solution of (3.5) in (0 , ∞ ). To show that it is a super-solution as well, it remains to consider the case when ρ µ − φ attains a strict local minimum pointat λ ′ ( µ ∗ ). In such an event, φ ′ = φ ′ ( λ ′ ( µ ∗ )) is nonnegative. Now, at the point s = λ ′ ( µ ∗ ), − λ ′ ( µ ∗ ) φ ′ + | φ ′ | − g ′ (0) + f ′ (0) Z τ Z R Γ( τ, y ) e − λ ′ ( µ ∗ ) φ ′ τ + φ ′ y dydτ = ∆ ( λ ′ ( µ ∗ ) φ ′ , φ ′ ) = ∆ (cid:18) λ ( µ ∗ ) µ ∗ φ ′ , φ ′ (cid:19) − ∆( λ ( φ ′ ) , φ ′ ) ≥ , λ ′ ( µ ∗ ) = λ ( µ ∗ ) /µ ∗ and ∆( λ ( φ ′ ) , φ ′ ) = 0 in the second equality; (3.2) and themonotonicity of λ ∆( λ, p ) for the last inequality. By the uniqueness proved in Proposition1.7(a), we deduce that the unique viscosity solution of (3.5) is given by (3.6). Hence,ˆ s µ = λ ′ ( µ ∗ ) = λ ( µ ∗ ) /µ ∗ = inf p> λ ( p ) p . (3.7)Case (iii). Let µ → ∞ in (3.6), then the sequence of viscosity solutions ρ µ converges, i.e. ρ µ ( s ) → ρ ∞ ( s ) := max { , s Ψ( s ) − λ (Ψ( s ) } in C loc ([0 , ∞ )) . By stability property of viscosity solution [3, Theorem 6.2], ρ ∞ is a viscosity solution of (3.5) in(0 , ∞ ). We claim that ρ ∞ satisties (1.11) for µ = ∞ . Indeed, ρ ∞ (0) = 0 andlim s →∞ ρ ∞ ( s ) s ≥ lim s →∞ ρ µ ( s ) s = µ for each µ ∈ [ µ ∗ , ∞ ) . Letting µ → ∞ , we verified (1.11). Hence, by the uniqueness result of Proposition 1.7(b), weconclude that ρ ∞ gives the unique viscosity solution of (3.5), and thus (3.7) is valid for µ = ∞ aswell. This completes the proof of Theorem 3. In this section, we consider environments with a single shifting speed, i.e. the hypotheses ofTheorem 4 and, in particular, the assumption (1.18) are enforced.For µ ∈ R , recall that λ − ( µ ) and λ + ( µ ) are defined by the implicit formula0 = ∆ ± ( λ, µ ) = − λ + µ + R , ± + R , ± Z τ Z R Γ( t, y ) e µy − λt dydτ. Then (1.10) can be re-written asmax { ρ, ˜ H ( s, ρ, ρ ′ ) } = 0 in (0 , ∞ ) where ˜ H ( s, ρ, ρ ′ ) = ∆ − ( ρ − sρ ′ , ρ ′ ) for s ≤ c , ∆ + ( ρ − sρ ′ , ρ ′ ) for s > c . (3.8)Since λ − (resp. λ + ) are coercive and strictly convex, λ − (resp. inverse of λ + ) is a homeomor-phism of R and we can similarly define Ψ − (resp. Ψ + ) to be the inverse of λ − (resp. inverse of λ + ). Next, define ( c ∗− , µ ∗− ) and ( c ∗ + , µ ∗ + ) by0 < c ∗± = inf µ> λ ± ( µ ) µ = λ ( µ ∗± ) µ ∗± . Since R , − + R , − < R , + + R , + , ∂ λ ∆ ± ( λ, µ ) < + ( λ − ( µ ) , µ ) > ∆ − ( λ − ( µ ) , µ ) = 0, we seethat λ − ( µ ) < λ + ( µ ) for each µ >
0. It then follows that c ∗− < c ∗ + and for any c > c Ψ − ( c ) − λ − (Ψ − ( c )) = max µ> { c µ − λ − ( µ ) } > max µ> { c µ − λ + ( µ ) } = c Ψ + ( c ) − λ + (Ψ + ( c )) . (3.9)33oreover, if c > c ∗ + , then c Ψ + ( c ) − λ + (Ψ + ( c )) ≥ c µ ∗ + − λ + ( µ ∗ + ) > c ∗ + µ ∗ + − λ + ( µ ∗ + ) = 0 . (3.10)In the remainder, we divide the proof of Theorem 4 into the following lemmas. Lemma 3.1. If µ ∈ (0 , µ ∗ + ] , then ˆ s µ ( c ) = λ + ( µ ) /µ if c ≤ λ + ( µ ) /µ,λ − ( p ( c , µ )) /p ( c , µ ) if c > λ + ( µ ) /µ and p ( c , µ ) < µ ∗− c ∗− , otherwise (3.11) where p ( c , µ ) is the smallest root of c p − λ − ( p ) = c µ − λ + ( µ ) . (3.12)We will postpone and sketch the proof once after the more delicate Lemma 3.2 is established. Lemma 3.2. If µ ∈ ( µ ∗ + , ∞ ) , then ˆ s µ ( c ) = c ∗ + if c ≤ λ ′ + ( µ ∗ + ) ,λ − (¯ p ( c )) / ¯ p ( c ) if λ ′ + ( µ ∗ + ) < c ≤ λ ′ + ( µ ) and ¯ p ( c ) < µ ∗− ,λ − ( p ) /p if c > λ ′ + ( µ ) and p ( c , µ ) < µ ∗− ,c ∗− , otherwise . (3.13) where c ∗ + = λ ′ + ( µ ∗ + ) = λ + ( µ ∗ + ) /µ ∗ + , p ( c , µ ) is the smallest root of (3.12) and ¯ p ( c ) is the smallestroot of c p − λ − ( p ) = c Ψ + ( c ) − λ + (Ψ + ( c )) (3.14)Finally, letting µ → ∞ in the above lemma, we have Lemma 3.3. ˆ s ∞ ( c ) = c ∗ + if c ≤ c ∗ + ,λ − (¯ p ( c )) / ¯ p ( c ) if c ∈ ( c ∗ + , ¯ c ) ,c ∗− if c ≥ ¯ c . (3.15) where c ∗ + = λ ′ + ( µ ∗ + ) = λ + ( µ ∗ + ) /µ ∗ + and ¯ c is the unique positive number such that ¯ p (¯ c ) = µ ∗− .Proof of Lemma 3.2. First, we claim that p ( c , µ ) < Ψ − ( c ) ∧ µ and ¯ p ( c ) < Ψ + ( c ) ∧ Ψ − ( c ), andthat both are increasing in c ∈ (0 , ∞ ).Indeed, define the auxiliary functions F ( c , µ, p ) = c p − λ − ( p ) − c µ + λ + ( µ ) and F ( c , p ) = c p − λ − ( p ) − c Ψ + ( c ) + λ + (Ψ + ( c )) . F i is increasing in p ∈ [0 , Ψ − ( c )] and decreasing in p ∈ [Ψ − ( c ) , ∞ ), and F ( c , µ, µ ) = − λ − ( µ ) + λ + ( µ ) > , F ( c , µ, < ,F ( c , Ψ + ( c )) = − λ − (Ψ + ( c )) + λ + (Ψ + ( c )) > , F ( c , < . It then follows that the smallest roots p ( c , µ ) ∈ (0 , µ ) and ¯ p ( c ) ∈ (0 , Ψ + ( c )). Moreover, ∂ p F ( c , µ, p ) > , ∂ µ F ( c , µ, p ) = − c + λ ′ + ( µ ) , ∂ c F ( µ, c , p ) = p − µ < . (3.16) ∂ p F ( c , ¯ p ) > , ∂ c F ( c , ¯ p ) = ¯ p − Ψ + ( c ) < p ( c , µ ) is increasing in c , and increasing in µ provided c > λ ′ + ( µ ). ¯ p ( c ) is increasingin c . Since F ( c ∗ + , µ ∗− ) = µ ∗− ( c ∗ + − c ∗− ) >
0, we obtain ¯ p ( c ∗ + ) < µ ∗− and ¯ p (+ ∞ ) = + ∞ . Thereexists a unique number ¯ c such that ¯ p (¯ c ) = µ ∗− . Second, if c ≥ λ ′ + ( µ ), we claim that p ( c , µ ) = µ ∗− defines a decreasing function c = g ( µ ) for µ ∗− < µ ≤ Ψ + (¯ c ) and ¯ c = g (Ψ + (¯ c )).In fact, c = g ( µ ) solves implicitly from F ( c , µ, µ ∗− ) = 0. It is decreasing for µ ∈ ( µ ∗− , Ψ + (¯ c ]due to (3.16). A direct computation gives F (¯ c , Ψ + (¯ c ) , µ ∗− ) = F (¯ c , µ ∗− ) = 0.Next, we divide the proof into the following five mutually exclusive cases: (i) c ≤ λ ′ + ( µ ∗ + ) ;(ii) λ ′ + ( µ ∗ + ) < c ≤ λ ′ + ( µ ) and ¯ p ( c ) < µ ∗− ; (iii) c > λ ′ + ( µ ) and p ( c , µ ) < µ ∗− ; (iv) λ ′ + ( µ ∗ + ) < c ≤ λ ′ + ( µ ) and ¯ p ( c ) ≥ µ ∗− ; (v) c > λ ′ + ( µ ) and p ( c , µ ) ≥ µ ∗− .Case (i). Since c ≤ c ∗ + , we can directly verify that the formula (3.6) (with λ, λ ′ , Ψ replaced by λ + , λ ′ + , Ψ + ) defines a viscosity solution of (1.10) in [0 , ∞ ) with R i given by (1.18), which satisfiesthe boundary conditions (1.11). Hence, the spreading speed ˆ s µ coincides with the homogeneousspreading speed c ∗ + in the ”+” environment.Case (ii). Note that λ ′ + ( µ ∗ + ) < c ≤ λ ′ + ( µ ). Define the function ρ µ ( s ) := µs − λ + ( s ) for s ∈ [ λ ′ + ( µ ) , + ∞ ) ,s Ψ + ( s ) − λ + (Ψ + ( s )) for s ∈ [ c , λ ′ + ( µ )) , max { ¯ ps − λ − (¯ p ) , } for s ∈ [0 , c ) . (3.17)Then ρ µ satisfies (1.11) and is a classical solution of (1.10) with R i ( s ) given in (1.18) in [0 , ∞ ) except { c , λ − (¯ p ) / ¯ p } . Since ρ ′ µ ( c − ) = ¯ p < Ψ + ( c ) = ρ ′ µ ( c +) and ρ ′ µ ( λ − (¯ p ) / ¯ p − ) = 0 < ρ ′ µ ( λ − (¯ p ) / ¯ p +) =¯ p , we conclude that ρ µ is automatically a viscosity sub-solution in (0 , ∞ ). It suffices to show it isalso a viscosity super-solution, it suffices to consider the two points s = c and s = λ − (¯ p ) / ¯ p wherethe ρ µ may not be differentiable. Since p < µ ∗− , the latter point can be treated as in case (i) ofproof of Theorem 3. Thus, it suffices to consider the case when ρ µ − φ , for some test function φ ,attains a strict local minimum at the point s = c . Now, at the point s = c , ρ µ ( c ) − c φ ′ + | φ ′ | + R , + + R , + Z τ Z R Γ( τ, y ) e ( ρ µ ( c ) − c φ ′ ) τ + φ ′ y dydτ = ∆ + ( c φ ′ − c Ψ + ( c ) + λ + (Ψ + ( c )) , φ ′ ) − ∆ + ( λ + ( φ ′ ) , φ ′ ) ≥ , φ ′ ∈ [¯ p, Ψ + ( c )], where we used ∆ + ( λ + ( p ) , p ) = 0 for all p for the first equality, and thefact that ∆ + ( λ, p ) is decreasing in λ and that c φ ′ − c Ψ + ( c ) + λ + (Ψ + ( c )) ≤ λ + ( φ ′ ) (see the lastequality of (3.9)) for the last inequality. Hence, ˆ s µ = λ − (¯ p ) / ¯ p .Case (iii). Define the function ρ µ ( s ) := µs − λ + ( s ) for s ∈ [ c , + ∞ ) , max { ps − λ − ( p ) , } for s ∈ [0 , c ) . (3.18)A direction computation gives ρ µ is a classical solution of (1.10) except { c , λ − ( p ) /p } . Similaras above, it is a viscosity sub-solution of (1.10) in (0 , ∞ ). To show it is also a super-solution, itsuffices to consider the two points s = c and s = λ − ( p ) /p where ρ µ is not differentiable. Since p < µ ∗− , the latter point can be treated as in case (i) of proof of Theorem 5. Thus, it suffices toconsider the case when ρ µ − φ , for some test function φ , attains a strict local minimum at the point s = c . At that point, we have ρ µ ( c ) − c φ ′ + | φ ′ | + R , + + R , + Z τ Z R Γ( τ, y ) e ( ρ µ ( c ) − c φ ′ ) τ + φ ′ y dydτ = ∆ + ( c φ ′ − c µ + λ + ( µ ) , φ ′ ) − ∆ + ( λ + ( φ ′ ) , φ ′ ) ≥ , holds for φ ′ ( c ) := φ ′ ∈ [ p, µ ], where we used the fact that ∆ + ( λ, p ) is decreasing in λ , and c s − λ + ( s ) is increasing on [0 , µ ]. Hence, ˆ s µ = λ − ( p ) /p .Case (iv). Denote the function ρ µ ( s ) := µs − λ + ( s ) for s ∈ [ λ ′ + ( µ ) , + ∞ ) ,s Ψ + ( s ) − λ + (Ψ + ( s )) for s ∈ [ c , λ ′ + ( µ )) , ¯ ps − λ − (¯ p ) for s ∈ [ λ ′− (¯ p ) , c ] ,s Ψ − ( s ) − λ − (Ψ − ( s )) for s ∈ [ c ∗− λ ′− (¯ p )] , s ∈ [0 , c ∗− ] . (3.19)Similarly as before, we see that (3.19) is a viscosity sub-solution. Furthermore, one might di-rectly adapt the argument in Case (i) of Theorem 3 to obtain it is also a viscosity super-solution.Therefore, we concludes ˆ s µ = c ∗− .Case (v). Now it is standard to check that ρ µ ∈ C ([0 , ∞ )] defined by ρ µ ( s ) := µs − λ + ( s ) for s ∈ [ c , + ∞ ) ,ps − λ − ( p ) , for s ∈ [ λ ′− ( p ) , c ] ,s Ψ − ( s ) − λ − (Ψ − ( s )) for s ∈ [ c ∗− , λ ′− ( p )] , s ∈ [0 , c ∗− ] . (3.20)is a unique viscosity solution of (1.10)-(1.11). Hence, ˆ s µ = c ∗− .36 ketch proof of Lemma 3.5. First, we emphasize that if µ ∗ + ≤ µ ∗− , then ¯ p ( c , µ ) < µ ∗− is alwaysvalid due to ¯ p < µ , so there will be only two mutually exclusive cases for µ ∈ (0 , µ ∗ + ), namely,(i) c ≤ λ + ( µ ) /µ ; (ii) c > λ + ( µ ) /µ and p ( c , µ ) < µ ∗− .Case (i). It follows from an easy modification of Case (i) in Theorem 3.Case (ii). This is already done by Case (iii) in the proof of Lemma 3.2.In addition, when µ ∗− < µ ∗ + . Case (iii) c > λ + ( µ ) /µ and p ( c , µ ) ≥ µ ∗− does exist. But then,one could check this by the similar strategy in case (v) in proof of Lemma 3.2. Remark 3.4. If c ≤
0, then the result is the same as that of homogeneous environment. When µ ∗− ≥ µ ∗ + > µ , λ − ( p ( c ,µ )) p ( c ,µ ) → λ − ( µ ) µ as c → ∞ . Proof of Theorem 5.
Let r > f ( t, x, u ) = u ( r ( t, x ) − u ) and f ≡
0. Then (H1)-(H6) are satisfied with R ( s ) = r a.e. in (0 , ∞ ) , R ≡ r and R ≡ R ≡ . And (1.17) takes the form ∆( λ, p ) = − λ + p + r = 0. Hence λ ( p ) = p + r , and the minimumpoint µ ∗ of λ ( p ) /p is √ r . Hence, the formula (1.25) follows from (1.16). Proof of Theorem 6.
We take f ≡ R , + = r > R , − = r > λ − ( p ) = p + r , λ + ( p ) = p + r , r < r (3.21)Moreover, c ∗− = 2 √ r < c ∗ + = 2 √ r and µ ∗− = √ r < µ ∗ + = √ r . First, we derive (1.26) and (1.27)from (1.19). Note that p is defined by the smallest root of (1.20), i.e. c p − p − r = c µ − µ − r , then we get p = c − p ( c − µ ) + 4( r − r )2 . When µ ≤ √ r , then p ≤ µ ∗− and (1.26) follows from the first two alternatives in (1.19). Note thatthe third alternative in (1.19) holds only when µ ∈ ( √ r , √ r ], when c ≥ µ + r − r µ − √ r = µ + √ r + r − r µ − √ r . Hence (1.26) and (1.27) follows from (1.19), by noting that c ∗− = 2 √ r ,λ + ( µ ) µ = µ + r µ and λ − ( p ) p = c − p ( c − µ ) + 4( r − r )2 + 2 r c − p ( c − µ ) + 4( r − r ) . (3.22)Next, let µ ≥ √ r . We derive (1.28) and (1.29) from (1.21). Here we note that p is the smallestroot of (1.22), i.e. c p − p − r = c − c − r . Hence,¯ p = c − √ r − r . µ ∈ [ √ r , √ r + √ r − r ] and c ∈ (2 √ r , √ µ ]. This divides (1.21) into the two cases (1.28) and (1.29). Notethat c ∗− = 2 √ r , c ∗ + = 2 √ r , λ , λ − ( p ) /p is given in (3.22), and λ − (¯ p )¯ p = c − √ r − r + r c − √ r − r . We omit the details.
A Proof of Proposition 2.10
In this section, we proof Proposition 2.10 by applying the comparison result in [37, Theorem A.1],which was inspired by the arguments developed by Ishii [27] and Tourin [44]. Consider the followingHamilton-Jacobi equation:min { w t + ˜ H ( t, x, D x w ) , w − Lt } = 0 in Ω for some L ∈ R . (A.1) Definition A.1.
We say that a lower semicontinuous (lsc) function w is a viscosity super-solutionof (A.1) if w − Lt ≥ ϕ ∈ C ∞ (Ω), if ( t , x ) ∈ Ω is a strict localminimum point of w − ϕ , then ∂ t ϕ ( t , x ) + ˜ H ∗ ( t , x , D x ϕ ( t , x )) ≥ w is a viscosity sub-solution of (A.1) if for alltest functions ϕ ∈ C ∞ (Ω), if ( t , x ) ∈ Ω is a strict local maximum point of w − ϕ such that w ( t , x ) − Lt >
0, then ∂ t ϕ ( t , x ) + ˜ H ∗ ( t , x , D x ϕ ( t , x )) ≤ w is a viscosity solution of (A.1) if and only if w is simultaneously a viscositysuper-solution and a viscosity sub-solution of (A.1).Let Ω be a domain in [0 , T ) × R N with some given T >
0. We impose additional assumptionsthe Hamiltonian ˜ H : Ω × R N → R . Namely, for each R > ω R : [0 , ∞ ) → [0 , ∞ ) such that ω R (0) = 0 and ω R ( r ) > r >
0, such that the following holds:(A1) For each ( t , x ) ∈ Ω, p ˜ H ( t , x , p ) is a continuous function from R N to R ;(A2) For each R > t , x ) ∈ Ω ∩ [(0 , T ) × B R (0)], there exist a constant δ > h , k ) ∈ R × R N such that˜ H ( s, y, p ) − ˜ H ( t, x, p ) ≤ ω R (( | x − y | + | t − s | )(1 + | p | ))for t, x, s, y, p such that p ∈ R N , and0 < k ( t, x ) − ( t , x ) k + k ( s, y ) − ( t , x ) k < δ , (cid:13)(cid:13)(cid:13)(cid:13) ( t − s, x − y ) k ( t − s, x − y ) k − ( h , k ) (cid:13)(cid:13)(cid:13)(cid:13) < δ . (A.2)38A3) There is a constant M ≥ λ ∈ [0 ,
1) and x ∈ R N , there exist constants¯ ǫ ( λ, x ) > C ( λ, x ) > H (cid:18) t, x, λp − ǫ ( x − x ) | x − x | + 1 (cid:19) − M ≤ λ ( ˜ H ( t, x, p ) − M ) + ǫ ¯ C ( λ, x )for all ( t, x, p ) ∈ Ω × R N and ǫ ∈ [0 , ¯ ǫ ( λ, x )]. Theorem A.1.
Suppose that ˜ H satisfies (A1)–(A3). Let ¯ w and w be a pair of super- and sub-solutions of (A.1) such that ¯ w ≥ w on ∂ p Ω , then ¯ w ≥ w in Ω . Proof.
This is [37, Theorem A.1], by taking the set Γ to be the entire Ω, the hypotheses (A1)–(A4)therein become (A1)–(A3) here.For our purpose, let H ( s, q, p ) = q + | p | + R ( s ) + R ( s ) Z τ Z R Γ( τ, y ) e τq + yp dydτ. Since H is strictly increasing in q , we define ˜ H ( s, p ) by H ( s, q, p ) = 0 if and only if − q = ˜ H ( s, p ) . (A.3)Define H ∗ and ˜ H ∗ to be the lower envelopes of H and ˜ H respectively: H ∗ ( s, q, p ) = lim inf ( s ′ ,q ′ ,p ′ ) → ( s,q,p ) H ( s ′ , q ′ , p ′ ) , and ˜ H ∗ ( s, p ) = lim inf ( s ′ ,p ′ ) → ( s,p ) ˜ H ( s ′ , p ′ ) . Similarly, define the upper envelopes H ∗ of H and ˜ H ∗ of ˜ H by replacing lim inf by lim sup in theabove. Lemma A.2.
We show that (A.3) holds for the lower and upper envelopes as well, i.e. H ∗ ( s , q , p ) ≤ if and only if q + ˜ H ∗ ( s , p ) ≤ ,H ∗ ( s , q , p ) ≥ if and only if q + ˜ H ∗ ( s , p ) ≥ Proof.
We only show the first part of (A.4), since the latter part is analogous. Let q = − ˜ H ∗ ( s , p ).By monotonicity of H in q , it remains to show that H ∗ ( s , q , p ) = 0.First, choose ( s n , p n ) → ( s , p ) such that q n := − ˜ H ( s n , p n ) → q . By definition of ˜ H , we have H ( s n , q n , p n ) = 0. Taking n → ∞ , we have0 ≥ H ∗ ( s , q , p ) . (A.5)Next, choose another sequence ( s ′ n , q ′ n , p ′ n ) → ( s , q , p ) such that H ( s ′ n , q ′ n , p ′ n ) → H ∗ ( s , q , p ) . H in q , we have0 = H ( s ′ n , − ˜ H ( s ′ n , p ′ n ) , p ′ n ) ≤ H ( s ′ n , − ˜ H ∗ ( s ′ n , p ′ n ) , p ′ n ) . and hence 0 ≤ H ( s ′ n , q ′ n , p ′ n ) + h H ( s ′ n , − ˜ H ∗ ( s ′ n , p ′ n ) , p ′ n ) − H ( s ′ n , q ′ n , p ′ n ) i . (A.6)We claim that lim sup n →∞ h H ( s ′ n , − ˜ H ∗ ( s ′ n , p ′ n ) , p ′ n ) − H ( s ′ n , q ′ n , p ′ n ) i ≤ . (A.7)Indeed, this is due to lim sup n →∞ [ − ˜ H ∗ ( s ′ n , p ′ n )] ≤ q and that ( s ′ n , q ′ n , p ′ n ) → ( s , q , p ), and that H ( s, q, p ) is continuous and monotone in q . Having proved (A.7), we can take n → ∞ in (A.6) toget 0 ≤ H ∗ ( s , q , p ) . Combining with (A.5), the first part of (A.4) is proved.
Lemma A.3.
Let ˜ H ( s, p ) be given in (A.3) . Then ˜ H is convex in p , and hypothesis (A3) holds.Proof. We first prove the convexity of ˜ H . It suffices to show that12 ˜ H ( s, p ) + 12 ˜ H ( s, p ) ≥ ˜ H (cid:18) s, p + p (cid:19) for any s, p , p . (A.8)For i = 1 ,
2, denote q i = ˜ H ( s, p i ), then by the convexity of H ( s, q, p ) in ( q, p ), then0 = 12 H ( s, − q , p ) + 12 H ( s, − q , p ) ≥ H (cid:18) s, − q + q , p + p (cid:19) . By the monotonicity of H in q , we may compare the above with H (cid:16) s, − ˜ H (cid:0) s, p + p (cid:1) , p + p (cid:17) = 0and deduce − q + q ≤ − ˜ H (cid:0) s, p + p (cid:1) . This proves (A.8).Next, the hypothesis (A3) follows as a consequence of the convexity. Indeed,˜ H ( x/t, λp − ǫψ ′ ( x − x )) ≤ λ ˜ H ( x/t, p ) + ǫ ˜ H ( x/t, ψ ′ ( x − x )) + (1 − λ − ǫ ) ˜ H ( x/t, ≤ λ ˜ H ( x/t, p ) + (1 − λ ) M + ǫC ( λ, x ) , where ψ ′ ( x − x ) = x − x | x − x | +1 , M = sup ˜ H ( x/t,
0) and C ( λ, x ) = sup ˜ H ( x/t, ψ ′ ( x − x )) . Proof of Proposition 2.10.
Let ρ and ρ be respectively sub- and super-solutions ofmin { H ( s, ρ − sρ ′ , ρ ′ ) , ρ } = 0 for s ∈ (0 , ∞ ) . It follows from Lemma A.2 that they are respectively sub- and super-solutions ofmin { ρ − sρ ′ + ˜ H ( s, ρ ′ ) , ρ } = 0 for s ∈ (0 , ∞ ) . Define w ( t, x ) = tρ ( x/t ) and w ( t, x ) = tρ ( x/t ) , w and w are respectively sub-and super-solutions ofmin { w t + ˜ H ( x/t, w x ) , w } = 0 for ( t, x ) ∈ (0 , ∞ ) × (0 , ∞ ) . (A.9)To apply Theorem A.1, we need to verify the boundary conditions. Now, w ( t,
0) = tρ (0) ≤ tρ (0) = w ( t,
0) for each t > , and for each x > w (0 , x ) ≤ lim sup t → + h tρ (cid:16) xt (cid:17)i = x lim s →∞ ρ ( s ) s ≤ x lim inf s →∞ ρ ( s ) s = lim inf t → + h tρ (cid:16) xt (cid:17)i ≤ w (0 , x ) . Moreover, for t > x > w ( t, x ) ≤ x sup s> ρ ( s ) s . Since the supremum is finite (see Lemma2.9(iv)), this means that w ( t, x ) is continuous at (0 ,
0) and w (0 ,
0) = 0 ≤ w (0 , R and R are both non-increasing, or both non-decreasing.(ii) R is continuous, and R is monotone.(iii) R is piecewise constant, and the functions R and R + R are locally monotone.First, we consider the case (ii), and assume for the moment that R is non-decreasing. Fix ( t , x )and let ( h , k ) = ( − x ,t ) k ( − x ,t ) k . Then for ( t, x, s, y ) satisfying (A.2), we have xt > ys , so that0 = H (cid:16) ys , − ˜ H (cid:16) ys , p (cid:17) , p (cid:17) ≤ R (cid:16) ys (cid:17) − R (cid:16) xt (cid:17) + H (cid:16) xt , − ˜ H (cid:16) ys , p (cid:17) , p (cid:17) . (A.10)Hence, ˜ H (cid:16) ys , p (cid:17) − ˜ H (cid:16) xt , p (cid:17) ≤ H (cid:16) xt , − ˜ H (cid:16) xt , p (cid:17) , p (cid:17) − H (cid:16) xt , − ˜ H (cid:16) ys , p (cid:17) , p (cid:17) ≤ R (cid:16) ys (cid:17) − R (cid:16) xt (cid:17) ≤ ω R ( | x − y | + | t − s | ) , (A.11)where we used the fact that ∂ q H ≥ H (cid:16) xt , − ˜ H (cid:0) xt , p (cid:1) , p (cid:17) = 0 for thesecond inequality, and the fact that R is continuous (and that t, s are bounded away from zero)for the last inequality.In case R is non-decreasing, the proof for case (i) is the same as case (ii), where the right handside of (A.11) is replaced by 0, since R has the same monotonicity of R . The proof for cases (i)and (ii) when R is non-increasing is similar, and we omit the details.It remains to verify (A2) for the case (iii). Since R is piecewise constant, there exists acountable set { s k } such that R is constant in each open interval in R \ { s k } . To verify (A3),41uppose first ( t , x ) is given so that s := x /t
6∈ { s k } . Then by the local monotonicity of R ,there exists a unit vector ( h , k ) = ( − x , t ) or ( x , − t ) such that for ( t, x, s, y ) satisfying (A.2),we have R (cid:16) xt (cid:17) = R (cid:16) ys (cid:17) and R (cid:16) ys (cid:17) − R (cid:16) xt (cid:17) ≤ ω R ( | x − y | + | t − s | ) . (A.12)Then again (A.10) and (A.11) holds.It remains to consider the case when s = s k for some k , i.e. R has a jump discontinuity.Assume, for definiteness, that R ( s +) > R ( s − ). First, we claim that there is δ > R ( s ′ ) = R ( s +) for s ′ ∈ [ s , s + δ ) , R ( s ′ ) = R ( s − ) for s ′ ∈ ( s − δ , s ) . (A.13)Indeed, R is piecewise constant, so the above holds for s ′ close to but not equal to s . Next, thefact that R + R is locally monotone implies that R ( s ) + R ( s ) ≤ lim sup s ′ → s ( R + R )( s ′ ) ≤ lim sup s ′ → s R ( s ′ ) + lim sup s ′ → s R ( s ′ ) . (A.14)Since R i are u.s.c., we have R i ( s ) ≥ lim sup s ′ → s R i ( s ) for i = 1 ,
2. Substituting these into (A.14), wehave R i ( s ) = lim sup s ′ → s R i ( s ′ ) for i = 1 , R ( s +) > R ( s − ), we must have R ( s ) = R ( s +). We have proved(A.13).Next, we claim that lim δ → sup | s i − s | <δs s ,j → s and s ,j → s such that s ,j < s ,j andlim δ → inf | s i − s | <δs R ( s ,j ) + δ . (A.17)By the first part of (A.17), we deduce that lim inf s ′ → s − R ( s ′ ) ≥ R ( s ). Since R is u.s.c., we deducethat R is left continuous at s , i.e. lim s ′ → s − R ( s ′ ) = R ( s ). In view of the second part of (A.17),it is impossible for both s ,j and s ,j to be less than equal to s for j ≫
1. i.e. we have s ,j > s for j ≫
1. Using also (A.15), we have R ( s ) ≥ lim inf j →∞ R ( s ,j ) ≥ lim inf j →∞ R ( s ,j ) + δ > lim inf s ′ → s + R ( s ′ ) . Combining with (A.13), we obtainlim s ′ → s − ( R + R )( s ′ ) < ( R + R )( s ) and ( R + R )( s ) > lim inf s ′ → s + ( R + R )( s ′ ) . R + R is locally monotone at s , this is impossible. We have proved (A.16).Having proved (A.13) and (A.16), we may again take ( h , k ) = ( − x ,t ) k ( − x ,t ) k and derive (A.10)and (A.11), so that the hypothesis (A2) can again be verified.Finally, if case (iii) and R ( s +) < R ( s − ) hold, then one can argue similarly that hypothesis(A2) holds with the choice of ( h , k ) = ( x , − t ) k ( x , − t ) k . We omit the details. In conclusion, we haveverified that w and w are respectively sub- and super-solutions of (A.9) in (0 , ∞ ) × (0 , ∞ ), andhypotheses (A1)-(A3) hold. Hence, we can apply Theorem A.1 to obtain w ( t, x ) ≤ w ( t, x ) in[0 , ∞ ) × [0 , ∞ ), i.e. ρ ( s ) ≤ ρ ( s ) in [0 , ∞ ). B Estimation for Proposition 2.2
In this section, we establish the upper estimate of w ∗ ( t, x ) and lower estimate of w ∗ ( t, x ). Lemma B.1.
Assume that φ satisfies ( IC µ ) for some µ ∈ (0 , ∞ ) . Let w ∗ ( t, x ) and w ∗ ( t, x ) begiven by (2.6) , then for any δ > , there exist positive numbers Q and Q such that max { ( µ − δ ) x − Q t, } ≤ w ∗ ( t, x ) ≤ w ∗ ( t, x ) ≤ ( µ + δ ) x + Q t in [0 , ∞ ) × [0 , ∞ ) . In particular, w ∗ (0 , x ) = w ∗ (0 , x ) = µx for each x > .Proof. Suppose k φ k ∞ ≤ L , by (H1), we see that u ǫ is a supersolution of ǫ∂ t u − ǫ ∂ xx u + Cu ≥ , ∞ ) × R , for some C >
0. By (H5), there exists δ >
0, such that u ( t, ≥ δ > t ≥ t . Note that w ǫ ( t, x ) = − ǫ log u ǫ ( t, x ) satisfies ∂ t w ǫ − ǫ∂ xx w ǫ + | ∂ x w ǫ | − C ≤ t , ∞ ) × (0 , ∞ ) ,w ǫ ( t, ≤ ǫ | log δ | in [ t , ∞ ) ,w ǫ ( t , x ) < + ∞ in [0 , ∞ ) . (B.1)In view of ( IC ) µ , we know for any small δ ∈ (0 , µ ), there exists 0 < C < C , such that C e − ( µ + δ ) x ≤ φ ( θ, x ) ≤ C e − ( µ − δ ) x for ( θ, x ) ∈ [ − τ , × [0 , ∞ ) . Therefore,( µ − δ ) x − ǫ log C ≤ w ǫ ( θ, x ) ≤ ( µ + δ ) x − ǫ log C , for ( θ, x ) ∈ [ − ǫτ , × [0 , ∞ ) . Next, we define ¯ z ǫ = ( µ + δ ) x + Q ( t + ǫ ) , where Q is chosen to be Q = max ( sup t ∈ [ − τ , ∞ ) [ − log u ( t, , | log C | , C ) , which is finite in view of (H5). Then we have w ǫ ( t, ≤ ¯ z ǫ ( t,
0) for all t ≥ − ǫτ , and w ǫ ( θ, x ) ≤ z ǫ ( θ, x ) , for all ( t, x ) ∈ [ − ǫτ , × R + .
43y comparison principle (¯ z ǫ is super-solution of the first equation in (B.1)), w ǫ ( t, x ) ≤ z ǫ ( t, x ) = ( µ + δ ) x + Q ( t + ǫ ) , for ( t, x ) ∈ [ − ǫτ , × R + . (B.2)Next, let p λ ( p ) be given by the implicit formula∆( λ, p ) := − λ + p + | ∂ v f ( · , · , | ∞ Z τ Z R Γ( τ, y ) e py − λτ dydτ = 0 . (B.3)Then one can similarly define z ǫ ( t, x ) = ( µ − δ ) x − Q ( t + ǫ ) with Q = max {| log C | , λ ( µ − δ ) } .By comparison, we havemax { z ǫ ( t, x ) , − ǫ log( k φ k ∞ + L ) } ≤ w ǫ ( t, x ) in [ − ǫτ , ∞ ) × R + . (B.4)Combining (B.2) and (B.4), and letting ǫ →
0, we havemax { ( µ − δ ) x − Q t, } ≤ w ∗ ( t, x ) ≤ w ∗ ( t, x ) ≤ ( µ + δ ) x + Q t in R . Setting t → δ →
0, it follows that µx ≤ w ∗ (0 , x ) ≤ w ∗ (0 , x ) ≤ µx for all x > Remark B.2.
By (B.4), there exists s > Q / ( µ − δ ) such that w ǫ ( t, x ) ≥ max { Q t, − O ( ǫ ) } when x ≥ st. This implies lim t →∞ sup x ≥ st u ( t, x ) = 0 . Lemma B.3.
Assume that φ satisfies ( IC ∞ ) . Let w ∗ ( t, x ) be given by (2.6) , then w ∗ (0 , x ) = w ∗ (0 , x ) = ∞ for each x > . (B.5) Proof.
Given a solution u ( t, x ) of (1.1) with compactly supported initial data φ . Fix µ >
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