Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line
AAccelerated nonlocal nonsymmetric dispersionfor monostable equations on the real line
Dmitri Finkelshtein Pasha Tkachov November 9, 2018
Abstract
We consider the accelerated propagation of solutions to equations witha nonlocal linear dispersion on the real line and monostable nonlinearities(both local or nonlocal, however, not degenerated at ), in the case wheneither of the dispersion kernel or the initial condition has regularly heavytails at both ±∞ , perhaps different. We show that, in such case, thepropagation to the right direction is fully determined by the right tailsof either the kernel or the initial condition. We describe both cases ofintegrable and monotone initial conditions which may give different ordersof the acceleration. Our approach is based, in particular, on the extensionof the theory of sub-exponential distributions, which we introduced earlyin [16]. Keywords: nonlocal diffusion; reaction-diffusion equation; front prop-agation; acceleration; monostable equation; nonlocal nonlinearity; long-time behavior; integral equation
We will study non-negative solutions u : R × R + → R + := [0 , ∞ ) to the equation ∂∂t u ( x, t ) = κ (cid:90) R a ( x − y ) u ( y, t ) dx − mu ( x, t ) − u ( x, t )( Gu )( x, t ) ,u ( x,
0) = u ( x ) . (1.1)Here κ , m > ; ≤ a ∈ L ( R ) ∩ L ∞ ( R ) with (cid:82) R a ( x ) dx = 1 ; and G is a nonnega-tive mapping on functions which is acting in x , i.e. ( Gu )( x, t ) := (cid:0) Gu ( · , t ) (cid:1) ( x ) ≥ for u ≥ .We will distinguish two cases for the initial condition u : R → R + : lim x →±∞ u ( x ) = 0 , (C1) Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.( [email protected] ). Fakultät für Mathematik, Universität Bielefeld, Postfach 110 131, 33501 Bielefeld, Ger-many ( [email protected] ). a r X i v : . [ m a t h . A P ] N ov nd lim x →∞ u ( x ) = 0 , inf x ≤− ρ u ( x ) > for some ρ ≥ . (C2)We will not assume any symmetricity of either a ( x ) or, in the case (C1), u ( x ) ;in particular, each of them may behave differently at ∞ and −∞ .The function u ( x, t ) may be interpreted as the local density of an evolving intime system of entities which reproduce themselves, compete, and die. The re-production appears according to the dispersion, which is realized via the fecun-dity rate κ and the density a of a probability dispersion distribution. The deathmay appear due the constant inner mortality m > within the system, as wellas due to the density dependent rate Gu ≥ , which describes a competitionwithin the system. For another interpretation for the equation (1.1) rewritten inthe reaction-diffusion form (1.2) and further references, see below and also [14].We consider (1.1) in the space E := L ∞ ( R ) with the standard ess sup -norm.By a solution to (1.1) on R + , we understand the so-called classical solution,that is a mapping u : R + → E which is continuous in t ∈ R + and continuouslydifferentiable (in the sense of the norm in E ) in t ∈ (0 , ∞ ) .We start with the following assumptions: β := κ − m > . ( A1 ) There exists θ > , such that, for each ≤ v ≤ θ, G ≤ Gv ≤ Gθ = β. ( A2 )Here and below we write v ≤ w for v, w ∈ E , if v ( x ) ≤ w ( x ) for a.a. x ∈ R .Note that we will often just write x ∈ R omitting ‘for a.a.’ before this.As a result, u ≡ and u ≡ θ are stationary solutions to (1.1). The rest ofassumptions, ( A3 )–( A11 ), are considered in Section 2 below. In particular, theyensure that, between and θ , there are not other constant stationary solutionsto (1.1); and also that u ≡ is an asymptotically unstable solution and u ≡ θ is an asymptotically stable one. Because of this, the equation (1.1) belongs tothe class of the so-called monostable equations, see e.g. [3]. One can also rewrite(1.1) in the so-called reaction-diffusion form ∂∂t u ( x, t ) = κ ( a ∗ u )( x, t ) − κ u ( x, t ) + ( F u )( x, t ) ,u ( x,
0) = u ( x ) , (1.2)where the symbol ∗ stands for the classical convolution on R , i.e. ( a ∗ v )( x ) := (cid:90) R a ( x − y ) v ( y ) dy, x ∈ R , (1.3)and the reaction F is given by F v := v ( β − Gv ) , v ∈ E. (1.4)Then, under assumptions ( A1 )–( A2 ), we will have that F θ = F ≤ F v ≤ βv, ≤ v ≤ θ. (1.5)The assumption ( A3 ) below yields, in particular, that G is continuous at on { v ∈ E : 0 ≤ v ≤ θ } ; as a result, we require that the reaction F in (1.2) is such2hat F vv → β > as v → (both convergences are in E ). Because of F ,we get then that the Fréchet derivative of F must be a (strictly positive) constantmapping. In particular, we do not allow the degenerate reaction F (cid:48) (0) = 0 ,see e.g. [1] and cf. Example 1.1 below. Therefore, we consider a sub-class ofmonostable reaction-diffusion equations of the form (1.2).The solution u to the equation (1.2) may be interpreted as a density ofa species which invades according to a nonlocal diffusion within the space R meeting a reaction F , see e.g. [12, 22]. In the recent decade, there is a growinginterest to the study of nonlocal monostable reaction-diffusion equations, seee.g. [4, 7, 8, 18, 23, 25, 29]; for the origins of the topic see also [2, 9, 24, 27].We will distinguish two main classes of the examples for G or F , which fulfillthe assumptions of Section 2; see [14, Examples 1.6–1.8] for further details andreferences. Note that, in both examples of F below, the mapping Gu = β − F uu is well-defined, cf. (1.4).
Example 1.1 (Reaction–diffusion equation with a local reaction) . Consider(1.2) with F ( u ) = f ( u ) for a function f : R → R which satisfies the followingassumptions, for some θ > , f is Lipschitz continuous on [0 , θ ];lim r → f ( r ) r = β ; f (0) = f ( θ ) = 0; 0 < f ( r ) ≤ βr, r ∈ (0 , θ ) . In particular, if f is differentiable at , then we require f (cid:48) (0) = β > . Example 1.2 (Spatial logistic equation and its generalizations) . Consider afunction ≤ a − ∈ L ( R ) with (cid:82) R a − ( x ) dx = 1 , such that, for some δ > , κ a ( x ) − βa − ( x ) ≥ δ B δ (0) ( x ) , x ∈ R . (1.6)Here and below, B r ( x ) := [ x − r, x + r ] , r > , x ∈ R . Take an arbitrary θ > and consider (1.2) with F u = γ k u ( θ − a − ∗ u ) k , γ k := βθ k , k ∈ N . (1.7)To formulate our main result, we start with the following definition. Definition 1.3.
Let β > be given by ( A1 ).1) Let b : R → R + be continuous and strictly decreasing on ( ρ, ∞ ) , for some ρ > , with lim x →∞ b ( x ) = 0 . Then, for some τ > , there exists a function r ( t ) = r ( t, b ) , t > τ , which uniquely solves the equation b (cid:0) r ( t ) (cid:1) = e − βt , t > τ, (1.8)and r ( t ) → ∞ , t → ∞ . 3) Similarly, if the function b is continuous and strictly increasing on ( −∞ , − ρ ) with lim x →−∞ b ( x ) = 0 , then one can define l ( t ) = l ( t, b ) → ∞ , t → ∞ as theunique solution to the equation b (cid:0) − l ( t ) (cid:1) = e − βt , t > τ. (1.9)In other words, r ( t ) and l ( t ) are given through the inverse functions to − log b ,namely, for t > τ , r ( t, b ) = (cid:0) − log b (cid:22) R + (cid:1) − ( βt ) , l ( t, b ) = (cid:0) − log b (cid:22) R − (cid:1) − ( βt ) . (1.10)We are going to find sufficient conditions on a and u , such that the cor-responding solution u to (1.1), in the case (C1), becomes arbitrary close to θ (as t goes to ∞ ) inside the (expanded) interval ( − l ( t ) , r ( t )) and becomes arbi-trary close to outside of this interval. In the case (C2), one has to considerthe interval ( −∞ , r ( t )) instead. Here l ( t ) = l ( t, b ) and r ( t ) = r ( t, b ) , where,cf. (1.10), log b ( x ) ∼ log max (cid:8) a ( x ) , u ( x ) (cid:9) , x → ∞ , if (C1) holds, (1.11) log b ( x ) ∼ log max (cid:26)(cid:90) ∞ x a ( y ) dy, u ( x ) (cid:27) , x → ∞ , if (C2) holds, (1.12)and we suppose that the function b has regularly heavy tails at ∞ , see Defini-tion 1.4 below. Here and below the notation f ( x ) ∼ g ( x ) , x → ∞ means that f ( x ) g ( x ) → , x → ∞ . In particular, for any small ε, δ > , we will have that (cid:8) x > (cid:12)(cid:12) u ( x, t ) ∈ ( δ, θ − δ ) (cid:9) ⊂ (cid:0) r ( t − tε ) , r ( t + tε ) (cid:1) for all t big enough; in the case (C1), the corresponding result also holds fornegative values of x and the function l ( t ) instead. Definition 1.4.
1) A bounded function b : R → R + is said to have a regu-larly heavy tail at ∞ in the sense of densities, if b ∈ L ( R + ) , b is decreasingto and convex on ( ρ, ∞ ) for some ρ > , and b (cid:0) x + y (cid:1) ∼ b ( x ) , y ∈ R , x → ∞ , (1.13) (cid:90) x b ( x − y ) b ( y ) dy ∼ (cid:18)(cid:90) R + b ( y ) dy (cid:19) b ( x ) , x → ∞ . (1.14)A bounded function b : R → R + is said to have a regularly heavy tail at −∞ in the sense of densities, if the function b ( − x ) has a regularly heavytail at ∞ in the sense of densities.2) A bounded function b : R → R + is said to have a regularly heavy tail at ∞ in the sense of distributions, if b is decreasing to on R , b is convex on ( ρ, ∞ ) for some ρ > , and − (cid:90) x b ( x − y ) db ( y ) ∼ b ( −∞ ) b ( x ) , x → ∞ , (1.15)where db ( y ) is the Lebesgue–Stieltjes measure associated with b .4 emark . By [17, Lemmas 3.2, 3.4 and Definition 2.21], (1.15) implies (1.13).
Remark . Note that if b : R → R + has a regularly heavy tail at ∞ in thesense of densities and b ∈ L ( R ) , then the function B ( x ) := (cid:90) ∞ x b ( y ) dy, x ∈ R (1.16)has a regularly heavy tail at ∞ in the sense of distribution. The inverse statementis not, in general, true, cf. [17, Section 4.2].Examples of functions with regularly heavy tails at ∞ in the sense of densitiesare the following: (log x ) µ x − q , (log x ) µ x ν exp (cid:0) − p (log x ) q (cid:1) , (log x ) µ x ν exp (cid:0) − x α (cid:1) , (log x ) µ x ν exp (cid:16) − x (log x ) q (cid:17) , (1.17)where p > , q > , α ∈ (0 , , ν, µ ∈ R . See also Lemma 3.3 below for a sufficientcondition, which can be checked for further ‘intermediate’ asymptotics at ∞ .To get examples of functions with regularly heavy tails at ∞ in the sense ofdistributions, one can use (1.16).Note that, see Lemma 3.3 for details, any b with a regularly heavy tail at ∞ in the sense of densities is such that, for each k > , e kx b ( x ) → ∞ , x → ∞ ; this explains the name: the tail of b at ∞ is ‘heavier’ than the tail of an expo-nential function. By Remark 1.5, the same property has each b with a regularlyheavy tail at ∞ in the sense of distributions.Now one can formulate our main result; recall that the exact formulationsfor the assumptions ( A3 )–( A11 ) are given in Section 2 below.
Theorem 1.
Let either ( A1 ) – ( A10 ) hold or ( A11 ) hold. Let ≤ u ≤ θ , u (cid:54)≡ and u be the corresponding solution to (1.1) .1) Let u satisfy (C1) and functions b, b , b : R → R + have regularly heavytails at both ±∞ in the sense of densities, and the following assumptionshold either u ( x ) ≥ b ( x ) or a ( x ) ≥ b ( x ) , x ∈ R , (1.18) max (cid:8) a ( x ) , u ( x ) (cid:9) ≤ b ( x ) , x ∈ R . (1.19) Suppose also that log b ( x ) ∼ log b ( x ) ∼ log b ( x ) , (1.20) as x → ±∞ . Then, for each ε ∈ (0 , , lim t →∞ ess inf [ − l ( t − εt,b ) ,r ( t − εt,b )] u ( x, t ) = θ, (1.21) lim t →∞ ess sup ( −∞ , − l ( t + εt,b )] ∪ [ r ( t + εt,b ) , ∞ ) u ( x, t ) = 0 . (1.22)5 ) Let u satisfy (C2) and functions b, b , b : R → R + have regularly heavytails at ∞ in the sense of distributions, and the following assumptions holdeither u ( x ) ≥ b ( x ) or (cid:90) ∞ x a ( y ) dy ≥ b ( x ) , x ∈ R , (1.23) max (cid:26)(cid:90) ∞ x a ( y ) dy, u ( x ) (cid:27) ≤ b ( x ) , x ∈ R . (1.24) Suppose also that (1.20) holds as x → ∞ . Then, for each ε ∈ (0 , , lim t →∞ ess inf ( −∞ ,r ( t − εt,b )] u ( x, t ) = θ, (1.25) lim t →∞ ess sup [ r ( t + εt,b ) , ∞ ) u ( x, t ) = 0 . (1.26) Remark . For a brevity of notations, we treat here and in the sequel thecondition u (cid:54)≡ as follows: there exist δ, ρ > and x ∈ R , such that u ( x ) ≥ δ for a.a. x ∈ B ρ ( x ) . Remark . We will see in Theorems 2 and 3 below, that the assumptions on b and b may be slightly weaken.Stress that the convergences in (1.21)–(1.22) or (1.25)–(1.26) are indeed ‘ac-celerated’ in t , since, because of (3.12), each b : R → R + with regularly heavytail(s) (in either of senses) satisfies, for each k > , r ( t, b ) − kt → ∞ , l ( t, b ) − kt → ∞ , t → ∞ . (1.27)The reason to introduce the function b in Theorem 1 is two-fold. First, weallow some flexibility in the choice of b and b and hence of a and u . Forexample, b may be a function from (1.17) with negative values of ν and µ ,whereas b may be ‘the same’ function, but with positive values of ν and µ ;then u and a (or (cid:82) ∞ x a in the second part of Theorem 1) may fluctuate betweensuch b and b . In this case, one can take b equal to ‘the same’ function, but with ν = µ = 0 , since then (1.20) evidently holds. Secondly, choosing such b , one canfind r ( t, b ) explicitly (i.e. (1.8) can be solved). Namely, cf. [14, Example 2.18],one has the following values of r ( t ) = r ( t, b ) : b ( x ) = x − q , r ( t ) = exp (cid:16) βq t (cid:17) ; b ( x ) = exp (cid:0) − p (log x ) q (cid:1) , r ( t ) = exp (cid:18)(cid:16) βp t (cid:17) q (cid:19) ; b ( x ) = exp (cid:0) − x α (cid:1) , r ( t ) = ( βt ) α ; b ( x ) = exp (cid:16) − x (log x ) q (cid:17) , r ( t ) ∼ βt (log t ) q , t → ∞ . (Recall that here q > , α ∈ (0 , , p > .) Note that, for b ( − x ) and l ( t ) = l ( t, b ) ,the same examples hold. Remark . In view of (1.10)–(1.12), the asymptotic of r ( t ) may be differentin the cases (C1) and (C2) for the same kernel a . For example, let a ( x ) = x − q , q > , for large x ; then (cid:82) ∞ x a ( y ) dy is proportional to x − q +1 for large x .Therefore, if u decays at + ∞ faster than x − q , then, in the case (C1), we willget r ( t ) = exp (cid:0) βq t (cid:1) , whereas, in the case (C2), we will get r ( t ) = exp (cid:0) βq − t (cid:1) .Hence the propagation in the latter case will be faster.6ur method is based on the usage of functions with regularly heavy tailsbecause of the following reasons. The conditions on G we require imply theeffect called the linear determinacy in e.g. [26], or linear selection in [20], cf.also the pulled fronts in [19]. The effect is that the long-time behavior of thesolutions to (1.1) is well-described by the solutions to the corresponding problemlinearized at the unstable stationary solution u ≡ , that is ∂∂t w ( x, t ) = κ (cid:90) R a ( x − y ) w ( y, t ) dx − mw ( x, t ) ,w ( x,
0) = u ( x ) . (1.28)Indeed, note that the term u ( x, t )( Gu )( x, t ) in (1.1) is small for ‘big’ values of x relatively to u ( x, t ) , provided that G is continuous at ∈ E . Next, becauseof ( A2 ), we have that u ( x, t ) ≤ w ( x, t ) for all x and t . The solution to (1.28)is given through a series of the convolution powers, and the main peculiarityof the functions with regularly heavy tails at infinity is that their convolutionpowers can be estimated by the functions themselves. For the monotone caserelated to (C2), it was the classical Kesten’s bound for distributions on R , seee.g. [17, Theorem 3.34] and Lemma 3.7 below. For the integrable case relatedto (C1), we used our extension of Kesten’s bound to the densities on R , see [16,Theorem 2.22] and Lemma 3.6 below. This is the main tool to get (1.22) and(1.26), see Theorem 3.In order to prove the convergence to θ as well, namely, to get (1.21) and(1.25), we construct in (4.3) a minorant g ( x, t ) to the solution u ( x, t ) to (1.1),which is a sub-solution to the linear equation (1.28) with m replaced by m + δ for a small δ > . The detailed realization of the lower estimates is given inSection 4.For an overview of the existing results about the propagation of solutionsto (1.1) (even over R d , d ≥ ), we refer the reader to [14, Subsection 1.5].In brief, for the case d = 1 considered in the present paper, the situation isthe following. If both the kernel a and the initial condition u are light-tailed,more precisely, if a is exponentially integrable and u is exponentially bounded,then, for example, (1.21)–(1.22) hold for linear r ( t ) = c + t and l ( t ) = c − t (withexplicit formulas for c ± ∈ R ). This case corresponds to the (non-accelerated)linear dispersion spreading.The accelerated case for the local non-linearity (see Example 1.1) was knownin the mathematical biology, see e.g. [21]. The first rigorous result in this direc-tion was done by Garnier [18], who proved an analogue of (1.21)–(1.22) for acompactly supported initial condition u and symmetric heavy-tailed kernel a ,such that ( A9 ) holds. However, in his approach, instead of the function r ( t + εt, b ) in (1.22) with arbitrary small ε > , appeared this function with an unknown ε > , i.e. the result was not sharp.Our results in [14], being rephrased for the case d = 1 , yield both (1.21)–(1.22) for (C1) and (1.25)–(1.26) for (C2), provided that the function b in The-orem 1 was symmetric for (C1) (and r ( t ) = l ( t ) then), and b was the antideriva-tive, cf. (1.16), of a symmetric function for (C2). Note that the functions a and u were not need to be symmetric, up to the equivalence (1.20) though. In par-ticular, either a or u (but not both) might be still light-tailed. Recently, analternative approach was proposed in [5], where an analogous result to (1.21)–(1.22) was obtained (for d = 1 and in the special case of Example 1.1), provided7hat both the kernel a and the initial condition u are symmetric, and a isheavy-tailed; the technique used there goes back to [10]. In another recent pa-per [1], also for the case of Example 1.1, a similar result was obtained for anon-necessary symmetric a and for u which satisfies (C2) with the additionalrestriction that u ( x ) = 0 for large x .Therefore, the present paper is seemed to be the first one which deals with thecase when either of the kernel and the initial condition has (perhaps different)heavy tails at both ±∞ in the case of an integrable initial condition, or considersa monotone-like initial condition which is not necessarily vanishing at + ∞ . Westress that Theorem 1 shows that the acceleration for the propagation of thesolution to (1.1) to the right direction is fully determined by the right tails ofeither a or u .Note also the effect similar to the observed in Remark 1.9 about the pos-sibility of different speeds for the cases (C1) and (C2) was also shown for ananalogue of (1.1) with the fractional Laplacian (in particular, when a is singularand non-integrable), see [6, 11]. Describe now the rest of our assumptions. The first ones guarantee the existence-uniqueness and comparison results of Proposition 2.1 below.
There exists l θ > , such that, for each ≤ v, w ≤ θ (cid:107) Gv − Gw (cid:107) ≤ l θ (cid:107) v − w (cid:107) . ( A3 ) For some p ≥ and for each ≤ v ≤ w ≤ θ, κ a ∗ v − v Gv + pv ≤ κ a ∗ w − w Gw + pw. ( A4 ) Proposition 2.1 ([15, Theorems 2.1, 2.2, Proposition 4.2]) . Let assumptions ( A1 ) – ( A4 ) hold, and ≤ u ≤ θ . Then, for each T > , there exists a uniquesolution u = u ( x, t ) to (1.1) for t ∈ [0 , T ] ; and ≤ u ( · , t ) ≤ θ for all t > . (2.1) Moreover, let ≤ v ≤ θ and v = v ( x, t ) be the corresponding solution to (1.1) ;then u ≤ v implies that ≤ u ( · , t ) ≤ v ( · , t ) ≤ θ for all t > . (2.2) Remark . Note that the assumption ( A1 ) excludes the trivial case when u ( x, t ) converges to as t → ∞ uniformly in x ∈ R . Next, we have shownin [15, Theorems 2.2] that the assumptions ( A1 )–( A4 ) are sufficient to get thecomparison principle for solutions to (1.1) with initial conditions ≤ u ≤ θ (for the exact formulation see also Lemma 4.5 below). For particular cases of G the assumption ( A4 ) is also a necessary condition for the comparison. Forinstance, in the case of Example 1.2 with k = 1 , the condition ( A4 ) reads as κ a ( x ) ≥ ( κ − m ) a − ( x ) , x ∈ R , cf. (1.6). It was shown in [13, Remark 3.6], that if the latter inequality fails, thesolution may not satisfy (2.1). 8he rest of assumptions we need, in particular, to show that a solution u = u ( x, t ) to (1.1) converges to θ locally in space, when time tends to ∞ , seeProposition 2.3 below for the exact formulation. There exist ρ, δ > , such that a ( x ) ≥ ρ for | x | ≤ δ. ( A5 ) For any ≤ v n , v ≤ θ , such that v n loc == ⇒ v , n → ∞ , Gv n loc == ⇒ Gv, n → ∞ , ( A6 )where loc == ⇒ means uniform convergence on all compact subsets of R . For each y ∈ R and ≤ v ≤ θ , ( T y Gv )( x ) = ( GT y v )( x ) for x ∈ R , ( A7 )where T y : E → E , y ∈ R is the translation operator, given by ( T y v )( x ) := v ( x − y ) , x ∈ R . (2.3)The condition ( A7 ) implies that, for any r ≡ const ∈ (0 , θ ) , Gr ≡ const .We will assume then also that Gr < β, r ∈ (0 , θ ) . ( A8 )Finally, we will distinguish two cases. If the condition (cid:90) R | y | a ( y ) dy < ∞ ( A9 )holds, then we set m := κ (cid:90) R ya ( y ) dy, (2.4)and assume, additionally to ( A4 ), that there exist p ≥ , ≤ b ∈ C ∞ ( R ) ∩ L ∞ ( R ) , δ > , such that a ( x ) − b ( x ) ≥ δ B δ (0) ( x ) , x ∈ R ,w Gw ≤ κ b ∗ w + pw for ≤ w ≤ θ. ( A10 )Otherwise, if ( A9 ) does not hold, then we assume that, for each n ∈ N , there exist ≤ a n ∈ L ( R ) , κ n > , G n : E → E, θ n ∈ (0 , θ ] which satisfy ( A1 ) – ( A10 ) instead of a , κ , G , θ ,correspondingly, such that m n := κ n (cid:90) R ya n ( y ) dy ∈ R , θ n > θ − n , n ∈ N , κ n a n ∗ w − wG n w ≤ κ a ∗ w − wGw for ≤ w ≤ θ n , n ∈ N . ( A11 )If ( A9 ) does not hold (e.g. a ( x ) ∼ | x | − as x → ∞ and/or x → −∞ , see themain results below), then, to fulfill ( A11 ) in Examples 1.1–1.2, we choose, for m ∈ (0 , κ ) , a sequence of sets Λ n ⊂ R , Λ n (cid:37) R , such that κ n := κ (cid:82) Λ n a ( x ) dx > , and define a n ( x ) := (cid:0)(cid:82) Λ n a ( x ) dx (cid:1) − Λ n ( x ) a ( x ) , x ∈ R . In Example 1.1, wetake G n := G , whereas, in Example 1.2 and k = 1 in (1.7) (the general k can beconsidered analogously), we set G n u := κ − a − n ∗ u , where a − n ( x ) := 11 Λ n ( x ) a − ( x ) , x ∈ R . Since θ n := ( κ n − m ) / (cid:0) κ − (cid:82) R a − n ( x ) dx (cid:1) → θ , n → ∞ , one can assumethat θ n > θ − n . Proposition 2.3.
Let ≤ u ≤ θ , u (cid:54)≡ , and let u = u ( x, t ) be the corre-sponding solution to (1.1) .1) ([15, Theorem 2.3]) Let ( A1 ) – ( A10 ) hold and m be given by (2.4) . Then,for each compact set K ⊂ R , lim t →∞ ess inf x ∈ K u ( x + t m , t ) = θ. (2.5)
2) (cf. [15, Theorem 2.5]) Let ( A11 ) hold. Then, for each compact set K ⊂ R and for each n ∈ N , lim inf t →∞ ess inf x ∈ K u ( x + t m n , t ) ≥ θ − n . (2.6) Remark . Note that the proof of the second statement in Proposition 2.3 isa straightforward modification of that in [15, Theorem 2.5].
Remark . Note also that we required in Definition 1.4 an additional convexityat infinity of a regularly heavy-tailed function just to cover the case m (cid:54) = 0 in(2.4) or m n (cid:54) = 0 in ( A11 ), see the usage of Proposition 3.12 in the proof ofTheorem 2 below.
Definition 3.1.
A function b : R → R + is said to be – (right-side) long-tailed , if there exists ρ ≥ , such that b ( x ) > for all x ≥ ρ ;and, for any y ≥ , lim x →∞ b ( x + y ) b ( x ) = 1; (3.1) – (right-side) tail-decreasing (tail-continuous, tail-convex, tail-log-convex) , if b ( x ) > , x ∈ ( ρ, ∞ ) , for some ρ ≥ , and b is strictly decreasing to (respectively, b is continuous, b is convex, log b is convex) on ( ρ, ∞ ) ; – sub-exponential on R + in the sense of densities, if b ∈ L ( R + ) ∩ L ∞ ( R + ) , b islong-tailed, and (cid:90) x b ( x − y ) b ( y ) dy ∼ (cid:18)(cid:90) R + b ( y ) dy (cid:19) b ( x ) , x → ∞ ; (3.2) – sub-exponential on R + in the sense of distributions, if b ∈ L ∞ ( R + ) , b isdecreasing to on R + , and − (cid:90) x b ( x − y ) db ( y ) ∼ b ( −∞ ) b ( x ) , x → ∞ , (3.3)where db ( y ) is the Lebesgue–Stieltjes measure associated with b .10 emark . If b is sub-exponential on R + in the sense of densities, then thefunction (cid:0) (cid:107) b (cid:107) L ( R + ) (cid:1) − b ( x ) is a sub-exponential probability density on R + , cf.e.g. [17, Definition 4.6].If b is sub-exponential on R + in the sense of distributions, the function R + ( x ) (cid:16) − (cid:0) b ( −∞ ) (cid:1) − b ( x ) (cid:17) is a sub-exponential probability distribution on R + , cf. e.g. [17, Definition 3.1].As it was point out in Remark 1.5, cf. [17, Lemmas 3.2, 3.4 and Defini-tion 2.21], if b is sub-exponential in the sense of distributions, then (3.1) holds. Lemma 3.3.
Let b : R → R + be (right-side) long-tailed.1) ([17, Lemma 2.17]) For each k > , lim x →∞ e kx b ( x ) = ∞ ; (3.4)
2) ([17, Lemma 2.19, Proposition 2.20]) There exists a non-decreasing func-tion h : (0 , ∞ ) → (0 , ∞ ) , with h ( x ) < x and lim x →∞ h ( x ) = ∞ , such that lim x →∞ sup | y |≤ h ( x ) (cid:12)(cid:12)(cid:12)(cid:12) b ( x + y ) b ( x ) − (cid:12)(cid:12)(cid:12)(cid:12) = 0; (3.5)
3) ([17, Theorem 4.15, Section 4.2]) Let, additionally, b be a tail-log-convexfunction, such that b ∈ L ( R + ) ; and suppose that the function h in (3.5) can be chosen such that lim x →∞ x b (cid:0) h ( x ) (cid:1) = 0 . (3.6) Then b is sub-exponential on R + in the sense of densities and the function (1.16) is sub-exponential on R + in the sense of distributions.Remark . By [16, Proposition 2.15], if b : R → R + is tail-decreasing and < h ( x ) < x with lim x →∞ h ( x ) = ∞ , then (3.5) is equivalent to lim x →∞ b ( x ± h ( x )) b ( x ) = 1 . Example 3.5 ([16, Subsection 3.2]) . Let b : R → R + be a bounded tail-decreasing and tail-log-convex function, such that, for some C > , the function Cb ( x ) has either of the asymptotics (1.17) as x → ∞ , where p > , q > , α ∈ (0 , , ν, µ ∈ R . Then h ( x ) in (3.5) can be chosen such that (3.6) holds; inparticular, then b is sub-exponential on R + . Note also that the functions (1.17)themselves are tail-decreasing and tail-log-convex. Lemma 3.6.
Let b ∈ L ( R , R + ) be sub-exponential on R + in the sense ofdensities. Suppose that there exist ρ, K > , such that b ( x + y ) ≤ Kb ( x ) , x > ρ, y > (3.7) (for example, let b be tail-decreasing). ) ([16, Theorem 2.19]; for n = 2 , see also [17, Lemma 4.13]) For each n ≥ , lim x →∞ b ∗ n ( x ) b ( x ) = n (cid:16)(cid:90) R b ( y ) dy (cid:17) n − , where b ∗ n ( x ) = ( b ∗ . . . ∗ b (cid:124) (cid:123)(cid:122) (cid:125) n )( x ) , x ∈ R , and ∗ is given by (1.3) .2) ([16, Theorem 2.22]) Let, additionally, b be bounded and there exist abounded d : R → R + which is sub-exponential on R + , such that (3.7) holds with b replaced by d , and, for some D > and ρ (cid:48) > , b ( − x ) ≤ D d ( x ) , x > ρ (cid:48) (3.8) (for example, let (3.8) hold with d ( x ) = | x | δ , x ∈ R , δ > ). Then, forany δ ∈ (0 , , there exist C δ , x δ > , such that b ∗ n ( x ) ≤ C δ (1 + δ ) n (cid:16)(cid:90) R b ( y ) dy (cid:17) n − b ( x ) (3.9) for all x > x δ , n ∈ N . Lemma 3.7.
Let b : R → R + be bounded, continuous, and decreasing on R to function, which is sub-exponential on R + in the sense of distributions. Denote b (cid:63) ( x ) := b ( x ) , x ∈ R . For each n ≥ , we consider the Lebesgue–Stieltjesintegral b (cid:63)n ( x ) := − (cid:90) R b ( x − y ) db (cid:63) ( n − ( y ) , x ∈ R . (3.10)
1) ([17, Corollary 3.20]) For each n ≥ , b (cid:63)n ( x ) ∼ n (cid:0) b ( −∞ ) (cid:1) n − b ( x ) , x → ∞ .
2) ([17, Theorem 3.34]) For each δ ∈ (0 , , there exists C δ > , such that b (cid:63)n ( x ) ≤ C δ (1 + δ ) n (cid:0) b ( −∞ ) (cid:1) n − b ( x ) , x ≥ . (3.11) Remark . Let b ∈ L ( R ) and B be given by (1.16). Then B (cid:63)n ( x ) = (cid:90) ∞ x b ∗ n ( y ) dy, x ∈ R . Recall that here by (cid:63) we denote the convolution (3.10) of decreasing boundedfunctions on the real line (e.g. tails of probability distributions), whereas by ∗ we denote the convolution (1.3) of integrable functions on the real line (e.g.probability densities). Lemma 3.9 ([14, Lemma 2.15]) . Let b : R → R + be (right-side) tail-decreasingand long-tailed function. Then, for any k > , r ( t, b ) − kt → ∞ , t → ∞ . (3.12)12vidently, if b ( − x ) is (right-side) tail-decreasing and long-tailed, (3.12) holdsfor l ( t, b ) .For Theorem 1, we will use the functions r ( t ± εt, b ) and l ( t ± εt, b ) for anarbitrary small ε > . This allow us to estimate r ( t ± εt, b ) , where, for example, b is given by (1.17) with µ, ν ∈ R by r ( t ± ˜ εt, b ) , where b corresponds to µ = ν = 0 . Namely, we start with the following definition. Definition 3.10.
Let b , b : R + → R + and, for some ρ ≥ , b i ( x ) > forall s ∈ [ ρ, ∞ ) , i = 1 , . The functions b and b are said to be (asymptotically)log-equivalent , if log b ( x ) ∼ log b ( x ) , x → ∞ . (3.13) Lemma 3.11 ([14, Proposition 2.16]) . Let b , b : R → R + be two tail-decreasingfunctions which are log-equivalent, i.e. (3.13) holds. Define η ± ε ( t, b ) := r ( t ± εt, b ) , t > τ. (3.14) Then, for any < ε < ε < ε < , there exists τ > , such that, for all t ≥ τ , η − ε ( t, b ) ≤ η − ε ( t, b ) ≤ η − ε ( t, b ) ≤ η + ε ( t, b ) ≤ η + ε ( t, b ) ≤ η + ε ( t, b ) . (3.15)Clearly, replacing b ( x ) on b ( − x ) in (3.14), one gets an analogue of (3.15) for l ( t ± εt ) . Proposition 3.12.
Let b : R → R + be (right-side) long-tailed, tail-decreasing,and tail-convex. Then for any < ε < ε < and k > , there exists τ = τ ( k, ε , ε ) > , such that r ( t − ε t, b ) ≥ r ( t − ε t, b ) + kt, t ≥ τ. (3.16) Proof.
Since b is decreasing and convex on ( ρ, ∞ ) for some ρ > , it is well-known that the inverse function b − is also convex on (0 , α ) for some α > . Since t (cid:55)→ e − β (1 − ε ) t is also a convex function, we conclude that, for each ε ∈ (0 , , thefunction [ τ (cid:48) , ∞ ) (cid:51) t (cid:55)→ η ( t ) := r ( t, b ) is convex (for big enough τ (cid:48) > ). Provethat f ( t ) := η (cid:0) (1 − ε ) t (cid:1) − η (cid:0) (1 − ε ) t (cid:1) , t ≥ τ (cid:48) is a non-decreasing function. Indeed, since η ( · ) is convex, we have that the func-tion η ( t ) − η ( s ) t − s , t, s ≥ τ (cid:48) , is non-decreasing in each of coordinates. Therefore,for each t > t > τ (cid:48) , we have (1 − ε ) t > (1 − ε ) t and then η (cid:0) (1 − ε ) t (cid:1) − η (cid:0) (1 − ε ) t (cid:1) (1 − ε )( t − t ) ≤ η (cid:0) (1 − ε ) t (cid:1) − η (cid:0) (1 − ε ) t (cid:1) (1 − ε ) t − (1 − ε ) t ≤ η (cid:0) (1 − ε ) t (cid:1) − η (cid:0) (1 − ε ) t (cid:1) (1 − ε )( t − t ) , Multiplying this on − ε ≤ − ε , one gets η (cid:0) (1 − ε ) t (cid:1) − η (cid:0) (1 − ε ) t (cid:1) < η (cid:0) (1 − ε ) t (cid:1) − η (cid:0) (1 − ε ) t (cid:1) , that implies f ( t ) > f ( t ) . We set ν := inf t ≥ τ (cid:48) f ( t ) = f ( τ (cid:48) ) > . (3.17)13ince b is long-tailed, one gets lim x →∞ sup ≤ y ≤ log b ( x + y ) b ( x ) = lim x →∞ log sup ≤ y ≤ b ( x + y ) b ( x ) = 0 . Therefore, for any δ > , there exists x = x ( δ ) ≥ ρ , such that sup ≤ y ≤ (log b ( x ) − log b ( x + y )) ≤ δ, x ≥ x . (3.18)Let τ = τ ( δ, ε , ε ) ≥ τ (cid:48) be such that η (cid:0) (1 − ε ) t (cid:1) ≥ x , for all t ≥ τ . For anyfixed t ≥ τ , consider N = N ( t ) , such that ∆ := 1 N f ( t ) ∈ (cid:104) min (cid:110) ν, (cid:111) , (cid:105) . Then, by (3.18), (3.17), for all t ≥ τ , one gets ( ε − ε ) βt = log b (cid:0) η (cid:0) (1 − ε ) t (cid:1)(cid:1) − log b (cid:0) η (cid:0) (1 − ε ) t (cid:1)(cid:1) = N − (cid:88) j =0 (cid:16) log b (cid:0) η (cid:0) (1 − ε ) t (cid:1) + j ∆ (cid:1) − log b (cid:0) η (cid:0) (1 − ε ) t (cid:1) + ∆ + j ∆ (cid:1)(cid:17) ≤ δN ≤ δ min (cid:8) ν, (cid:9) f ( t ) . Hence, for any k > , it is sufficient to choose δ ≤ βν ( ε − ε ) k . The proof isfulfilled.Clearly, the corresponding analogue to (3.16) holds for l ( · ) as well. Definition 4.1.
1) Let L ( R + ) denote the set of all right-side long-tailed,tail-decreasing and tail-continuous bounded functions b : R → R + . Let L ( R − ) be the set of all b : R → R + , such that b ( − x ) belongs to L ( R + ) .We set also L ( R ) := L ( R + ) ∩ L ( R − ) .
2) Let P ( R − ) denote the set of all bounded functions b : R → R + such inf x ≤− ρ b ( x ) > for some ρ > . Let P ( R + ) denote the set of all boundedfunctions b : R → R + such inf x ≥ ρ b ( x ) > for some ρ > . We set PL ( R ) := P ( R − ) ∩ L ( R + ) . Let ( A1 ) hold. It will be convenient for us to extend Definition 1.3 by setting l ( t, b ) := ∞ , b ∈ P ( R − ) , r ( t, b ) := ∞ , b ∈ P ( R + ) , (4.1)for t > τ with a needed τ > . 14 roposition 4.2. Let ( A1 ) hold. Let b : R → R + be such that b ∈ L ( R + ) ∪P ( R + ) and b ∈ L ( R − ) ∪ P ( R − ) . Let ε ∈ (0 , be fixed, and τ = τ ( ε ) > besuch that both l t := l ( t − εt, b ) , r t := r ( t − εt, b ) (4.2) are well-defined for t > τ ; cf. also (4.1) . Let λ > be arbitrary. For t > τ and x ∈ R , we define g ( x, t ) = g ε ( x, t ) := λ ( − l t ,r t ) ( x ) + λb ( x ) e β − ε t R \ ( − l t ,r t ) ( x ) ∈ (0 , λ ] . (4.3) Then, for each δ ∈ (0 , εβ ) , there exists t = t ( ε, δ ) > τ , such that, for all t ≥ t ,the function g is a sub-solution to the equation ∂v∂t ( x, t ) = κ ( a ∗ v )( x, t ) − ( m + δ ) v ( x, t ) . Namely, for all t ≥ t and x ∈ R , ( F δ g )( x, t ) := ∂g∂t ( x, t ) − κ ( a ∗ g )( x, t ) + ( m + δ ) g ( x, t ) ≤ . (4.4) Proof.
It is sufficient to prove (4.4) for x ≥ ; indeed, then the result for x < may be obtained by replacing b ( x ) on b ( − x ) . Since b is long-tailed, (3.5) yieldsthat, for any δ ∈ (cid:0) , βε − δ κ (cid:1) , there exists x = x ( δ ) , such that sup | y |≤ h ( x ) b ( x + y ) b ( x ) ≥ − δ , x ≥ x . (4.5)In the sequel, to keep unified notations, we assume that both h ( r t ) and r t − h ( r t ) are equal to ∞ when r t = ∞ , t > τ , cf. (4.1) (remember that, by Lemma 3.3, h ( x ) < x ).Note also, that, by the above, lim t →∞ r t = lim t →∞ l t = ∞ (4.6)(it may be, see (4.1), that either of, or both, r t and l t are equal to ∞ for all t > τ ).Prove, first, that there exists t = t ( ε, δ ) > τ , such that ( a ∗ g )( x, t ) g ( x, t ) ≥ (1 − δ ) (cid:90) l t − h ( r t ) a ( y ) dy (4.7)for all x ≥ and t ≥ t . Note that, clearly, ( a ∗ g )( x, t ) ≥ (cid:90) l t − h ( r t ) a ( y ) g ( x − y, t ) dy (4.8)for x ∈ R and t > τ .1. Let x ∈ [0 , r t − h ( r t )) , t > τ . Then − h ( r t ) ≤ y ≤ l t yields − l t ≤ x − y < r t and hence, by (4.8), (4.3), ( a ∗ g )( x, t ) g ( x, t ) ≥ λ λ (cid:90) l t − h ( r t ) a ( y ) dy, b ∈ L ( R + ) , i.e. if r t < ∞ for t > τ , then we consider alsotwo other possibilities.2. Let x ∈ [ r t − h ( r t ) , r t ) , t > τ . Then it is straightforward to get from (4.8)and (4.3), that ( a ∗ g )( x, t ) g ( x, t ) ≥ (cid:90) l t x − r t a ( y ) dy + (cid:90) x − r t − h ( r t ) a ( y ) b ( x − y ) b ( r t ) dy, (4.9)where we used also that b ( r t ) = e − β − ε t for t > τ . Next, for the considered x , − h ( r t ) ≤ y ≤ x − r t yields ≤ x − y − r t < h ( r t ) , and hence, by (4.5), thereexists t > τ such that for all t ≥ t and x ∈ [ r t − h ( r t ) , r t ] b ( x − y ) b ( r t ) = b (cid:0) r t + ( x − y − r t ) (cid:1) b ( r t ) ≥ − δ , that, together with (4.9), implies (4.7).3. Let x ≥ r t , t > τ . Then, by (4.8) and (4.3), ( a ∗ g )( x, t ) g ( x ) ≥ λe − β − ε t λb ( x ) (cid:90) l t x − r t a ( y ) dy + (cid:90) x − r t − h ( r t ) a ( y ) b ( x − y ) b ( x ) dy. (4.10)Next, e − β − ε t = b ( r t ) ≥ b ( x ) for t > τ , since b is decreasing on [ r t , ∞ ) . The latteralso implies that b ( x − y ) ≥ b ( x ) if only ≤ y ≤ x − r t . Finally, by (4.5), thereexists t > t , such that b ( x − y ) ≥ (1 − δ ) b ( x ) , if only − h ( r t ) ≤ y < , x ≥ r t , t ≥ t . As a result, (4.10) implies (4.7), which is proved hence for all x ≥ and t ≥ t .Note that, by (4.3), ∂g∂t ( x, t ) = β − ε b ( x ) e β − ε t R \ ( − l t ,r t ) ( x ) ≤ β − ε g ( x, t ) . (4.11)Then, combining (4.7) and (4.11), one gets − ( F δ g )( x, t ) g ( x, t ) ≥ − β − ε + κ (1 − δ ) (cid:90) l t − h ( r t ) a ( y ) dy − ( m + δ ) (4.12)for all x ≥ and t ≥ t . By (4.6), we have that − β − ε + κ (1 − δ ) (cid:90) l t − h ( r t ) a ( y ) dy − ( m + δ ) → ε ( κ − m ) − δ κ − δ > , as t → ∞ . Combining this with (4.12), we conclude that there exists t ≥ t ,such that (4.4) holds for x ≥ and t ≥ t . Remark . It is worth noting that, indeed, in the proof of Proposition 4.2,both t and t and hence t do not depend on λ . Proposition 4.4.
Let ( A1 ) – ( A3 ) hold and b : R → R + be such that b ∈ L ( R + ) ∪P ( R + ) and b ∈ L ( R − ) ∪ P ( R − ) . Then, for each ε ∈ (0 , , there exist λ = λ ( ε ) ∈ (0 , θ ) and τ = τ ( ε ) > , such that, for each λ ∈ (0 , λ ) , the function g = g ( x, t ) , given by (4.3) , is a sub-solution to (1.1) . Namely, for all t ≥ τ and x ∈ R , ( F g )( x, t ) := ∂g∂t ( x, t ) − κ ( a ∗ g )( x, t )+ mg ( x, t ) + g ( x, t )( Gg )( x, t ) ≤ . (4.13)16 roof. Take any ε ∈ (0 , and δ ∈ (0 , εβ ) . By ( A1 )–( A3 ), there exists λ = λ ( δ ) = λ ( ε ) ∈ (0 , θ ) , such that ≤ u ≤ λ implies ≤ Gu ≤ δ. (4.14)By Proposition 4.2 and Remark 4.3, for each λ ∈ (0 , λ ] , the function g = g ( x, t ) ,given by (4.3), satisfies (4.4) for all x ∈ R and t > τ for some τ > . Since(4.3) yields g ≤ λ , then (4.14) and (4.4) imply −F g = − ∂g∂t + κ ( a ∗ g ) − mg − g ( Gg ) ≥ − ∂g∂t + κ ( a ∗ g ) − mg − δg = −F δ g ≥ , that yields (4.13).To proceed further we will need the following generalization of the compar-ison (2.2) for solutions to (1.1). Lemma 4.5 ([15, Theorems 2.2]) . Let ( A1 ) – ( A4 ) hold. Let T > be fixed.Suppose that u , u : [0 , T ] → E are continuous mappings, continuously differ-entiable in t ∈ (0 , T ] , and such that, for ( x, t ) ∈ R × (0 , T ] , ∂u ∂t − κ a ∗ u + mu + u Gu ≤ ∂u ∂t − κ a ∗ u + mu + u Gu ,u ( x, t ) ≥ , u ( x, t ) ≤ θ, ≤ u ( x, ≤ u ( x, ≤ θ. Then u ( x, t ) ≤ u ( x, t ) for ( x, t ) ∈ R × [0 , T ] . Theorem 2.
Let either ( A1 ) – ( A10 ) hold or ( A11 ) hold. Let ≤ u ≤ θ , u (cid:54)≡ (cf. Remark 1.7), and let u = u ( x, t ) be the corresponding solution to (1.1) .Suppose also that there exist b : R → R + and D, ρ > , such that1) either (C1) holds, b ∈ L ( R ) , the inequality ( a ∗ u )( x ) ≥ Db ( x ) , (4.15) holds for all | x | > ρ , and b is convex on ( −∞ , ρ ) and on ( ρ, ∞ ) .2) or (C2) holds, b ∈ PL ( R ) , the inequality (4.15) holds for all x > ρ , and b is convex on ( ρ, ∞ ) .Then, for each ε ∈ (0 , , lim t →∞ ess inf x ∈ Λ − ε ( t,b ) u ( x, t ) = θ, (4.16) where Λ − ε ( t, b ) := (cid:40) [ − l ( t − εt, b ) , r ( t − εt, b )] , b ∈ L ( R ) , ( −∞ , r ( t − εt, b )] , b ∈ PL ( R ) . (4.17)17 roof. First, we note that, by Proposition 2.1, ≤ u ≤ θ implies ≤ u ( · , t ) ≤ θ for t > .Let (C1) hold and b ∈ L ( R ) . Since u (cid:54)≡ in the sense of Remark 1.7, thereexists a continuous function ˜ u : R → R + , such that ˜ u ( x ) ≤ u ( x ) , x ∈ R and ˜ u ( x ) ≥ δ for all x ∈ B ρ ( x ) with some x ∈ R , δ, ρ > . Let ˜ u ( x, t ) be the corresponding solution to (1.1). Then by [15, Theorem 2.1], ˜ u ( · , t ) is acontinuous function for all t > . We set also I ρ := [ − ρ, ρ ] .Let (C2) hold and b ∈ PL ( R ) . Then there exists a non-increasing continuousfunction ˜ u : R → R + which is strictly decreasing on ( −∞ , − ρ ) for some ρ > ,such that ˜ u ( x ) ≤ u ( x ) , x ∈ R and ˜ u ( x ) ≥ δ , x < − ρ for some δ > .Let ˜ u ( x, t ) be the corresponding solution to (1.1). Then by [15, Theorem 2.1,Proposition 5.7], ˜ u ( · , t ) is a continuous and non-increasing function for all t > .We set then I ρ := ( −∞ , ρ ] .In both cases, by Proposition 2.1, ˜ u ( x, t ) ≤ u ( x, t ) , x ∈ R , t ≥ . (4.18)Moreover, by [15, Proposition 5.3], ˜ u ( x, t ) > inf y ∈ R s> ˜ u ( y, s ) ≥ , x ∈ R , t > . (4.19)Fix an arbitrary ε ∈ (0 , and take any δ ∈ (0 , ε ) . Consider λ = λ ( δ ) > and τ = τ ( δ ) > , both given by Proposition 4.4. Then, by (4.19), γ := min x ∈ I ρ ˜ u ( x, τ ) > , and, by (4.18), u ( x, τ ) ≥ γ, x ∈ I ρ . (4.20)By ( A2 ), ≤ u ≤ θ implies Gu ≤ β . Rewrite (1.1) in the form (1.2) with F given by (1.4), then, by (1.5), F u ≥ . Then, it is straightforward to show byDuhamel’s principle (see [14, formula (4.16)]), that, for all t > , u ( x, t ) ≥ κ te − κ t ( a ∗ u )( x ) , x ∈ R . (4.21)Let us re-define the given function b by setting b ( x ) := γD for x ∈ I ρ . Notethat, by Definition 4.1, the re-defined function will still belong to either L ( R ) or PL ( R ) , and, by Definition 1.3 and (4.17), for big enough t , the set Λ − ε ( t, b ) will remain the same for the new b . For the new b , (4.15), (4.20), (4.21) yield u ( x, τ ) ≥ D κ te − κ t b ( x ) , x ∈ R . (4.22)Next, combining again (4.18) and (4.19), we will get, cf. (1.8), (1.9), (4.2),(4.17), λ := ess inf x ∈ Λ − δ ( τ ,b ) u ( x, τ ) > . (4.23)Set now λ := min (cid:8) λ , λ , D κ τ e − ( κ + β − δ ) τ (cid:9) . x ∈ R , u ( x, τ ) ≥ g δ ( x, τ ) . (4.24)Since, by Proposition 4.4, g δ is a sub-solution to (1.1), we immediately concludefrom Lemma 4.5 and (4.24), that, for each τ ≥ , u ( x, τ + τ ) ≥ g δ ( x, τ + τ ) , for a.a. x ∈ R . In particular, cf. (4.3), (4.17), u ( x, τ + τ ) ≥ λ for a.a. x ∈ Λ − δ ( τ + τ, b ) , τ ≥ . (4.25)Fix an arbitrary τ ≥ , such that r (( τ + τ )(1 − δ ) , b ) > , l (( τ + τ )(1 − δ ) , b ) > for the latter, see also (4.1) in the case b ∈ PL ( R ) . Set (cid:101) Λ := (cid:2) − l (( τ + τ )(1 − δ ) , b ) + 1 , r (( τ + τ )(1 − δ ) , b ) − (cid:3) , b ∈ L ( R ) , (cid:0) −∞ , r (( τ + τ )(1 − δ ) , b ) − (cid:3) , b ∈ PL ( R ) . Clearly, Λ − δ ( τ + τ, b ) = (cid:91) y ∈ (cid:101) Λ B ( y ) . (4.26)Take and fix now an arbitrary y ∈ (cid:101) Λ . Then, by (4.25), u ( x, τ + τ ) ≥ λ B ( y ) ( x ) , x ∈ R . (4.27)Consider the equation (1.1) with the initial condition v ( x ) = λ B ( y ) ( x ) , x ∈ R ; let v ( x, t ) be the corresponding solution to (1.1). By the uniqueness andcomparison (2.2) in Proposition 2.1, (4.27) yields u ( x, τ + τ + t ) ≥ v ( x, t ) , x ∈ R , t ∈ R + . (4.28)Let, first, ( A1 )–( A10 ) hold. Take an arbitrary µ ∈ (0 , θ ) . Apply Proposi-tion 2.3 to the solution v and K = B ( y ) ; then there exists t µ ≥ , such that v ( x + t µ m , t µ ) ≥ µ for a.a. x ∈ B ( y ) . As a result, by (4.28), u ( x + t µ m , τ + τ + t µ ) ≥ µ, (4.29)for each τ ≥ and a.a. x ∈ B ( y ) . Stress that, by ( A7 ), t µ does not dependon a y ∈ R ; therefore, beside y ∈ (cid:101) Λ = (cid:101) Λ( τ ) , t µ does not depend on τ . As aresult, by (4.26) for any δ ∈ (0 , and µ ∈ (0 , θ ) , there exist λ = λ ( δ ) > , τ = τ ( δ ) > , and t µ ≥ , such that, for all τ ≥ and for a.a. x ∈ Λ − δ ( τ + τ, b ) ,the inequality (4.29) holds.Take any ˜ ε ∈ ( δ, ε ) . Apply now [14, Lemma 3.1] for ε := ˜ ε > δ =: ε , t = τ , t = τ + t ; cf. also (3.14). One gets that there exists τ ≥ , such that,for all τ ≥ τ , r (( τ + τ + t µ )(1 − ˜ ε ) , b ) ≤ r (( τ + τ )(1 − δ ) , b ) , l ( · , b ) . As a result, (4.29) holds for all τ ≥ τ and a.a. x ∈ Λ − ˜ ε ( τ + τ + t µ , b ) ⊂ Λ − δ ( τ + τ, b ) .In particular, for all τ > , u ( x, τ + τ + t µ ) ≥ µ, (4.30)provided that − l (( τ + τ + t µ )(1 − ˜ ε ) , b ) − t µ m < x < r (( τ + τ + t µ )(1 − ˜ ε ) , b ) − t µ m , cf. also (4.1) for the case b ∈ PL ( R ) . Denote T := τ + τ + t µ . Let m ≥ (theopposite case may be considered analogously). Then, in particular, (4.30) holdsfor all − l ( T (1 − ε ) , b ) < x < r ( T (1 − ˜ ε ) , b ) − T m , as l ( · , b ) is increasing. By Proposition 3.12, r ( T (1 − ε ) , b ) ≤ r ( T (1 − ˜ ε ) , b ) − T m for T big enough. As a result, (4.30) holds for all − l ( T (1 − ε ) , b ) < x < r ( T (1 − ε ) , b ) , and big enough T . In other words, we have then that (4.30) holds for all x ∈ Λ − ε ( τ + τ + t µ , b ) and τ > τ for some τ > τ . Since µ ∈ (0 , θ ) was arbitrary,the latter fact yields (4.16).Let now ( A11 ) hold. Then, for a sufficiently large n ∈ N , we will take anarbitrary µ ∈ (cid:0) , θ − n (cid:1) , and, using (2.6) and the same arguments as the above,we will show that (4.30) holds for all x ∈ Λ − ε ( τ + τ + t µ , b ) and big enough τ .Then, the arbitrariness of n and µ yields (4.16) as well.The following result is a simple modification of [14, Proposition 3.17]. Proposition 4.6.
Let f ∈ L ( R , R + ) and g ∈ L ( R ) ∪ PL ( R ) . Then there exists D > such that ( g ∗ f )( x ) ≥ Dg ( x ) (4.31) for all | x | ≥ ρ if g ∈ L ( R ) and for all x > ρ if g ∈ PL ( R ) .Proof. For any r > , we have that ( g ∗ f )( x ) g ( x ) ≥ (cid:90) | y |≤ r g ( x − y ) g ( x ) f ( y ) dy ≥ (cid:18) − sup | y |≤ r (cid:12)(cid:12)(cid:12) g ( x − y ) g ( x ) − (cid:12)(cid:12)(cid:12)(cid:19) (cid:90) | y |≤ r f ( y ) dy. By Lemma 3.3, item 2, and Definition 4.1, one gets that the latter expressionin brackets converges to as x → ±∞ for the case g ∈ L ( R ) or x → ∞ for thecase g ∈ PL ( R ) . Therefore, there exists D > , such that ( g ∗ f )( x ) g ( x ) ≥ D . Corollary 4.7.
Let either ( A1 ) – ( A10 ) hold or ( A11 ) hold. Let u ∈ E + θ , u (cid:54)≡ , cf. Remark 1.7; and let u be the corresponding solution to (1.1) . ) Let u ∈ L ( R ) . Suppose that, for some b ∈ L ( R ) , ρ > ,either u ( x ) ≥ b ( x ) or a ( x ) ≥ b ( x ) , | x | > ρ. Then (4.16) holds.2) Let u is non-increasing on R , and lim x →∞ u ( x ) = 0 . Suppose that, forsome b ∈ PL ( R ) , ρ > , either u ( x ) ≥ b ( x ) , x > ρ, or, for some δ > , x ∈ R , u ( x ) ≥ δ ( −∞ ,x ) ( x ) , x ∈ R , and (cid:90) ∞ x a ( y ) dy ≥ b ( x ) , x > ρ. (4.32) Then (4.16) holds.Proof.
1) The statement is a straightforward application of Theorem 2 andinequality (4.31), applied for either f = a, g = u ( − ρ,ρ ) + b R \ ( − ρ,ρ ) ∈ L ( R ) or f = u , g = a ( − ρ,ρ ) + b R \ ( − ρ,ρ ) ∈ L ( R )
2) The first case is also followed from Theorem 2 and inequality (4.31) with f = a, g = u ( −∞ ,ρ ) + b [ ρ, ∞ ) ∈ PL ( R ) f = a and g = u ∈ PL ( R ) . The second case follows from the followingchain of inequalities: first, because of (4.32) and (3.1), ( a ∗ u )( x ) ≥ δ (cid:90) ∞ x − x a ( y ) dy ≥ δb ( x − x ) , x > ρ + x , (assuming, without loss of generality that ρ + x > ); and, because of(3.1), for a small δ (cid:48) > , b ( x − x ) ≥ (1 − δ (cid:48) ) b ( x ) , x > ρ δ (cid:48) for some ρ δ (cid:48) > ρ + x . Proposition 5.1.
The following statements hold.1) Let u satisfy (C1) . Suppose that there exists p ∈ L ( R ) , such that both p ( s ) , p ( − s ) are sub-exponential on R + in the sense of densities, and thereexist ρ, K > , such that p ( x + τ ) ≤ Kp ( x ) , p ( − x − τ ) ≤ Kp ( − x ) , x ≥ ρ, τ ≥ , (5.1) max (cid:8) a ( x ) , u ( x ) (cid:9) ≤ p ( x ) , | x | ≥ ρ. (5.2) Then, for any ε ∈ (0 , , there exist C ε , x ε > , such that ( a ∗ n ∗ u )( x ) ≤ C ε (1 + ε ) n p ( x ) , | x | > x ε , n ∈ N . (5.3)21 ) Let u satisfy (C2) and be of a bounded variation on R . Suppose that thereexists q ∈ L ∞ ( R ) which is decreasing to on R and is sub-exponential on R + in the sense of distributions, and there exists ρ > , such that q iscontinuous on [ ρ, ∞ ) and max (cid:26)(cid:90) ∞ x a ( y ) dy, u ( x ) (cid:27) ≤ q ( x ) , x ≥ ρ. (5.4) Then, for any ε ∈ (0 , , there exist C ε , x ε > , such that ( a ∗ n ∗ u )( x ) ≤ C ε (1 + ε ) n q ( x ) , x > x ε , n ∈ N . (5.5) Proof.
1) Define, for x ∈ R , ˜ a ( x ) := 11 ( −∞ ,R ) ( x ) a ( x ) + 11 [ R, ∞ ) ( x ) p ( x ) , ˜ u ( x ) := 11 ( −∞ ,R ) ( x ) u ( x ) + 11 [ R, ∞ ) ( x ) p ( x ) , where R = R ( ε ) > is chosen such that max (cid:8) (cid:107) ˜ a (cid:107) , (cid:107) ˜ u (cid:107) (cid:107) u (cid:107) (cid:9) ≤ √ ε . (Here thesub-index denotes the norm in L ( R ) .) Then a ≤ ˜ a , u ≤ ˜ u , and ˜ a, ˜ u are sub-exponential on R + in the sense of densities (cf. [16, Corollary 2.18]).By (3.9), which we apply for δ = √ ε − , C ε = ˜ c ε √ ε , b ( x ) = (cid:107) ˜ a (cid:107) ˜ a ( x ) , x ∈ R , there exists ˜ x = ˜ x ( ε ) > , such that ˜ a ∗ n ( x ) ≤ ˜ c ε (1 + ε ) n ˜ a ( x ) , x ≥ ˜ x . (5.6)Let us estimate ( a ∗ n ∗ u )( x ) = (cid:90) ∞ ˜ x a ∗ n ( y ) u ( x − y ) dy + (cid:90) ˜ x −∞ a ∗ n ( y ) u ( x − y ) dy =: I ( x ) + I ( x ) . (5.7)By (5.1), (5.6), the following estimate holds for x ≥ ˜ x + max { ρ, R } =: ˜ x , I ( x ) ≤ (cid:90) ∞ ˜ x ˜ a ∗ n ( y )˜ u ( x − y ) dy ≤ (cid:90) ∞ ˜ x ˜ c ε (1 + ε ) n ˜ a ( y )˜ u ( x − y ) dy ≤ ˜ c ε (1 + ε ) n (˜ a ∗ ˜ u )( x ); I ( x ) ≤ (cid:90) ˜ x −∞ ˜ a ∗ n ( y )˜ u ( x − y ) dy = (cid:90) ˜ x −∞ ˜ a ∗ n ( y ) p ( x − y ) dy ≤ Kp ( x − ˜ x ) (cid:90) ˜ x −∞ ˜ a ∗ n ( y ) dy ≤ K (1 + ε ) n p ( x − ˜ x ) . By [16, Proposition 2.17] and since p is long-tailed, there exists x ε ≥ ˜ x , suchthat for all x ≥ x ε , (˜ a ∗ ˜ u )( x ) ≤ (1 + ε )2˜ a ( x ) = (1 + ε )2 p ( x ) ,p ( x − ˜ x ) ≤ (1 + ε ) p ( x ) . Hence, (5.3) holds for all x > x ε and C ε := 2(1 + ε ) max { ˜ c ε , K } . Redefining x ε and C ε if needed, we prove similarly (5.3) for all x < − x ε .22) Define, for x ∈ R , A ( x ) := (cid:90) ∞ x a ( y ) dy, ˜ A ( x ) := 11 ( −∞ ,R ) ( x ) (cid:18)(cid:90) Rx a ( y ) dy + q ( R ) (cid:19) + 11 [ R, ∞ ) ( x ) q ( x ) , ˜ u ( x ) := 11 ( −∞ ,R ) ( x ) (cid:0) u ( x ) − u ( R ) + q ( R ) (cid:1) + 11 [ R, ∞ ) ( x ) q ( x ) , where R = R ( ε ) > is such that max (cid:8) ˜ u ( −∞ ) , ˜ A ( −∞ ) A ( −∞ ) (cid:9) ≤ √ ε . Then A ≤ ˜ A, u ≤ ˜ u , and ˜ A, ˜ u are sub-exponential on R + in a sense of distributions.By (3.11), which we apply for δ = √ ε − , C δ = ˜ c ε √ ε , b ( x ) = A ( −∞ ) ˜ A ( x ) , x ∈ R , there exists ˜ x = ˜ x ( ε ) ≥ (in fact, one can put ˜ x = 0 ), such that (cid:90) ∞ x a ∗ n ( y ) dy = A (cid:63)n ( x ) ≤ ˜ A (cid:63)n ( x ) ≤ ˜ c ε (1 + ε ) n ˜ A ( x ) , x ≥ ˜ x . (5.8)Let us estimate (5.7) for chosen ˜ x . By (5.8) and [28, Ch.I Theorem 4b], wehave for x ≥ ˜ x + max { ρ, R } =: ˜ x , I ( x ) = − (cid:90) ∞ ˜ x u ( x − y ) dA (cid:63)n ( y ) = u ( x − ˜ x ) A (cid:63)n (˜ x ) + (cid:90) ∞ ˜ x A (cid:63)n ( y ) du ( x − y ) ≤ u ( x − ˜ x ) + ˜ c ε (1 + ε ) n (cid:90) ∞ ˜ x ˜ A ( y ) du ( x − y ) ≤ u ( x − ˜ x ) + ˜ c ε (1 + ε ) n ˜ A (˜ x ) u ( x − ˜ x ) − ˜ c ε (1 + ε ) n (cid:90) ∞ ˜ x u ( x − y ) d ˜ A ( y ) ≤ q ( x − ˜ x ) + ˜ c ε (1 + ε ) n ( q ( x − ˜ x ) + ˜ A (cid:63) ( x )); I ( x ) ≤ q ( x − ˜ x ) (cid:90) ˜ x −∞ a ∗ n ( y ) dy ≤ q ( x − ˜ x ) . Since ˜ A is sub-exponential and long-tailed, there exists x ε ≥ ˜ x , such that forall x ≥ x ε , ˜ A (cid:63) ( x ) ≤ ε ) ˜ A ( x ) = 2(1 + ε ) q ( x ) , ˜ A ( x − ˜ x ) = q ( x − ˜ x ) ≤ (1 + ε ) q ( x ) . Hence, (5.5) holds for C ε = (4 + 3 ε )˜ c ε , and the proof is fulfilled. Definition 5.2.
1) Let S ( R + ) ⊂ L ( R + ) denote the set of all bounded func-tions b : R → R + which are sub-exponential on R + in the sense of densities,tail-decreasing and tail-continuous. Let S ( R − ) ⊂ L ( R − ) be the set of allbounded b : R → R + , such that b ( − x ) belongs to S ( R + ) . We set also S ( R ) := S ( R + ) ∩ S ( R − ) .
2) Let M ( R ) denote the set of all bounded monotone functions b : R → R + such that lim x →∞ b ( x ) = 0 . We set also MS ( R ) := M ( R ) ∩ S ( R + ) . roposition 5.3. Let ( A1 ) hold and ≤ u ∈ E . Suppose that u (cid:54)≡ , cf.Remark 1.7. Let w = w ( x, t ) be the corresponding solution to (1.28) . Let either (C1) hold and b ∈ S ( R ) be such that max (cid:8) a ( x ) , u ( x ) (cid:9) ≤ b ( x ) , | x | > ρ, (5.9) or (C2) hold and b ∈ MS ( R ) be such that max (cid:26)(cid:90) ∞ x a ( y ) dy, u ( x ) (cid:27) ≤ b ( x ) , x > ρ, (5.10) for some ρ > . Then, for each ε ∈ (0 , , lim t →∞ ess inf x ∈ R \ Λ + ε ( t,b ) w ( x, t ) = 0 , (5.11) where Λ + ε ( t, b ) := (cid:40) [ − l ( t + εt, b ) , r ( t + εt, b )] , b ∈ S ( R ) , ( −∞ , r ( t + εt, b )] , b ∈ MS ( R ) . (5.12) Proof.
The solution to (1.28) is given by w ( x, t ) = e − mt u ( x ) + e − mt ∞ (cid:88) n =1 ( κ t ) n n ! ( a ∗ n ∗ u )( x ) , x ∈ R . By Proposition 5.1 with p = b for b ∈ S ( R ) or q = b for b ∈ MS ( R ) , one getsthat, for any δ ∈ (0 , , there exist C δ , x δ > , such that, for all | x | > x δ > ρ ,in the case b ∈ S ( R ) , and for all x > x δ > ρ , in the case b ∈ MS ( R ) , w ( x, t ) ≤ e − mt u ( x ) + e − mt ∞ (cid:88) n =1 ( κ t ) n n ! C δ (1 + δ ) n b ( x ) and since, in both cases (5.9) and (5.10), u ( x ) ≤ b ( x ) for the considered valuesof x , one can continue ≤ max { C δ , } e κ (1+ δ ) t − mt b ( x ) . Take any ε ∈ (0 , and δ (cid:48) ∈ (0 , ε ( κ − m )) , and set δ = ( κ − m ) ε − δ (cid:48) κ ∈ (0 , , thatensures κ (1 + δ ) − m = ( κ − m )(1 + ε ) − δ (cid:48) . Therefore, w ( x, t ) ≤ max { C δ , } e − δ (cid:48) t e ( κ − m )(1+ ε ) t b ( x ) , again, for either all | x | > x δ or for all x > x δ , depending on the class to that b belongs. By (4.17) and Definition 1.3, we conclude then that there exists τ > ,such that, for all t > τ , w ( x, t ) ≤ max { C δ , } e − δ (cid:48) t , x ∈ R \ Λ + ε ( t, b ) , that implies the statement.Now we can easily get the corresponding result for the solution to (1.1).24 heorem 3. Let ( A1 ) – ( A4 ) hold. Let ≤ u ≤ θ , u (cid:54)≡ (cf. Remark 1.7), andlet u = u ( x, t ) be the corresponding solution to (1.1) . Let a, u and b : R → R + satisfy the assumptions of Proposition 5.3. Then, for each ε ∈ (0 , , lim t →∞ ess inf x ∈ R \ Λ + ε ( t,b ) u ( x, t ) = 0 , (5.13) where Λ + ε is given by (5.12) .Proof. By Proposition 2.1, ≤ u ≤ θ implies ≤ u ( · , t ) ≤ θ for t > ; and then,by ( A2 ), Gu ≥ . Then, it is straightforward to show by Duhamel’s principle,that u ( · , t ) ≤ w ( · , t ) , t > , where w solves (1.28). Hence the proof follows fromProposition 5.3.Finally, one can prove the main Theorem 1. Proof of Theorem 1.
Let ε ∈ (0 , , b and b be fixed. Take any ε ∈ (0 , ε ) .Apply Corollary 4.7 for the function b and ε , and apply Theorem 3 for thefunction b and ε . By (1.20), one can apply Lemma 3.11, then, for big t , r ( t − tε, b ) ≤ r ( t − tε , b ) , r ( t + tε , b ) ≤ r ( t + tε, b ) , and the same holds for l ( · ) . Therefore, for big t , Λ − ε ( t, b ) ⊂ Λ − ε ( t, b ) , R \ Λ + ε ( t, b ) ⊂ R \ Λ + ε ( t, b ) , and hence θ ≥ ess inf x ∈ Λ − ε ( t,b ) u ( x, t ) ≥ ess inf x ∈ Λ − ε ( t,b ) u ( x, t ) → θ, t → ∞ and ≤ ess sup x ∈ R \ Λ + ε ( t,b ) u ( x, t ) ≤ ess sup x ∈ R \ Λ + ε ( t,b ) u ( x, t ) → , t → ∞ , that completes the proof. Acknowledgments
Authors gratefully acknowledge the financial support by the DFG through CRC701 “Stochastic Dynamics: Mathematical Theory and Applications” (DF and PT),the European Commission under the project STREVCOMS PIRSES-2013-612669(DF), and the “Bielefeld Young Researchers” Fund through the Funding LinePostdocs: “Career Bridge Doctorate – Postdoc” (PT).
References [1] M. Alfaro and J. Coville. Propagation phenomena in monostable integro-differential equations: Acceleration or not?
J. Differential Equations , 263(9):5727–5758, 2017.[2] D. G. Aronson. The asymptotic speed of propagation of a simple epidemic.In
Nonlinear diffusion (NSF-CBMS Regional Conf. Nonlinear DiffusionEquations, Univ. Houston, Houston, Tex., 1976) , pp. 1–23. Res. NotesMath., No. 14. Pitman, London, 1977.253] H. Berestycki and F. Hamel. Front propagation in periodic excitable media
Comm. Pure Appl. Math. , 55(8):949–1032, 2002.[4] O. Bonnefon, J. Coville, J. Garnier, and L. Roques. Inside dynamics of so-lutions of integro-differential equations.
Discrete Contin. Dynam. Systems- Ser. B , 19(10):3057–3085, 2014.[5] E. Bouin, J. Garnier, C. Henderson, and F. Patout. Thin front limitof an integro–differential Fisher–KPP equation with fat–tailed kernels.arXiv:1705.10997, 2017.[6] X. Cabré and J.-M. Roquejoffre. The influence of fractional diffusion inFisher-KPP equations.
Comm. Math. Phys. , 320(3):679–722, 2013.[7] J. Coville, J. Dávila, and S. Martínez. Nonlocal anisotropic dispersalwith monostable nonlinearity.
J. Differential Equations , 244(12):3080–3118, 2008.[8] J. Coville and L. Dupaigne. Propagation speed of travelling fronts in nonlocal reaction-diffusion equations.
Nonlinear Anal. , 60(5):797–819, 2005.[9] O. Diekmann. On a nonlinear integral equation arising in mathemati-cal epidemiology. In
Differential equations and applications (Proc. ThirdScheveningen Conf., Scheveningen, 1977) , volume 31 of
North-HollandMath. Stud. , pp. 133–140. North-Holland, Amsterdam–New York, 1978.[10] L. C. Evans and P. E. Souganidis. A PDE approach to geometric opticsfor certain semilinear parabolic equations.
Indiana Univ. Math. J. , 38(1):141–172, 1989.[11] P. Felmer and M. Yangari. Fast propagation for fractional KPP equationswith slowly decaying initial conditions.
SIAM J. Math. Anal. , 45(2):662–678, 2013.[12] P. C. Fife.
Mathematical aspects of reacting and diffusing systems , vol-ume 28 of
Lecture Notes in Biomathematics . Springer-Verlag, Berlin-NewYork, 1979. ISBN 3-540-09117-3. iv+185 pp.[13] D. Finkelshtein, Y. Kondratiev, and P. Tkachov. Traveling wavesand long-time behavior in a doubly nonlocal Fisher–KPP equation.arXiv:1508.02215, 2015.[14] D. Finkelshtein, Y. Kondratiev, and P. Tkachov. Accelerated front propa-gation for monostable equations with nonlocal diffusion. arXiv:1611.09329,2016.[15] D. Finkelshtein and P. Tkachov. The hair-trigger effect for a class of non-local nonlinear equations. arXiv:1702.08076, 2017.[16] D. Finkelshtein and P. Tkachov. Kesten’s bound for sub-exponential densi-ties on the real line and its multi-dimensional analogues. arXiv:1704.05829,2017.[17] S. Foss, D. Korshunov, and S. Zachary.
An introduction to heavy-tailed andsubexponential distributions . Springer Series in Operations Research andFinancial Engineering. Springer, New York, second edition, 2013. ISBN978-1-4614-7100-4; 978-1-4614-7101-1. xii+157 pp.[18] J. Garnier. Accelerating solutions in integro-differential equations.
SIAMJ. Math. Anal. , 43(4):1955–1974, 2011.[19] J. Garnier, T. Giletti, F. Hamel, and L. Roques. Inside dynamics of pulledand pushed fronts.
J. Math. Pures Appl. , 98(4):428–449, Oct. 2012.2620] M. Lucia, C. B. Muratov, and M. Novaga. Linear vs. nonlinear selection forthe propagation speed of the solutions of scalar reaction-diffusion equationsinvading an unstable equilibrium.
Comm. Pure Appl. Math. , 57(5):616–636,2004.[21] J. Medlock and M. Kot. Spreading disease: integro-differential equationsold and new.
Math. Biosci. , 184(2):201–222, Aug. 2003.[22] J. D. Murray.
Mathematical biology. II , volume 18 of
Interdisciplinary Ap-plied Mathematics . Springer-Verlag, New York, third edition, 2003. ISBN0-387-95228-4. xxvi+811 pp. Spatial models and biomedical applications.[23] B. Perthame and P. E. Souganidis. Front propagation for a jump processmodel arising in spatial ecology.
Discrete Contin. Dyn. Syst. , 13(5):1235–1246, 2005.[24] K. Schumacher. Travelling-front solutions for integro-differential equa-tions. I.
J. Reine Angew. Math. , 316:54–70, 1980.[25] Y.-J. Sun, W.-T. Li, and Z.-C. Wang. Traveling waves for a nonlocalanisotropic dispersal equation with monostable nonlinearity.
NonlinearAnal. , 74(3):814–826, 2011.[26] H. Weinberger. On sufficient conditions for a linearly determinate spreadingspeed.
Discrete Contin. Dyn. Syst. - Ser. B , 17(6):2267–2280, 2012.[27] H. F. Weinberger. Asymptotic behavior of a model in population genetics.pages 47–96. Lecture Notes in Math., Vol. 648, 1978.[28] D. V. Widder.
The Laplace Transform . Princeton Mathematical Series, v.6. Princeton University Press, Princeton, N. J., 1941. x+406 pp.[29] H. Yagisita. Existence and nonexistence of traveling waves for a nonlocalmonostable equation.