A study of nonlocal spatially heterogeneous logistic equation with harvesting
aa r X i v : . [ m a t h . A P ] N ov A STUDY OF NONLOCAL SPATIALLY HETEROGENEOUS LOGISTICEQUATION WITH HARVESTING
ANUP BISWAS AND MITESH MODASIYA
Abstract.
We study a class of nonlocal reaction-diffusion equations with a harvesting term wherethe nonlocal operator is given by a Bernstein function of the Laplacian. In particular, it includesthe fractional Laplacian, fractional relativistic operators, sum of fractional Laplacians of differentorder etc. We study existence, uniqueness and multiplicity results of the solutions to the steadystate equation. We also consider the parabolic counterpart and establish the long time asymptoticof the solutions. Our proof techniques rely on both analytic and probabilistic arguments. Introduction
One of the most celebrated reaction-diffusion models was introduced by Fisher [29] and Kol-mogorov, Petrovsky, and Piskunov [42] in 1937 (popularly known as Fisher-KPP model). Sincethen, it has been widely used to model spatial propagation or spreading of biological species intohomogeneous environments (see books [48, 49] for a review). The corresponding equation is givenby ( ∂ t − ν ∆) u ( x, t ) = au (1 − uN ) in D × (0 , T ) , u ( x, t ) = 0 on ∂D × [0 , T ] , where u = u ( x, t ) represents the population density at the space-time point ( x, t ), ν is the diffusionparameter, N > D corresponds to a confinement situation, for instance in a hostile environment.Various generalizations to the above model have been studied both in bounded and unboundeddomains.However, it is recently observed that the heat operator may be too restrictive to describe thespreading of species and for this reason a nonlocal operator may be more useful than a local one,see for instance Berestycki-Coville-Vo [5], Humphries et al. [34], Huston et al. [35], Massaccesi-Valdinoci [47], Viswanathan et al. [58]. On the other hand, starting from the seminal work ofCaffarelli-Silvestre [17] the theory of fractional Laplacian has significantly expanded in many direc-tions and there is a large existing literature for this operator. The fractional Laplacian operatorshave been extensively used for mathematical modelling, for instance anomalous diffusion [16, 57],crystal dislocation [27], water waves [15]. However, there are other types of nonlocal operatorsthat are also of importance. For instance, relativistic operators appearing in quantum mechanics[1, 28], sum of fractional Laplacians of different order appearing in the modelling of acoustic wavepropagation in attenuating media [59]. This calls for consideration of a general family of L´evyoperators (including the above mentioned nonlocal operators) for which a unified theory can bedeveloped. This motivates us to study positive solutions to the following nonlocal logistic equationΨ(–∆) u = au − f ( x, u ) − ch ( x, u ) in D ,u = 0 in D c , Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi BhabhaRoad, Pune 411008, India. Email: [email protected]; [email protected]
Mathematics Subject Classification.
Primary: 35R11, 35S15, 35K57 Secondary: 35J60, 92D25.
Key words and phrases.
Bernstein functions of the Laplacian, nonlocal semipositone problems, long time behaviour,nonlocal Fisher-KPP, bifurcation, variable order nonlocal kernel. where a, c ∈ R , h represents the harvesting term and Ψ( − ∆) denotes the generator of a subordinateBrownian motion and the subordinator is unique determined by its Laplace exponent Ψ. For moredetails about Ψ(–∆) please see Subsection 1.1. One of our main goals in this article is to studyexistence and multiplicity of solutions for different values of a and c . For Ψ(–∆) = − ∆ similarproblems have been studied widely in literature (cf. [19, 20, 23, 26, 32, 43, 50, 54]). But for nonlocalsituation there are only few results and to the best of our knowledge, all of them consider the caseΨ(–∆) = ( − ∆) α / , the fractional Laplacian (cf. [4, 16, 25, 44, 52]). Our results not only generalizesthe existing works but also introduces several new methods. Recently, there have been quite afew works studying pde involving Ψ(–∆) (cf. [7, 8, 9, 10, 12, 36, 37, 38, 41]). We also mentionthe recent work Biswas-L˝orinczi [11] where several maximum principles and generalized eigenvalueproblems for Ψ(–∆) have been studied. Our novelty in this work also comes from the study of thelong time asymptotic of the parabolic pde( ∂ t − Ψ(–∆)) u + au − f ( x, u ) = 0 in D × [0 , T ) ,u ( x, T ) = u ( x ) and u ( x, t ) = 0 in D c × [0 , T ] . We use several potential theoretic tools to establish this long time behaviour.Before we conclude this section let also also mention another type of nonlocal kernel, knowndispersal nonlocal kernel, widely used to model nonlocal reaction-diffusion equations ( cf. [5, 18,31, 35] and references therein). It should be noted that dispersal nonlocal kernels are quite differentfrom the nonlocal kernels of Ψ(–∆) and therefore, the proof techniques involved in these modelsare different from ours.1.1.
A quick introduction to
Ψ(–∆)The class of non-local operators we would be interested in are generators of a large family ofL´evy processes, known as subordinate Brownian motions. These processes are obtained by a timechange of a Brownian motion by independent subordinators. In this section we briefly recall theessentials of the subordinate process which will be particularly used in this article.A Bernstein function is a non-negative completely monotone function, that is, an element of theset B = (cid:26) f ∈ C ∞ ((0 , ∞ )) : f ≥ − n d n f d x n ≤ , for all n ∈ N (cid:27) . In particular, Bernstein functions are increasing and concave. We will consider the following subset B = (cid:26) f ∈ B : lim x ↓ f ( x ) = 0 (cid:27) . For a detailed discussion of Bernstein functions we refer to the monograph [53]. Bernstein functionsare closely related to subordinators. Recall that a subordinator { S t } t ≥ is a one-dimensional, non-decreasing L´evy process defined on some probability space (Ω S , F S , P S ) . The Laplace transformof a subordinator is given by a Bernstein function, i.e., E P S [ e − xS t ] = e − t Ψ( x ) , t, x ≥ , where Ψ ∈ B . In particular, there is a bijection between the set of subordinators on a givenprobability space and Bernstein functions with vanishing right limits at zero.Let B be an R d -valued Brownian motion on the Wiener space (Ω W , F W , P W ), running twice asfast as standard d -dimensional Brownian motion, and let S be an independent subordinator withcharacteristic exponent Ψ. The random processΩ W × Ω S ∋ ( ω , ω ) B S t ( ω ) ( ω ) ∈ R d , is called subordinate Brownian motion under S . For simplicity, we will denote a subordinateBrownian motion by { X t } t ≥ , its probability measure for the process starting at x ∈ R d by P x , and ONLOCAL LOGISTIC EQUATION WITH HARVESTING 3 expectation with respect to this measure by E x . Note that the characteristic exponent of a purejump process { X t } t ≥ is given byΨ( | z | ) = Z R d \{ } (1 − cos( y · z )) j ( | y | ) d y, where the L´evy measure of { X t } t ≥ has a density y j ( | y | ), j : (0 , ∞ ) → (0 , ∞ ), with respect tothe Lebesgue measure, given by j ( r ) = Z ∞ (4 πt ) − d/ e − r t m (d t ) , (1.1)where m is the unique measure on (0 , ∞ ) satisfying [53, Theorem 3.2]Ψ( s ) = Z (0 , ∞ ) (1 − e − st ) m (d t ) . In particular, we have Z R d ( | y | ∧ j ( | y | ) d y < ∞ . In this article we impose the following weak scaling condition on the subordinators.There are 0 < κ ≤ κ < ≤ b such that 1 b (cid:16) Rr (cid:17) κ ≤ Ψ( R )Ψ( r ) ≤ b (cid:16) Rr (cid:17) κ for 1 ≤ r ≤ R < ∞ , (A1)and, there is b > j ( r ) ≤ b j ( r + 1) for r ≥
1. (A2)There is large family of subordinators that satisfy (A1) (see [10, 38]). Moreover, any completeBernstein function (see [53, Definition 6.1]) satisfying (A1) also satisfies (A2) ([39, Theorem 13.3.5],[40]). The conditions (A1)-(A2) are imposed throughout this article without any further mention.It is also helpful to keep in mind that for any c > j ( r ) ≍ Ψ( r − ) r d for 0 < r < c , where the comparison constants might depend on c and whenever (A1) holds for all R ≥ r > c = ∞ (see [14]). Example 1.1.
Some important examples of complete Bernstein functions Ψ satisfying (A1) aregiven by(i) Ψ( x ) = x α/ , α ∈ (0 , κ = κ = α ;(ii) Ψ( x ) = ( x + m /α ) α/ − m , m > α ∈ (0 , κ = κ = α ;(iii) Ψ( x ) = x α/ + x β/ , α, β ∈ (0 , κ = α ∧ β , and κ = α ∨ β ;(iv) Ψ( x ) = x α/ (log(1 + x )) − β/ , α ∈ (0 , β ∈ [0 , α ) with κ = α − β and κ = α ;(v) Ψ( x ) = x α/ (log(1 + x )) β/ , α ∈ (0 , β ∈ (0 , − α ), with κ = α and κ = α + β .Corresponding to the examples above, the related processes are (i) α -stable subordinator, (ii)relativistic α -stable subordinator, (iii) sums of independent subordinators of different indices, etc.The operator − Ψ(–∆) is defined by − Ψ(–∆) f ( x ) = 12 Z R d ( f ( x + y ) + f ( x − y ) − f ( x )) j ( | y | ) d y (1.2)= Z R d ( f ( x + y ) − f ( x ) − {| y |≤ } y · ∇ f ( x )) j ( | y | ) d y, which is classically defined for f ∈ C b ( R d ). Here C b ( R d ) denotes the space of all bounded continuousfunction in R d that are twice continuously differentiable. Also, − Ψ(–∆) is the generator of the
NONLOCAL LOGISTIC EQUATION WITH HARVESTING strong Markov process { X t } t ≥ we introduced above. In connection to the examples above, therelated − Ψ(–∆) operators are (i) α -fractional Laplacian, (ii) α -relativistic operator, (iii) sum offractional Laplacians etc.1.2. Problem and main results
Let D ⊂ R d be a bounded C , domain. For positive constants a, c we consider the followingnonlocal logistic equation with a harvesting termΨ(–∆) u = au − f ( x, u ) − ch ( x, u ) in D ,u >
D ,u = 0 in D c , (1.3)where f : ¯ D × [0 , ∞ ) → [0 , ∞ ) , h : ¯ D × [0 , ∞ ) → [0 , ∞ ) are given continuous functions satisfying s f ( x, s ) , h ( x, s ) are continuously differentiable , f ( x,
0) = f s ( x,
0) = 0 , dd s (cid:20) f ( x, s ) s (cid:21) > s > , lim s →∞ inf x ∈ D f ( x, s ) s = ∞ , and h is bounded with max ¯ D h ( x, > . (A3)A typical example for f is b ( x ) u where b in a positive continuous function. By a solution of(1.3) we mean viscosity solution. For a definition and regularity properties of viscosity solutionssee Section 2 below. As well known, existence of solutions to (1.3) is closely connected with theprincipal eigenvalue of the operator − Ψ(–∆). It is also known that there are only countably manyeigenvalues 0 < λ < λ ≤ λ → ∞ satisfying (see [9]) − Ψ(–∆) ϕ n + λ n ϕ n = 0 in D, and ϕ n = 0 in D c . The first eigenvalue λ is simple and ϕ > D . The principal eigenvalue λ also satisfies aBerestycki-Nirenbarg-Varadhan [3] type characterization, that is, λ = sup { λ : ∃ ψ ∈ C b, + ( D ) such that − Ψ(–∆) ψ + λψ ≤ D } , (1.4)where C b, + ( D ) denotes the collection of all bounded, non-negative continuous functions on R d thatare positive inside D . Before we state our fist main result we recall the notion of stability for asolution u to the boundary value problem − Ψ(–∆) u + g ( x, u ) = 0 in D ,u = 0 in D c . (1.5)A solution u of (1.5) is said to be a stable solution if the Dirichlet principal eigenvalue of theoperator − Ψ(–∆) + g s ( x, u ) is positive, otherwise we say u is an unstable solution . Our first resultis about the logistic equation (i.e., h = 0) Theorem 1.1.
The logistic equation
Ψ(–∆) u = au − f ( x, u ) in D ,u > in D ,u = 0 in D c , (1.6) has no solution for a ≤ λ and has exactly one solution v a for a > λ . Furthermore, the function ( λ , ∞ ) ∋ a v a is continuous, increasing and v a is stable. When Ψ(–∆) = − ∆, Theorem 1.1 is well known. See for instance Oruganti, Shi and Shivaji[50, Theorem 2.5]. For Ψ(–∆) = ( − ∆) α / (i.e., the fractional Laplacian), similar result (withoutstability analysis of solutions) is obtained recently by Marinelli-Mugani [44, Proposition 4.2] using ONLOCAL LOGISTIC EQUATION WITH HARVESTING 5 a variational technique (see also Chhetri-Girg-Hollifield [25, Theorem 2.8]). We also refer to thework of Berestycki, Roquejoffre and Rossi [4, Theorem 1.2] which establishes a similar result for thefractional Laplacian for a periodic patch model in R d . We not only obtain uniqueness of solutionsbut also establish the result for a large class of L´evy operators. It should also be noted that wework in the framework of viscosity solution and therefore, the standard variational technique (asused in [4, 25, 50]) does not work here. Also, our approach is quite robust in the sense that it canalso be applied to non-translation invariant operators and non self-adjoint operators.Next we consider the harvesting term h and study existence of positive solutions. Note that weallow h to depend on u . One such popular example is the predation function h ( x, s ) = s s , althoughour approach does not cover this particular function. The case h ( x, s ) = h ( x ) is known as constantyield harvesting. Letting F ( x, u ) = au − f ( x, u ) − ch ( x, u ) in (1.3) we see that F ( x, ≤
0. Suchproblems are known as semipositone problems, see [19, 20, 26, 54] and references therein. WhenΨ(–∆) = − ∆, existence and multiplicity of solutions to (1.3) have been widely studied; see forinstance, Korman-Shi [43], Oruganti-Shi-Shivaji [50], Costa-Dr´abek-Tehrani [23], Gir˜ao-Tehrani[32] and references therein. We obtain the following bifurcation result for equation (1.3). Theorem 1.2.
Suppose that a > λ and inf s ∈ [0 ,K ] h ( · , s ) (cid:13) in D for every K > . Then thefollowing hold. (i) There exists a positive constant c ◦ such that (1.3) has a maximal solution u ( x, c ) for c < c ◦ . (ii) There is no solution for c > c ◦ . (iii) There exist positive δ, ˜ c such that for every a ∈ ( λ , λ + δ ) there exists a solution u ( x, c ) to (1.3) for each c ∈ (0 , ˜ c ) and u (cid:12) u . Furthermore, lim c → k u ( · , c ) k C ( D ) = 0 . (iv) There exists b c ∈ (0 , ˜ c ) so that for any a ∈ ( λ , λ + δ ) , u , u are the only solutions to (1.3) for < c ≤ b c . Remark 1.1.
The condition inf s ∈ [0 ,K ] h ( · , s ) (cid:13) c . This condition does not have any influence on Theorem 1.2(iii) and (iv).The above result should be compared with [50, Theorem 3.2 and 3.3] which establish a similarresult for Ψ(–∆) = − ∆ and h ( x, u ) = h ( x ). To our best knowledge, there are no similar existingresults for nonlocal operators. For the fractional Laplacian operators only existence of a solutionis obtained for c > a > λ in [25, Theorem 2.9]. The main idea in obtaining Theorem 1.2(iii)is to apply the implicit function theorem of Crandall and Rabinowitz [24]. In case of the Laplacianthis is applied on the forward operator [50, Theorem 3.3]. But the same method can not applied fornonlocal operators due to lack of appropriate Schauder estimates. We instead consider the inverseoperator (see (2.3) below) and establish appropriate estimates so that the implicit function theoremcan be applied.As a corollary to the proof of Theorem 1.2 we get the following uniqueness result which generalizes[50, Theorem 3.4]. In the following result V denotes the potential measure function of ladder-heightprocess corresponding to { X t } (see Section 2). Corollary 1.1.
Suppose that sup s ∈ [0 ,k ] sup D (cid:12)(cid:12)(cid:12) h ( x, s ) V ( δ D ( x )) (cid:12)(cid:12)(cid:12) < ∞ , (1.7) for every finite k . Then for every a > λ , there exists a ˘ c ∈ (0 , c ◦ ) so that for every c ∈ (0 , ˘ c ) ,there exists a unique solution u to (1.3) satisfying λ u ( x ) ≥ c h ( x, u ( x )) , x ∈ R d . (1.8) NONLOCAL LOGISTIC EQUATION WITH HARVESTING
Next we discuss the long time behaviour of the parabolic nonlocal equation. Consider theterminal value problem ( ∂ t − Ψ(–∆)) u + au − f ( x, u ) = 0 in D × [0 , T ) ,u ( x, T ) = u ( x ) and u ( x, t ) = 0 in D c × [0 , T ] . (1.9)By a solution of (1.9) we mean a potential theoretic solution. More precisely, we say u ∈ C ( R d × [0 , T ]) is a solution to ( ∂ t − Ψ(–∆)) u + ℓ ( x, t ) = 0 in D × [0 , T ) ,u ( x, T ) = g ( x ) and u ( x, t ) = 0 in D c × [0 , T ] , (1.10)if u ( x, t ) = E x [ g ( X ( T − t ) ∧ τ )] + E x "Z ( T − t ) ∧ τ ℓ ( X s , t + s ) ds , ( x, t ) ∈ D × [0 , T ] , (1.11)where τ denotes the first exit time of X from D . It can be shown that potential theoretic solutionsare same as viscosity solution of (1.10) (see Lemma 4.1 below). The benefit of working with (1.11)is that it allows us to make use of the underlying probabilistic structure of the model. Our nextmain result is the following Theorem 1.3.
Let u T be the positive and bounded solutions of (1.9) in [0 , T ] . Then the followinghold. (a) For a > λ , we have lim T →∞ u T ( x, → v a , uniformly in D , where v a is the unique solutionof (1.6) . (b) For a ≤ λ , we have lim T →∞ u T ( x, → , uniformly in D . To the best of our knowledge, there are no available results similar to Theorem 1.3 in nonlocalsetting. However, there are quite a few works on the fractional Fisher-KPP equation in R d ; seefor instance, Berestycki-Roquejoffre-Rossi [4], Cabr´e-Roquejoffre [16], L´eculier [46] and referencestherein. For nonlocal dispersal operators in R d large time behaviour has been studied by Berestycki-Coville-Vo [5], Cao-Du-Li-Li [18], Su-Li-Lou-Yang [56] and references therein. The method used inthese works are not applicable for our model. Since our nonlocal operator is quite general in naturethere are no existing parabolic pde estimate (other than fractional Laplacian) that can be used toobtain our result. So we rely on the heat-kernel estimates of the underlying stochastic process X ,and hence the reason to use probabilistic representation of the solution.The rest of the article is organized as follows: In Section 2 we introduce the relation betweenviscosity solution and the Green function representation. We also gather few known results in thissection which is used later in our proofs. Theorems 1.1 and 1.2 are proved in Section 3 whereasSection 4 contains the proof of Theorem 1.3.2. Preliminaries
In this section we recall the notion of nonlocal viscosity solutions, introduced by Caffarelli andSilvestre [17], and its connection with potential theory. We also gather few results which will laterbe used to prove our main results. Denote by C b ( x ) the space of all bounded continuous functionsin R d that are twice continuously differentiable in some neighbourhood around x . Definition 2.1.
A function u : R d → R , upper-semicontinuous in ¯ D , is said to be a viscositysubsolution of Ψ(–∆) u = f in D , if for every x ∈ D and a test function ξ ∈ C b ( x ) satisfying ξ ( x ) = u ( x ) and ξ ( y ) > u ( y ) for y ∈ R d \ { x } we have Ψ(–∆) ξ ( x ) ≤ f ( x ).We say u is a viscosity super-solution of Ψ(–∆) u = f , if − u is a viscosity subsolution ofΨ(–∆) u = − f in D . Furthermore, u is said to be a viscosity solution if it is both a viscositysub- and super-solution. ONLOCAL LOGISTIC EQUATION WITH HARVESTING 7
The viscosity solution of Ψ(–∆) u = f in D, and u = 0 in D c , (2.1)can be represented using Green function and this representation is going to play a key role in thisarticle. We need few notations to introduce this representation. Let τ be the first exit time of X from D i.e., τ = inf { t > X t / ∈ D } . We define the killed process { X Dt } by X Dt = X t if t < τ , and X Dt = ∂ if t ≥ τ , where ∂ denotes a cemetery point. X Dt has transition density p D ( t, x, y ) and its transition semigroup { P Dt } t ≥ is given by P Dt f ( x ) = E x [ f ( X t ) { t< τ } ] = Z D f ( y ) p D ( t, x, y ) d y. (2.2)The Green function of X D is defined by G D ( x, y ) = Z ∞ p D ( t, x, y ) d t . Then the solution of (2.1) can be represented as (see [6, Section 3.1],[38]) u ( x ) = G f ( x ) := Z D G D ( x, y ) f ( y ) d y = E x (cid:20)Z τ f ( X t ) d t (cid:21) , (2.3)where the last equality follows from (2.2).For some of our proofs below we will use some information on the normalized ascending ladder-height process of { X t } t ≥ , where X t denotes the first coordinate of X t . Recall that the ascendingladder-height process of a L´evy process { Z t } t ≥ is the process of the right inverse { Z L − t } t ≥ , where L t is the local time of Z t reflected at its supremum (for details and further information we referto [2, Chapter 6]). Also, we note that the ladder-height process of { X t } t ≥ is a subordinator withLaplace exponent ˜Ψ( x ) = exp (cid:18) π Z ∞ log Ψ( x y )1 + y d y (cid:19) , x ≥ . Consider the potential measure V ( x ) of this process on the half-line ( −∞ , x ). Its Laplace transformis given by Z ∞ V ( x ) e − sx d x = 1 s ˜Ψ( s ) , s > . It is also known that V = 0 for x ≤
0, the function V is continuous and strictly increasing in (0 , ∞ )and V ( ∞ ) = ∞ (see [30] for more details). As shown in [13, Lemma 1.2] and [14, Corollary 3],there exists a constant C = C ( d ) such that1 C Ψ( r − ) ≤ V ( r ) ≤ C Ψ( r − ) , r > . (2.4)This function V will appear in several places of this article. Let us recall the following up to theboundary regularity result from [38, Theorem 1.1 and 1.2] Theorem 2.1.
Assume (A1) - (A2) and f ∈ C ( D ) . Let u be the solution of (2.1) . Then for someconstant C , dependent on d, D, Ψ , we have k u k C φ ( D ) ≤ C k f k L ∞ ( D ) , (2.5) NONLOCAL LOGISTIC EQUATION WITH HARVESTING where φ = Ψ( r − ) − and k u k C φ ( D ) := k u k C ( D ) + sup x,y ∈ D,x = y | u ( x ) − u ( y ) φ ( | x − y | ) . Furthermore, there exists α , dependent on d, D, Ψ , satisfying (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) uV ( δ D ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C α ( D ) ≤ C k f k L ∞ ( D ) , (2.6) where δ D denotes the distance function from ∂D . Using (A1), φ ( r ) ≤ κr κ for r ≤
1, for some constant κ , and thus, it follows from (2.5) that u is κ -H¨older continuous upto the boundary. (2.6) provides a fine boundary decay estimate and thisshould be compared with the results in [51]. Our next result is the Hopf’s lemma which we borrowfrom [11, Theorem 3.3]. Theorem 2.2.
Let u ∈ C b ( R d ) be a non-negative viscosity solution of − Ψ(–∆) u + c ( x ) u ≤ in D, where c is a bounded function. Then either u ≡ in R d or u > in D . Furthermore, if u > in D , then there exists η > satisfying u ( x ) V ( δ D ( x )) > η for x ∈ D . (2.7)To introduce our next results we required the principal eigenvalue for the operator − Ψ(–∆) + c where c is a continuous and bounded function in D . The principal eigenvalue is defined in the samefashion as in [3] and given by λ ( c ) = sup { λ : ∃ ψ ∈ C b, + ( D ) such that − Ψ(–∆) ψ + ( c ( x ) + λ ) ψ ≤ D } . (2.8)Note that for c = 0 we have λ (0) = λ . Next we recall the following refined maximum principlefrom [11, Theorem 3.4 and Lemma 3.1]. Theorem 2.3.
Suppose that λ ( c ) > and v ∈ C b ( R d ) be a solution to − Ψ(–∆) v + cv ≥ in D, v ≤ in D c . Then we have v ≤ .Again, if w ∈ C b ( R d ) is a solution to − Ψ(–∆) w + ( c ( x ) + λ ( c )) w ≥ in D, w ≤ in D c , w ( x ) > , for an x ∈ D , then w = tϕ ∗ for some t > , where ϕ ∗ denotes the positive principal eigenfunctioncorresponding to λ ( c ) . The next result is an anti-maximum principle which is slightly stronger than [11, Theorem 3.5].
Theorem 2.4.
Let f ∈ C ( ¯ D ) and f (cid:12) . Then there exists a δ > such that for every λ ∈ ( λ ( c ) , λ ( c ) + δ ) if u is a solution of − Ψ(–∆) u + ( c ( x ) + λ ) u = f in D, and u = 0 in D c , (2.9) then sup D u ( x ) V ( δ D ( x )) < .Proof. Using [11, Theorem 3.5] we have a δ > λ ∈ ( λ ( c ) , λ ( c ) + δ ) if u is asolution to (2.9) then u < D . Now suppose, on the contrary, that the conclusion of the theoremdoes not hold. Then we find a sequence of δ n → u n < ∂D u n ( x ) V ( δ D ( x )) = 0 . (2.10) ONLOCAL LOGISTIC EQUATION WITH HARVESTING 9
First we observe that k u n k L ∞ → ∞ as n → ∞ . Otherwise, using the argument of Step 1 in [11,Theorem 3.5] we obtain a solution u (cid:12) − Ψ(–∆) u + ( c ( x ) + λ ∗ ) u = f ( x ) in D, u = 0 in D c . In view of Theorem 2.3, we must have u = tϕ ∗ for some t <
0, where ϕ ∗ is the positive Dirichletprincipal eigenfunction of − Ψ(–∆) + c in D . This is not possible since f = 0. Thus we must have k u n k L ∞ → ∞ . Define v n = u n k u n k L ∞ . Then the argument of Step 2 in [11, Theorem 3.5] gives usmax ¯ D (cid:12)(cid:12)(cid:12) v n V ( δ D ( x )) − tϕ ∗ V ( δ D ( x )) (cid:12)(cid:12)(cid:12) → , as n → ∞ , for some t <
0. Combining with (2.10) we must find a point x ∈ ∂D such that ϕ ∗ ( x ) V ( δ D ( x )) = 0.But ϕ ∗ V ( δ D ) can be continuously extended in ¯ D (by Theorem 2.1) and the extension is positive in¯ D , by Theorem 2.2. Thus we arrive at a contradiction. Hence we have a δ > (cid:3) Before we conclude this section let us also mention the following implicit function theorem from[24, Appendix]. In the following theorem X , Y denote Banach spaces. Theorem 2.5.
Let ( s , u ) ∈ R × X and F : R × X → Y be continuously differentiable in somesome neighbourhood of ( s , u ) . Assume that F ( s , u ) = 0 . Suppose that F u ( s , u ) is a linearhomeomorphism of X onto Y . Then there is exactly one C function z : ( s − ε, s + ε ) → X with z ( s ) = 0 satisfying F ( s, u + z ( s )) = 0 for s ∈ ( s − ε, s + ε ) where ε is some positive number. Proof of Theorems and
Lemma 3.1.
Suppose that g : ¯ D × [0 , ∞ ) → R is a continuous function, locally Lipschitz in thesecond variable uniformly with respect to the first, such that g ( x, s ) s is strictly decreasing for s > at each x ∈ D . In addition, also assume that g ( x,
0) = 0 and g s ( x, is continuous in ¯ D . Let u, v ∈ C b ( R d ) be such that (1) − Ψ(–∆) v + g ( x, v ) ≤ ≤ ˜ g ( x ) = − Ψ(–∆) u + g ( x, u ) in D , where ˜ g is a continuousfunction. (2) v > , u (cid:13) in D and v ≥ u = 0 in D c .Then we have v ≥ u in R d .Proof. Let ̺ = sup { t : tu < v in D } . Clearly, ̺ < ∞ . Also, ̺ >
0. Note that by Hopf’s lemma,Theorem 2.2, we have inf D v ( x ) V ( δ D ( x )) ≥ η > , and by (2.6) sup x ∈ D (cid:12)(cid:12)(cid:12) u ( x ) V ( δ D ( x )) (cid:12)(cid:12)(cid:12) ≤ η (3.1)for some η >
0. Thus for some small t > v > t u in D , giving us ̺ ≥ t >
0. Tocomplete the proof it is enough to show that ̺ ≥
1. On the contrary, we suppose that ̺ <
1. Let w = ̺u . Since g ( x,s ) s is strictly decreasing for s > − Ψ(–∆) w + g ( x, w ) = − Ψ(–∆) w + g ( x, ̺u ) ̺ ̺ (cid:13) ̺ [ − Ψ(–∆) u + g ( x, u )] ≥ D , (3.2)Applying [17, Lemma 5.8] we then have − Ψ(–∆)( v − w ) + g ( x, v ) − g ( x, w ) ≤ D, which in turn, gives − Ψ(–∆)( v − w ) + (cid:18) g ( x, v ) − g ( x, w ) v − w (cid:19) ( v − w ) ≤ D. Applying Hopf’s lemma, Theorem 2.2, we have either v − w = 0 in R d or inf D v ( x ) − w ( x ) V ( δ D ( x )) > η . Thefirst option is not possible due to (3.2). Again, if the second option holds, then using (3.1) we canfind t > u − w > t u in D implying v > ( ̺ + t ) u in D . This contradicts the definitionof ̺ . Hence we must have ̺ ≥ (cid:3) Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1.
Recall that ( λ , ϕ ) is the Dirichlet principal eigenpair, that is, − Ψ(–∆) ϕ + λ ϕ = 0 in D,ϕ = 0 in D c . (3.3)Suppose that a < λ and v is a positive solution of (1.6). Then − Ψ(–∆) v + av = f ( x, v ) v v ≥ D since f ( x,s ) s ≥ s ≥
0. Applying the refined maximum principle Theorem 2.3 we get v ≤ R d which is a contradiction.Similarly, if v is a positive solution with a = λ , we obtain − Ψ(–∆) v + λ v = f ( x,v ) v v ≥ D .Applying second part of Theorem 2.3 we have v = tϕ for some t > − Ψ(–∆) ϕ + λ ϕ − t − f ( x, tϕ ) = 0 in D, giving us − f ( x, tϕ ) = 0 in D .
This is not possible since tϕ > D . Thus we have established that no positive solution ispossible for a ≤ λ .Next we consider the case where a > λ . Existence of solution would be proved using a standardmonotone iteration method. To do so we need to construct a subsolution and supersolution. Let u = kϕ where k ∈ (0 , − Ψ(–∆) u + au − f ( x, u ) = ( a − λ ) u − f ( x, u )= u (cid:18) ( a − λ ) − f ( x, kϕ ) kϕ (cid:19) in D .
Since by mean value theorem f ( x,q ) q = f s ( x, r ) for some r ∈ (0 , q ) and f s ( x,
0) = 0, by choosing k small we would easily have (cid:18) ( a − λ ) − f ( x, kϕ ) kϕ (cid:19) > D. Thus we obtain a subsolution u . Again, sincelim s →∞ inf x ∈ D f ( x, s ) s = ∞ , there exist large M > k u k C ( D ) satisfying f ( x,M ) M ≥ a for all x in D . Fixing v = M we get − Ψ(–∆) v + av − f ( x, v ) ≤ D .
ONLOCAL LOGISTIC EQUATION WITH HARVESTING 11
Thus v is a super-solution. Now the existence of a solution is standard using monotone iterationmethod. Let us just sketch the argument. Define H ( x, u ) = au − f ( x, u ) and let θ > H ( x, · ) on the interval [0 , M ], i.e., | H ( x, q ) − H ( x, q ) | ≤ θ | q − q | for q , q ∈ [0 , M ] , x ∈ D .
Now consider the solutions of the following family of problems: − Ψ(–∆) u n +1 − θu n +1 = − H ( x, u n ) − θu n x ∈ D,u n +1 = 0 x ∈ D c , with u = u . It is standard to check that u ≤ u ≤ u ≤ · · · ≤ v . Applying Theorem 2.1 andArzel`a-Ascoli thereom it can be shown that the sequenece converges uniformly in R d to a limit v a ≥ u and v a is a viscosity solution to (1.6). See [7, Lemma 3.3] for more details. Uniqueness ofsolution to (1.6) follows from Lemma 3.1.Next we prove stability of the solution v a for a > λ . Note that given a ≥ a > λ we haveΨ(–∆) v a ≥ a v a − f ( x, v a ) in D .
Therefore, by Lemma 3.1, we have v a ≤ v a . Again, due to Theorem 2.1, it can easily be shownthat a v a is continuous.Fix a > λ and define w = (1 + h ) v a for h >
0. Since(1 + h ) f ( x, s ) < f ( x, (1 + h ) s ) for s ≥ , x ∈ D , we have − Ψ(–∆) w + aw − f ( x, w ) ≤ D. Using [17, Lemma 5.8] we then obtain − Ψ(–∆)( hv a ) + a ( hv a ) − f ( x, w ) + f ( x, v a ) = − Ψ(–∆)( w − v a ) − a ( w − v a ) − f ( x, w ) + f ( x, v a ) ≤ D. Dividing by h on both sides we get − Ψ(–∆) v a + av a − (cid:20) f ( x, w ) − f ( x, v a ) hv a (cid:21) v a ≤ D. Letting h → − Ψ(–∆) v a + av a − f s ( x, v a ) v a ≤ D. Then it follows from (2.8) that the principal eigenvalue λ ∗ of the operator − Ψ(–∆) + a − f s ( x, v a )is non-negative. Now suppose λ ∗ = 0. Then from the proof of [11, Theorem 3.2] (see the last partof the proof) we get that v a is a principal eigenfunction i.e., − Ψ(–∆) v a + av a − f s ( x, v a ) v a = 0 in D. Combining with (1.6) we have f s ( x, v a ) v a = f ( x, v a ) for all x ∈ D . But by (A3) we have sf s ( x, s ) − f ( x, s ) > s >
0. Thus we have a contradiction, giving us λ ∗ >
0. This completes theproof. (cid:3)
For the remaining part of this section we consider the equation with the harvesting term h :Ψ(–∆) u = au − f ( x, u ) − ch ( x, u ) in D ,u >
D ,u = 0 in D c , (3.4)where h satisfies the conditions in (A3). We start with the following lemma about non-existence. Lemma 3.2.
The following hold. (i) If a ≤ λ and c ≥ then equation (3.4) has no non negative solution except u = 0 when c = 0 . (ii) Suppose that inf s ∈ [0 ,K ] h ( · , s ) (cid:13) for any K > . Then for a > λ , there exists M > suchthat equation (3.4) has no nonzero non-negative solution when c > M .Proof. First we consider (i). Note that − Ψ(–∆) u + au = f ( x, u ) + ch ( x, u ) ≥ D. Then the arguments of Theorem 1.1 shows that there is no non-negative u satisfying above equationwhen a ≤ λ .(ii) Fix a > λ . We will prove theorem by contradiction. Assume that there exists positiveincreasing sequence c n → ∞ and solution u n (cid:13) u to (3.4), we have k u k L ∞ ≤ K, (3.5)for some K and all c ≥
0. Since lim s →∞ inf x ∈ D f ( x,s ) s = ∞ there exist large K > f ( x,K ) K ≥ a for all x in D . Taking v = K we get − Ψ(–∆) v + av − f ( x, v ) = aK − f ( x, K )= K (cid:18) a − f ( x, K ) K (cid:19) ≤ D . So v is a super-solution. Thus − Ψ(–∆) v + av − f ( x, v ) ≤ ≤ ch ( x, u ) = − Ψ(–∆) u + au − f ( x, u ) in D .
Using Lemma 3.1 we obtain (3.5). Now dividing both sides of (3.4) by c n we have − Ψ(–∆) (cid:18) u n c n (cid:19) + a u n c n − f ( x, u n ) c n + min s ∈ [0 ,K ] h ( x, s ) ≤ − Ψ(–∆) (cid:18) u n c n (cid:19) + a u n c n − f ( x, u n ) c n + h ( x, u n ) = 0 in D .
Since u n c n and f ( x,u n ) c n converges to 0 as c n → ∞ we get from above that min s ∈ [0 ,K ] h ( x, s ) = 0 whichis a contradiction. Hence the result. (cid:3) Next we prove existence of solution for small values of c . Lemma 3.3.
Fix a > λ . Then there exist c such that for c ∈ (0 , c ) equation (3.4) has a solution u satisfying u ≥ mβϕ where m, β are independent of c ∈ (0 , c ) .Proof. We will prove existence of a positive solution using a monotone iteration method. Let v bethe unique solution of Ψ(–∆) v = 1 in D ,v = 0 in D c . From maximum principle it is evident that v > D . Also, recall the principal eigenfunction ϕ from (3.3). Using Theorems 2.1 and 2.2 we obtain that (cid:12)(cid:12)(cid:12) v ( x ) V ( δ D ( x )) (cid:12)(cid:12)(cid:12) ≤ η , η ≤ ϕ ( x ) V ( δ D ( x )) ≤ η , x ∈ D , η , η , η > . (3.6)Thus ϕ ( x ) ≥ η η v ( x ) x ∈ D .
ONLOCAL LOGISTIC EQUATION WITH HARVESTING 13
Taking ε ( β ) = (1 − β ) η η we get ϕ − εv ≥ βϕ . Define φ = m ( ϕ − εv ). Note that φ ≥ mβϕ . Now − Ψ(–∆) φ + aφ − f ( x, φ ) − ch ( x, φ ) = − λ mϕ + mε + aφ − f ( x, φ ) − ch ( x, φ ) ≥ − λ β φ + aφ − f ( x, φ ) + mε − c k h k L ∞ ≥ (cid:18) a − λ β − f ( x, φ ) φ (cid:19) φ + mε − c k h k L ∞ . Now choose β ∈ ( λ a ,
1) and then choose m small so that f ( x, φ ) φ ≤ a − λ β in D. Then for any c ≤ m k h k − L ∞ ε = k h k − L ∞ (1 − β ) η η m := c we have − Ψ(–∆) φ + aφ − f ( x, φ ) − ch ( x, φ ) ≥ D .
Thus we have a subsolution for all c ≤ c . Again, as shown in Lemma 3.2, we can choose a K toserve a supersolution. Then using a standard monotone iteration method (same as in Theorem 1.1)we can obtain a solution u to (3.4) satisfying u ≥ φ . (cid:3) Using Lemmas 3.2 and 3.3 we obtain the following.
Theorem 3.1.
Assume the setting of Theorem . Suppose that a > λ . Then there exists c ◦ ≥ c such that (i) for < c < c ◦ , (3.4) has a maximal positive solution u ( x, c ) such that for any solution v ( x, c ) of (3.4) we have u ≥ v . Furthermore, lim c → k u ( · , c ) − v a k C ( D ) = 0; (3.7)(ii) for c > c ◦ , (3.4) has no positive solution.Proof. (i) From Theorem 1.1, we know that (1.6) has a unique positive solution v a when a > λ .Let u be any nonnegative solution of (3.4). Then − Ψ(–∆) v a + av a − f ( x, v a ) = 0 < ch ( x, u ) = − Ψ(–∆) u + au − f ( x, u ) in D .
Since u = v a = 0 in D c , using Lemma 3.1 we have that u ≤ v a in R d . Thus whenever (3.4)has a nonnegative solution for some c , we can construct maximal solution of u ( · , c ) for the sameparameter c as follows: we take v a as a supersolution of (3.4), any solution u as a subsolution, andstart the monotone iteration sequence starting from v a . Then we obtain a solution u in between v a and u ; in particular, u ≥ u . Since u can be any solution, the limit of the iterated sequencestarting from v a is the maximal solution. c ◦ = sup { c > c } . From Lemma 3.3 it is clear that c ◦ ≥ c . Now we show that for any c ∈ (0 , c ◦ ), (3.4) has a solution.Then from previous argument we can construct maximal solution for any c ∈ (0 , c ◦ ). Fix c ∈ (0 , c ◦ ).By definition of c ◦ we can find c ′ > c such that (3.4) has a solution u for c ′ . This also implies − Ψ(–∆) u + au − f ( x, u ) − ch ( x, u ) ≥ D .
Since v a ≥ u using a monotone iteration argument we can find a solution for the parameter c . Nowto show (3.7) we observe from Lemma 3.3 that for c ∈ (0 , c ) mβϕ ≤ u ( x, c ) ≤ v a in D. (3.8)Applying Theorem 2.1 we see that the family { u ( · , c ) } c ≤ c is equicontinuous and any limit point ξ ∈ C ( R d ) ,as c → ξ = aξ − f ( x, ξ ) in D. From (3.8) it follows that ξ > D . Thus, by Theorem 1.1, ξ = v a . This gives us (3.7).(ii) follows from the definition of c ◦ . (cid:3) Now we complete the proof of Theorem 1.2. By C ( D ) we denote the space of all continuousfunctions in ¯ D vanishing on the boundary. Proof of Theorem 1.2. (i) and (ii) follows from Theorem 3.1. So we consider (iii). The mainidea of this proof is to use Theorem 2.5 but due to lack of appropriate Schauder type estimate wecan not apply the theorem on the forward operator. Recall the Green operator G associated to theDirichlet problem (2.1), that is, G f ( x ) := Z D G D ( x, y ) f ( y ) d y = E x (cid:20)Z τ f ( X t )d t (cid:21) . (3.9)In view of (2.5), G : C ( D ) → C ( D ) is a compact, bounded linear operator. Now extend h on ¯ D × R by defining h ( x, s ) = h ( x,
0) + sh s ( x, s h ( x, s ) is C . We define F : R × C ( D ) → C ( D )by F ( c, u ) = G ( au − f ( x, u ) − ch ( x, u )) − u. Since G is linear, it is clear that F is continuously differentiable in a neighbourhood of (0 , DF ( c, u )( c , w ) = G ( aw − f s ( x, u ) w − c h ( x, u ) − ch s ( x, u ) w ) − w. Also, F (0 ,
0) = 0. Define
T w := F u (0 , w = G ( aw ) − w . It is clear that T is a bounded linearoperator. Furthermore, T w = 0 implies G ( aw ) = w giving us w ∈ C ( D ) and − Ψ(–∆) w + aw = 0.Since a is not an eigenvalue, we must have w = 0. Thus T is injective. Since G is compact,by Fredholm alternative on Banach spaces T is also surjective and T − is also bounded linear.Therefore, we can apply the implicit theorem Theorem 2.5 to obtain a C curve ( c, z ( c )) in ( − ε, ε ),with z (0) = 0 and F ( c, z ( c )) = 0. In other words,Ψ(–∆) z ( c ) = az ( c ) − f ( x, z ( c )) − ch ( x, z ( c )) in D ,z ( c ) = 0 in D c . (3.10)To complete the proof we only need to show that there exists ˜ c such that z ( c ) > D . Considering c = 0 and f ( x ) = − h ( x,
0) in Theorem 2.4 we choose the corresponding δ from Theorem 2.4. Fix a ∈ ( λ , λ + δ ). Since c z ( c ) is C we have | c | k z ( c ) k C ( D ) ≤ K for some K and all small c .Defining U c = z c c we obtain from (3.10) thatΨ(–∆) U c = aU c − F c ( x ) U c − h ( x, z ( c )) in D ,U c = 0 in D c , (3.11)where F c ( x ) = f ( x,z ( c )) z ( c ) . Note that the rhs of (3.11) is uniformly bounded for all c small. Thusapplying Theorem 2.1 we find that { U c } , { U c V ( δ D ) } are uniformly H¨older continuous in D . In partic-ular, the sequences are pre-compact. Now suppose that there exists c n → z ( c n ) ≯ D . Then we can extract a subsequence n k satisfyingsup x ∈ D (cid:12)(cid:12)(cid:12) U c nk ( x ) V ( δ D ( x )) − W ( x ) V ( δ D ( x )) (cid:12)(cid:12)(cid:12) → , as n k → , (3.12)for some W ∈ C ( D ). Furthermore, from the stability property of viscosity solution [17, Corol-lary 4.7] we obtain Ψ(–∆) W = aW − h ( x,
0) in
D , W = 0 in D c . Using Theorem 2.4 we have
W > D and inf D WV ( δ D ) >
0. From (3.12) we then have U c nk > D for all large n k which contradicts the fact z ( c n ) ≯ D for all n . Hence we can find ˜ c > ONLOCAL LOGISTIC EQUATION WITH HARVESTING 15 such that u ( c ) := z ( c ) > D . Moreover,lim c → k u ( c ) k C ( D ) = 0 . (iv) Suppose, on the contrary, that there exists a sequence c n → v ( · , c n ) of(3.4) corresponding to c n and v ( · , c n ) = u ( · , c n ) and v ( · , c n ) = u ( · , c n ). To simplify the notationwe denote by v n = v ( · , c n ) , u n = u ( · , c n ) , u n = u ( · , c n ). Since, by Theorem 2.1, { v n } is equi-conitnuous, from Theorem 1.1 one of the following hold.(a) There exists a subsequence { n k } satisfying k v n k − v a k C ( D ) = 0, as n k → ∞ .(b) There exists a subsequence { n k } satisfying k v n k k C ( D ) = 0, as n k → ∞ .We arrive a contradiction below in each of the cases. Consider (a) first. Since u n is the maximalsolution we have v n ≤ u n ≤ v a . Thus, by Theorem 3.1, we havelim n k →∞ k u n k − v n k k C ( D ) = 0 . Defining w n = u n − v n and using (3.4) we getΨ(–∆) w n k = aw n k − f ( x, u n k ) − f ( x, v n k ) w n k w n k − c n k h ( x, u n k ) − h ( x, v n k ) w n k w n k in D . (3.13)Since w n k (cid:13) D , by Theorem 2.2, we have w n k > D . Normalize w n k by defining ξ n k = k w nk k C ( D ) w n k . From (3.13) we then haveΨ(–∆) ξ n k = aξ n k − f ( x, u n k ) − f ( x, v n k ) w n k ξ n k − c n k h ( x, u n k ) − h ( x, v n k ) w n k ξ n k in D ,ξ n k = 0 in D c ,ξ n k > D . (3.14)Using Theorem 2.1, we see that { ξ n k } is equicontinuous and then passing to the limit along somesubsequence and using stability property of the viscosity solution in (3.14), we find a solution ξ ∈ C ( R d ) with ξ > D (due to Theorem 2.2) satisfyingΨ(–∆) ξ = aξ − f u ( x, v a ) ξ in D ,ξ = 0 in D c ,ξ > D .
But this contradicts the fact v a is a stable solution (see Theorem 1.1). Thus (a) is not possible. Sowe consider (b). Defining w n = u n − v n = 0 and ξ n = k w n k C ( D ) w n and repeating a similar argumentas above, we get a non-zero ξ satisfyingΨ(–∆) ξ = aξ in D , ξ = 0 in D c , which is a contradiction since a is not an eigenvalue of Ψ(–∆). Thus (b) is also not possible. Thiscompletes the proof of the theorem. (cid:3) Next we establish Corollary 1.1
Proof of Corollary 1.1.
First we show existence. Recall from Lemma 3.3 and Theorem 1.2(i)that for any c < c there exists a maximal solution u ( c ) = u ( · , c ) ofΨ(–∆) u = au − f ( x, u ) − ch ( x, u ) in D ,u >
D ,u = 0 in D c , (3.15)satisfying mβϕ ≤ u ( c ) ≤ v a . Using Theorem 2.1 we see that { u ( c ) } c
An upper (lower) semicontinuous function u is said to be a viscosity subsolution(supersolution) of (4.1) if for every ( x, t ) ∈ D × [0 , T ) and ϕ ∈ C , b ( x, t ) satisfying ϕ ( x, t ) = u ( x, t ) , ϕ ( y, s ) ≥ u ( y, s ) for y ∈ R d , t ≤ s < t + δ, (cid:16) ϕ ( x, t ) = u ( x, t ) , ϕ ( y, s ) ≤ u ( y, s ) for y ∈ R d , t ≤ s < t + δ, respectively, (cid:17) ONLOCAL LOGISTIC EQUATION WITH HARVESTING 17 for some δ >
0, we have ( ∂ t − Ψ(–∆)) ϕ ( x, t ) + ℓ ( x, t ) ≥ , (( ∂ t − Ψ(–∆)) ϕ ( x, t ) + ℓ ( x, t ) ≤ , respectively ) . The time derivative ∂ t can also be replaced by the derivative in parabolic topology i.e., ∂ t + ϕ ( x, t ) = lim h → ϕ ( x, t + h ) − ϕ ( x, t ) h . Let us first show that potential theoretic solution is also a viscosity solution.
Lemma 4.1.
Let u ∈ C b ( R d × [0 , T ]) satisfy (4.2) . Assume that ℓ, g are continuous. Then u is theunique viscosity solution of (4.1) .Proof. Let x ∈ B ⊂ D . By τ B we denotes the exit time from B i.e., τ B = inf { t > X t / ∈ B } . It is evident that τ B ≤ τ . First we show that for any δ < T − tu ( x, t ) = E x [ u ( X δ ∧ τ B , t + δ ∧ τ B )] + E x (cid:20)Z δ ∧ τ B ℓ ( X s , t + s ) ds (cid:21) . (4.3)Using (4.2) we write u ( x, t ) = E x [ g ( X T − t ) { T − t< τ } ] + E x (cid:20)Z T − t ℓ ( X s , t + s ) { s< τ } ds (cid:21) = E x [ g ( X T − t ) { δ ∧ τ B ≤ τ } { ( T − t ) < τ } ] + E x (cid:20) { δ ∧ τ B ≤ τ } Z T − tδ ∧ τ B ℓ ( X s , t + s ) { s< τ } ds (cid:21) + E x (cid:20) { δ ∧ τ B ≤ τ } Z δ ∧ τ B ℓ ( X s , t + s ) { s< τ } ds (cid:21) = E x h { δ ∧ τ B ≤ τ } E X δ ∧ τ B (cid:2) g ( X ( T − t − δ ∧ τ B ) ) { T − t − δ ∧ τ B < τ } (cid:3)i + E x " { δ ∧ τ B ≤ τ } E X δ ∧ τ B "Z ( T − t − δ ∧ τ B )0 ℓ ( X s , t + δ ∧ τ B + s ) { s< τ } ds + E x (cid:20) { δ ∧ τ B ≤ τ } Z δ ∧ τ B ℓ ( X s , t + s ) ds (cid:21) = E x (cid:2) { δ ∧ τ B ≤ τ } u ( X δ ∧ τ B , t + δ ∧ τ B ) (cid:3) + E x (cid:20) { δ ∧ τ B ≤ τ } Z δ ∧ τ B ℓ ( X s , t + s ) ds (cid:21) = E x [ u ( X δ ∧ τ B , t + δ ∧ τ B )] + E x (cid:20)Z δ ∧ τ B ℓ ( X s , t + s ) ds (cid:21) , where in the third line we use strong Markov property and in the last line we use the fact that P x ( δ ∧ τ B ≤ τ ) = 1 . This proves (4.3). This relation is key to show that u is also a viscositysolution. We only check that u is a viscosity subsolution and the other part would be analogous.Consider ( x, t ) ∈ D × [0 , T ) and ϕ ∈ C , b ( x, t ) satisfying ϕ ( x, t ) = u ( x, t ) , ϕ ( y, s ) ≥ u ( y, s ) for y ∈ R d , t ≤ s < t + δ. Choose a ball B , centered at x , small enough so that ϕ is C , in ¯ B × [ t, t + δ ]. Let δ < δ . Thenapplying Dynkin-Itˆo formula we know that E x (cid:20)Z δ ∧ τ B ( ∂ t − Ψ(–∆)) ϕ ( X s , t + s )d s (cid:21) = E x [ ϕ ( X δ ∧ τ B , t + δ ∧ τ B )] − ϕ ( x, t ) ≥ E x [ u ( X δ ∧ τ B , t + δ ∧ τ B )] − u ( x, t ) = − E x (cid:20)Z δ ∧ τ B ℓ ( X s , t + s ) ds (cid:21) , using (4.3). Since P x ( τ B >
0) = 1, dividing both sides by δ and letting δ → ∂ t − Ψ(–∆)) ϕ ( x, t ) + ℓ ( x, t ) ≥ . Similarly, we can show that u is a supersolution.The uniqueness part follows using a similar argument as in [55, Lemma 3.3] (Note that the proofof [55, Lemma 3.2] is based on the ideas from [17] which works for general nonlocal operators). (cid:3) Our next lemma concerns representation of Schr¨odinger equation.
Lemma 4.2.
Suppose that ℓ, V are continuous and bounded in D and g ∈ C ( D ) . Define ϕ ( x, t ) = E x h e R ( T − t ) ∧ τ V ( X s ,t + s ) d s g ( X ( T − t ) ∧ τ ) i + E x "Z ( T − t ) ∧ τ e R s V ( X k ,t + k ) d k ℓ ( X s , t + s ) d s . Then ϕ solves ( ∂ t − Ψ(–∆)) ϕ + ℓ + V ϕ = 0 in D × [0 , T ) ,ϕ ( x, T ) = g ( x ) and ϕ ( x, t ) = 0 in D c × [0 , T ] . (4.4) Proof.
It is routine to check ϕ is continuous (cf. [9, Lemma 3.1]) and ϕ ( · , t ) = 0 in D c . It alsofollows from the definition ϕ ( x, T ) = g ( x ). Now fix any t ∈ [0 , T ) and δ < T − t . Since g and ϕ vanish outside D we obtain that ϕ ( x, t )= E x h { T − t< τ } e R T − t V ( X s ,t + s ) d s g ( X T − t ) i + E x "Z ( T − t )0 { s< τ } e R s V ( X k ,t + k ) d k ℓ ( X s , t + s )d s = E x h { δ< τ } e R δ V ( X s ,t + s )d s E X δ h { T − t − δ< τ } e R T − t − δ V ( X s ,t + δ + s )d s g ( X T − t − δ ) ii + E x (cid:20)Z δ { s< τ } e R s V ( X k ,t + k )d k ℓ ( X s , t + s )d s (cid:21) + E x (cid:20) { δ< τ } e R δ V ( X s ,t + s )d s E X δ (cid:20)Z T − t − δ { s< τ } e R s V ( X k ,t + δ + k ) d k ℓ ( X s , t + s )d s (cid:21)(cid:21) = E x h { δ< τ } e R δ V ( X s ,t + s ) d s ϕ ( X δ , t + δ ) i + E x (cid:20)Z δ { s< τ } e R s V ( X k ,t + k ) d k ℓ ( X s , t + s )d s (cid:21) , (4.5)where the second equality follows from the strong Markov property. Now fix x ∈ D and define ξ ( p ) = E x [ ϕ ( X p ∧ τ , t + p ∧ τ )] . Then, using (4.5) we note that ξ ( p ) − ξ ( p − δ )= E x [ ϕ ( X p ∧ τ , t + p ∧ τ )] − E x h { p − δ< τ } E X p − δ h { δ< τ } e R δ V ( X s ,t + p − δ + s )d s ϕ ( X δ , t + p ) ii − E x (cid:20) { p − δ< τ } E X p − δ (cid:20)Z δ { s< τ } e R s V ( X k ,t + p − δ + k )d k ℓ ( X s , t + p − δ + s )d s (cid:21)(cid:21) = E x [ ϕ ( X p ∧ τ , t + p ∧ τ )] − E x h { p − δ< τ } E x h { p< τ } e R δ V ( X s + p − δ ,t + p − δ + s )d s ϕ ( X p , t + p ) (cid:12)(cid:12)(cid:12) F τ ∧ ( p − δ ) ii − E x (cid:20) { p − δ< τ } E x (cid:20)Z δ { s + p − δ< τ } e R s V ( X p − δ + k ,t + p − δ + k )d k ℓ ( X p − δ + s , t + p − δ + s )d s (cid:12)(cid:12)(cid:12) F τ ∧ ( p − δ ) (cid:21)(cid:21) ONLOCAL LOGISTIC EQUATION WITH HARVESTING 19 = E x [ ϕ ( X p , t + p ) { p< τ } ] − E x h { p< τ } e R δ V ( X s + p − δ ,t + p − δ + s )d s ϕ ( X p , t + p ) i − E x (cid:20)Z δ { τ >s + p − δ } e R s V ( X p − δ + k ,t + p − δ + k )d k ℓ ( X p − δ + s , t + p − δ + s )d s (cid:21) . Then, using the quasi-continuity property of X , we obtainlim δ → δ ( ξ ( p ) − ξ ( p − δ )) = − E x [ V ( X p , t + p ) ϕ ( X p , t + p ) { τ >p } ] − E x [ { τ ≥ p } ℓ ( X p , t + p )]= − E x [ V ( X p , t + p ) ϕ ( X p , t + p ) { τ >p } ] − E x [ { τ >p } ℓ ( X p , t + p )] := − ζ ( p ) , where the last line follows since P x ( τ = p ) = 0. Hence the left derivative of ξ exists and given by ζ which is continuous. Therefore, ξ is a C function. Now using the fundamental theorem of calculuswe obtain ϕ ( x, t ) = ξ ( T − t ) + Z T − t ζ ( s ) d s = E x [ ϕ ( X T − t ) { T − t< τ } ] + E s (cid:20)Z T − t ( V ( X s , t + s ) ϕ ( X s , t + s ) + ℓ ( X s , t + s )) { s< τ } d s (cid:21) . Thus ϕ solves (4.4). (cid:3) Next we get a parabolic comparison principle. Let q : ¯ D × [0 , ∞ ) → [0 , ∞ ) be a continuousfunction, C in its second variable and q s : ¯ D × [0 , ∞ ) → R is also continuous. Also, assume that q ( x,
0) = 0 and s q ( x, s ) s is decreasing . Lemma 4.3.
Let u, v be two positive solutions of ( ∂ t − Ψ(–∆)) w + q ( x, w ) = 0 in D × [0 , T ) , w = 0 in D c × [0 , T ] . If u ( x, T ) ≤ v ( x, T ) in R d , then we also have u ≤ v in R d × [0 , T ] .Proof. Let G ( x, t ) = q ( x,u ( x,t )) u ( x,t ) and H ( x, t ) = q ( x,v ( x,t )) v ( x,t ) . Then using Lemma 4.2 we obtain u ( x, t ) = E x h e R T − t G ( X s ,t + s )d s u ( X T − t , T ) { T − t< τ } i , ( x, t ) ∈ D × [0 , T ] , (4.6) v ( x, t ) = E x h e R T − t H ( X s ,t + s )d s v ( X T − t , T ) { T − t< τ } i , ( x, t ) ∈ D × [0 , T ] . (4.7)Note that without loss of generality we may assume u ( · , T ) (cid:13)
0, otherwise from above we get u = 0and then, there is nothing to prove. Let K = max ¯ D × [0 ,T ] ( | G | + | H | ). Then it is evident from abovethat v ( x, t ) ≥ e − KT u ( x, t ) for all x, t. (4.8)Define β = sup { t : tu ≤ v in D × [0 , T ] } . Using (4.8) we get that β ≥ e − KT . To complete the proof we need to show that β ≥
1. Suppose,on the contrary, that β <
1. Denote by u = βu . Then for w = v − u ≥ w ( x, t ) = E x (cid:2) w ( X T − t , T ) { T − t< τ } (cid:3) + E x "Z ( T − t ) ∧ τ (cid:16) q ( X s , v ( X s , t + s )) − βq ( X s , u ( X s , t + s )) (cid:17) d s . For any δ ∈ (0 , T − t ), we can repeat the calculation of (4.5) with V = 0 to arrive at w ( x, t ) = E x (cid:2) { δ< τ } w ( X δ , t + δ ) (cid:3) + E x (cid:20)Z δ { s< τ } (cid:16) q ( X s , v ( X s , t + s )) − βq ( X s , u ( X s , t + s )) (cid:17) d s (cid:21) . By our assumption on q , q ( x, v ) − βq ( x, u ) ≥ q ( x, v ) − q ( x, βu ) ≥ − M w , for some constant M .Thus defining ξ ( s ) = E x [ { τ >s } w ( X s , t + s )] we obtain ξ ( δ ) ≤ ξ (0) + M Z δ ξ ( s )d s. Applying Gronwall’s inequality we then have ξ ( T − t ) ≤ Cw ( x, t ) , for some constant C , independent of ( x, t ) ∈ D × [0 , T ]. Since w ( x, T ) (cid:13)
0, we must have w ( x, t ) > t < T . Furthermore, w ( x, T ) ≥ (1 − β ) u ( x, T ), implying Cw ( x, t ) ≥ ξ ( T − t ) ≥ (1 − β ) E x [ { T − t< τ } u ( X s , T )] , which combined with (4.6) gives κu ( x, t ) ≤ w ( x, t ) for ( x, t ) ∈ D × [0 , T ] and for some κ >
0. Thiscertainly contradicts the definition of β . Hence β ≥
1, completing the proof. (cid:3)
Next we establish a regularity property in space up to the boundary.
Lemma 4.4.
Suppose that g, ℓ be such that k g k L ∞ , k ℓ k ∞ ≤ K . Then for any u satisfying u ( x, t ) = E x [ g ( X T − t ) { T − t< τ } ] + E x (cid:20)Z T − t ℓ ( X s , t + s ) { s< τ } d s (cid:21) , ( x, t ) ∈ D × [0 , T ] , we have, for t < T | u ( x, t ) − u ( y, t ) | ≤ C V ( | x − y | ) , x, y ∈ D, for a constant C dependent on t, T, K where V is the potential measure introduced in Section .We can also choose the constant C uniformly in t varying in a compact subset of [0 , T ) .Proof. Denote by R ( x ) = E x [ g ( X T − t ) { T − t< τ } ] = Z D g ( y ) p D ( T − t, x, z ) d z, R ( x ) = E x (cid:20)Z T − t ℓ ( X s , t + s ) { s< τ } d s (cid:21) = Z T − t ℓ ( x, t + s ) p D ( s, x, z ) d z, where p d ( t, x, z ) denotes the transition density of the killed process X D (see (2.2)). Then from thearguments of [38, Proposition 3.5] (see (3.13) of that paper) one can find a constant C , independentof t, T , satisfying | R ( x ) − R ( y ) | ≤ C V ( | x − y | ) x, y ∈ D . (4.9)To calculate R we recall the following result from [45, Theorem 1.1] and [33, Theorem 1.3] (seealso [22],[38, Theorem 3.1]) |∇ x p D ( t, x, y ) | ≤ C (cid:18) δ D ( x ) ∧ ∨ V − ( √ t ) (cid:19) p D ( t, x, y ) x, y ∈ D, (4.10) p D ( t, x, y ) ≤ C (cid:18) ∧ V ( δ D ( x )) √ t (cid:19) (cid:18) ∧ V ( δ D ( y )) √ t (cid:19) p ( t, | x − y | ) x, y ∈ D , (4.11)for t ∈ (0 , T ] and some constants C , C , dependent on T , where p denotes the transition densityof X . Using (A1) and (2.4) we also have, for any κ >
0, that C − (cid:18) Rr (cid:19) κ ≤ V ( R ) V ( r ) ≤ C (cid:18) Rr (cid:19) κ < r ≤ R ≤ κ , (4.12) ONLOCAL LOGISTIC EQUATION WITH HARVESTING 21 where C depends on κ . Take x, y ∈ D . Suppose 2 | x − y | ≤ max { δ D ( x ) , δ D ( y ) } . With no loss ofgenerality we may assume that y ∈ B ( x, δ D ( x )). Note that for any point z on the line joining x and y we get from (4.10)-(4.11) | x − y ||∇ x p D ( T − t, z, y ) | ≤ C | x − y | (cid:18) δ D ( z ) ∧ ∨ V − ( √ T − t ) (cid:19) p D ( T − t, z, y ) ≤ C | x − y | (cid:18) δ D ( z ) ∧ ∨ V − ( √ T − t ) (cid:19) · C (cid:18) ∧ V ( δ D ( z )) √ T − t (cid:19) (cid:18) ∧ V ( δ D ( y )) √ T − t (cid:19) p ( T − t, | z − y | ) ≤ C | x − y | δ D ( z ) V ( δ D ( z )) , for some C >
0. Since | x − y | ≤ δ D ( x ) ≤ δ D ( z ), using (4.12) with κ = diam( D ) we then obtain | x − y | V ( | x − y | ) |∇ x p D ( T − t, z, y ) ≤ C | x − y | δ D ( z ) V ( δ D ( z )) V ( | x − y | ) ≤ C C (cid:18) | x − y | δ D ( z ) (cid:19) − κ ≤ C C . Thus | R ( x ) − R ( y ) | ≤ Z D | g ( y ) || p D ( T − t, x, z ) − p D ( T − t, y, z ) | d z ≤ C C V ( | x − y | ) k g k L ∞ | D | . (4.13)Now we consider the situation 2 | x − y | ≥ max { δ D ( x ) , δ D ( y ) } . Then using (4.11)-(4.12) | R ( x ) − R ( y ) | ≤ C ( V ( δ D ( x )) + V ( δ D ( y ))) ≤ C ( V (2 | x − y | ) + V (2 | x − y | )) ≤ C V ( | x − y | ) , (4.14)for some constants C , C . Combining (4.9), (4.13) and (4.14) we get the result. (cid:3) Now we are ready to prove an existence result.
Lemma 4.5.
Let q be same as in Lemma . Also, assume that q ( x, · ) is C , uniformly withrespect to x . Let ℓ : ¯ D → ( −∞ , , ℓ : ¯ D → [0 , ∞ ) be be two continuous functions. Let v i , i = 1 , , be a non-negative solution satisfying ( ∂ t − Ψ(–∆)) v i + q ( x, v i ) + ℓ i ( x ) = 0 in D × [0 , T ) , v ( x, t ) = 0 in D c × [0 , T ] , and let g be such that v ( x, T ) ≤ g ( x ) ≤ v ( x, T ) . Then there exists a unique solution u ( v ≤ u ≤ v ) to ( ∂ t − Ψ(–∆)) u + q ( x, u ) = 0 in D × [0 , T ) ,u ( x, T ) = g ( x ) and u ( x, t ) = 0 in D c × [0 , T ] . (4.15) Proof.
The idea is similar to the elliptic case where we use monotone iteration method. Let m beLipschitz constant of s q ( x, s ) in [0 , k v k ], that is, | q ( x, s ) − q ( x, s ) | ≤ m | s − s | for s , s ∈ [0 , k v k ] , x ∈ ¯ D. Let F ( x, s ) = q ( x, s ) + ms and u = v . Define u to be the solution of( ∂ t − Ψ(–∆)) u − mu + F ( x, u ) = 0 in D × [0 , T ) ,u ( x, T ) = g ( x ) and u ( x, t ) = 0 in D c × [0 , T ] . By Lemma 4.2 we then have u ( x, t ) = E x h e − m ( T − t ) g ( X T − t ) { T − t< τ } i + E x (cid:20)Z T − t e − ms F ( X s , u ( X s , t + s )) { s< τ } d s (cid:21) . (4.16) Another use of Lemma 4.2 gives v i ( x, t ) = E x h e − m ( T − t ) v i ( X T − t , T ) { T − t< τ } i + E x (cid:20)Z T − t e − ms ˜ F i ( X s , v i ( X s , t + s )) { s< τ } d s (cid:21) , (4.17)where ˜ F i ( x, s ) = F ( x, s ) + ℓ i ( x ). Since F is non-decreasing in s , we have F ( x, v ) + ℓ ( x ) ≤ F ( x, v ) ≤ F ( x, v ) + ℓ ( x ) . Therefore, comparing (4.16) and (4.17) we have v ≤ u ≤ u = v in R d × [0 , T ]. Now we find aniterative sequence of solutions as follows: u k +1 is a solution to( ∂ t − Ψ(–∆)) u − mu + F ( x, u k ) = 0 in D × [0 , T ) , u = 0 in D c × [0 , T ] , u ( x, T ) = g ( x ) . In other words, u k +1 ( x, t ) = E x h e − m ( T − t ) g ( X T − t ) { T − t< τ } i + E x (cid:20)Z T − t e − ms F ( X s , u k ( X s , t + s )) { s< τ } d s (cid:21) . (4.18)The above argument shows that v ≤ u k +1 ≤ u k ≤ · · · ≤ v in R d × [0 , T ] . Furthermore, applying Lemma 4.4, we see that lim k →∞ u k ( · , t ) = u ( · , t ) uniformly in x , for each t ∈ [0 , T ]. Thus, using dominated convergence theorem, we can pass to the limit in (4.18) to obtain u ( x, t ) = E x h e − m ( T − t ) g ( X T − t ) { T − t< τ } i + E x (cid:20)Z T − t e − ms F ( X s , u ( X s , t + s )) { s< τ } d s (cid:21) . (4.19)From (4.19) it is easy to show that u is continuous in R d × [0 , T ] (cf. [9, Lemma 3.1]). Indeed,since x u ( x, t ) is continuous uniformly for t in compact subsets of [0 , T ) and t p D ( t, x, y ) iscontinuous in (0 , ∞ ), ( x, t ) u ( x, t ) is continuous in [0 , T ) × R d . To examine the continuity at T consider a sequence ( x n , t n ) → ( x, T ). Note that the second term in the above display goes to 0.Again, if x ∈ ∂D then E x n [ | g ( X T − t n ) | { T − t n < τ } ] ≤ E x n [ | g ( X T − t n ) | ] → , as n → ∞ , we get u ( x n , t n ) →
0. Also, if x ∈ D , since p D ( T − t n , x n , y )d y → δ x , we get u ( x n , t n ) → g ( x ). This gives continuity. Applying Lemma 4.2 we see that u is a solution to (4.15).Uniqueness of solution follows from Lemma 4.3. This completes the proof. (cid:3) Next we prove a sharp boundary behaviour for the solution of the parabolic equation.
Lemma 4.6.
Consider q from Lemma . Let u be a bounded solution of ( ∂ t − Ψ(–∆)) u + q ( x, u ) = 0 in D × [0 , T ) , u = 0 in D c × [0 , T ] , where u ( x, T ) (cid:13) . Then for every t < T there exists a constant C , dependent on t, T and u | R d × [ t,T ] ,satisfying C V ( δ D ( x )) ≤ u ( x, t ) ≤ C V ( δ D ( x )) x ∈ D .
Proof.
Denote by H ( x, t ) = q ( u ( x, t )) u ( x, t ) . Then H is a bounded, continuous function. Using Lemma 4.2 we then have u ( x, t ) = E x h e R T − t H ( X s ,t + s ) d s u ( X T − t , T ) { T − t< τ } i . Thus, for some constant C , get e − C T E x (cid:2) u ( X T − t , T ) { T − t< τ } (cid:3) ≤ u ( x, t ) ≤ e C T E x (cid:2) u ( X T − t , T ) { T − t< τ } (cid:3) . (4.20) ONLOCAL LOGISTIC EQUATION WITH HARVESTING 23
Using (4.11) and (4.20) we obtain u ( x, t ) ≤ C V ( δ D ( x )) , which gives the upper bound. Now from [13, Theorem 4.5] we know that p D ( t, x, y ) ≥ κ P x ( τ > t/ P y ( τ > t/ p ( t ∧ V ( r ) , | x − y | )and P x ( τ > t/ ≥ κ V ( δ D ( x )) p t ∧ V ( r ) ∧ ! , where D satisfies the inner and outer ball condition with radius r . Now let K ⋐ D be such thatmin K u ( x, T ) ≥ κ >
0. Using the lower bound in (4.20) and estimates above we then find u ( x, t ) ≥ e − C T Z D u ( y, T ) p D ( T − t, x, y )d y ≥ C κ P x ( τ > ( T − t ) / Z K u ( y, T ) p (( T − t ) ∧ V ( r ) , | x − y | ) P y ( τ > ( T − t ) / y ≥ C κ κ V ( δ D ( x )) p (( T − t ) ∧ V ( r ) , diam( D )) Z K P y ( τ > ( T − t ) / dy ≥ C − V ( δ D ( x )) , for some constants C , C , C . This gives the lower bound. Hence the proof. (cid:3) Now we are ready to prove Theorem 1.3. Recall that given an interval [0 , T ], u T solves( ∂ t − Ψ(–∆)) u T + au T − f ( x, u T ) = 0 in D × [0 , T ) ,u T ( x, T ) = u ( x ) and u T ( x, t ) = 0 in D c × [0 , T ] , (4.21)where 0 (cid:12) u ∈ C ( D ). Proof of Theorem 1.3.
First consider (i). We divide the proof in two steps.
Step 1.
First we note that u T ( x, T −
1) = E x (cid:2) { < τ } u ( X ) (cid:3) + E x (cid:20)Z { s< τ } F ( X s , u T ( X s , T − s )) d s (cid:21) , where F ( x, s ) = as − f ( x, s ). Thus u T ( x, T −
1) is independent of T (by Lemma 4.3). In fact, it issame as v ( x,
0) where v solves (4.21) in [0 , δ = T − − t ) we note thatfor t ≤ T − u T ( x, t ) = E x (cid:2) { T − − t< τ } u T ( X T − − t , T − (cid:3) + E x (cid:20)Z T − − t { s< τ } F ( X s , u T ( X s , t + s ))d s (cid:21) . Thus without any less of generality we may assume u = u T ( x, T − C − V ( δ D ( x )) ≤ u ( x ) ≤ C V ( δ D ( x )) , for x ∈ D. (4.22) Step 2.
Let v a be the unique positive solution (see Theorem 1.1) to − Ψ(–∆) v a + av a − f ( x, v a ) = 0 in D, v a = 0 in D c , v a > D . (4.23)Using (4.22), Theorems 2.1 and 2.2, we choose κ > ϕ ( x ) := κ − v a ( x ) ≤ u ( x ) ≤ κv a ( x ) := ˆ ϕ ( x ) , x ∈ D .
Note that ˘ ϕ is subsolution to (4.23) and ˆ ϕ is a supersolution to (4.23). Setting ˆ ϕ as the terminalcondition at time T we construct a solution ˆ w T in [0 , T ] with ˆ w T ≤ ˆ ϕ . This can be done using Lemma 4.5. Next we observe that ˆ w is increases with t . For instance, take t ≤ t ≤ T with T − t = t − t . Observe that ξ ( x, t ) = ˆ w T ( x, t − t + t ) is a solution to( ∂ t − Ψ(–∆)) u + au − f ( x, u ) = 0 in D × [ t , T ) ,u ( x, T ) = u T ( x, t ) and u ( x, t ) = 0 in D c × [ t , T ] . Using the uniqueness of solutions and comparison principle (Lemma 4.3) we see that ˆ w T ( x, t ) ≤ ˆ w T ( x, t ). For any pair t ≤ t ≤ T the same comparison holds due to continuity with respect to t and a density argument. Another application of Lemma 4.3 gives that u T ( x, ≤ ˆ w T ( x, ≤ ˆ ϕ ( x ).Now apply Lemma 4.4 to invoke equi-continuity and show that ˆ w T ( x, → ˆ w as T → ∞ . Thenpassing limit inˆ w T ( x,
0) = E x (cid:2) { T < τ } ˆ ϕ ( X T ) (cid:3) + E x (cid:20)Z T { s< τ } F ( X s , ˆ w T ( X s , s )) ds (cid:21) , as T → ∞ , we obtain ˆ w ( x ) = E x (cid:20)Z τ F ( X s , ˆ w ( X s , s )) ds (cid:21) = G F ( · , ˆ w )( x ) . This, in particular, implies − Ψ(–∆) ˆ w + a ˆ w − f ( ˆ w ) = 0 in Ω , ˆ w = 0 in Ω c . From uniqueness we must have ˆ w = v a .Follow a similar argument to construct a sequence of solution ˘ w (decreasing in t ) satisfying˘ ϕ ≤ ˘ w T ( x, ≤ u T ( x, . Argument similar to above shows thatlim T →∞ sup D | ˘ w T ( x, − v a | = 0 . Combining these two observations we complete the proof of (i).(ii) Proof is similar to (i). For a ≤ λ , we take ϕ as the super-solution to (4.23). Then repeatinga same argument we can conclude the proof. (cid:3) Acknowledgments
This research of Anup Biswas was supported in part by DST-SERB grants EMR/2016/004810and MTR/2018/000028. Mitesh Modasiya is partially supported by CSIR PhD fellowship (File no.09/936(0200)/2018-EMR-I).
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