Strong Convergence of Solutions and Attractors for Reaction-Diffusion Equations Governed by a Fractional Laplacian

Strong convergence of solutions and attractors for reaction-diffusion equations governed by a fractional Laplacian Jiaohui Xu Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi’an 710127, P. R. China Tom´ as Caraballo1 2 Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, P. R. China Jos´e Valero Centro de Investigaci´on Operativa, Universidad Miguel Hern´andez de Elche, Avenida de la Universidad s/n, 03202-Elche, Spain Abstract A nonlocal reaction-diffusion equation governed by a fractional Laplace operator on a bounded domain is studied in this paper. First, the strong convergence of solutions of the equations governed by fractional Laplacian to the solutions of the classical equations governed by a standard Laplace operator is proved, when the fractional parameter grows to 1. Second, for the autonomous case, the upper semicontinuity of global attractors with respect to the attractors of the limit problem is established. Apparently, these are the first results for this kind of problems on bounded domains. Keywords: Fractional Laplacian; Strong convergence of solutions; Global attractors. AMS subject classifications. 35R11, 35A15, 35B41, 35K65 1. Introduction In this paper, we study the problem        ∂u ∂t + (−∆)γu=f(u) + h(t),in O× (τ, ∞), u= 0,on ∂O, u(x, τ) = uτ(x),in O, (1) where (−∆)γ,γ∈(0,1), stands for the fractional Laplace operator (fractional Laplacian), Ois a bounded subset of Rmwith sufficiently smooth boundary, τ∈R,h∈L2 loc(R;L2(O)) and fis a continuous function satisfying some appropriate assumptions as specified later. First of all, we need to decide which definition of the fractional Laplacian is more appropriate for our analysis. It is well known that, in the unbounded domain case, namely in Rm, there are several equivalent definitions of fractional Laplacian (see, for instance, [13] for ten equivalent definitions), but in the bounded domain case, there exist several definitions which are not equivalent. We refer the reader to the papers [6, 16, 19] for more details. We will not discuss about them in this paper, but will simply choose one of those definitions for our investigation, more precisely the so called “regional”definition (see [6]), in such a way that we are able to prove the convergence of solutions to (1) to the corresponding ones of the limiting equation with standard Laplacian when γ→1−, ensuring this convergence holds in the strong sense, what somehow justifies the suitability and accuracy of the chosen definition of fractional Laplacian. 1Corresponding author 2E-mail addresses: [email protected] (J. Xu), [email protected] (T. Caraballo), jv[email protected] (J. Valero). 1 2 The properties of solutions and attractors to problem (1) when γis fixed have been studied widely in the literature. For example, the well-posedness and the existence of global attractors (in both deterministic and random setting, with bounded or unbounded domains) of problem (1) have been extensively investigated in the literature (see, e.g., [8, 17, 27, 28, 29, 30, 31] and the references therein. Additionally, for a stochastic version of (1), the upper semicontinuity of the attractors was established in [27, 28, 29] when a parameter in the noise term varies. It is remarkable that the definition of fractional Laplacian used in these stochastic papers is exactly the “regional”one we will use in our current analysis, what reinforces the idea that the definition we have chosen is appropriate. Our aim in this manuscript is to analyze the behavior of solutions to problem (1) when the parameter γgoes to 1−, and the motivation is based on the properties of fractional Laplace operator (−∆)γas γ→1−, (see, for example, [8] and [21]). In [21, Proposition 4.4], the authors proved that (−∆)γu(x) converges to −∆u(x) as γ→1 for all x∈Rmwhen u∈C∞ 0(Rm). In [8, Proposition 2.3], the authors proved that, for every u∈C∞ 0(Rm) and v∈W1,2 0(O), it holds lim γ→1−ZO v(−∆)γudx =−ZO v∆udx. Notice that in the papers [3], [4] this question was studied for an abstract parabolic equation and the Schr¨odinger equation. However, as far as we are aware, there are no results in the literature concerning the convergence of solutions of (1) to the solutions of the following limit problem,        ∂u ∂t −∆u=f(u) + h(t),in O× (τ, ∞), u= 0,on ∂O, u(x, τ) = uτ(x),in O, (2) as the parameter γgoes to 1−. In [33], we completed a first step by proving the convergence of solutions with respect to the weak topology in L2(Rm) in the case that the external function fis sublinear. In this manuscript, we will go much more further to study the convergence of solutions in the strong topology sense for more general functions f. Moreover, the upper semicontinuity of the global attractors will be established as well. The interest in studying problem (1) and its stationary version comes from different fields as has been described in our paper [33] (see also [1, 2, 7, 9, 10, 11, 12, 18, 20, 23, 24, 25, 26, 32, 34, 35] and the references therein). This paper is organized as follows. In Section 2, we recall the basic definitions and properties concerning the fractional Laplace operators, establish the setting of the problem we are tackling and impose the conditions that ensure the existence and uniqueness of weak solutions to problem (1). In Section 3, we improve first the known results about the convergence of the operator (−∆)γ by showing that limγ→1−(−∆)γu=−∆ustrongly in Lp((τ, τ +T)×Rm),for any p≥1 and u∈C∞ 0((τ, τ +T)×Rm), where τ∈R,T > 0. Also, limγ→1−(−∆)γu=−∆ustrongly in Lp(Rm),for any p≥1 and u∈C∞ 0(Rm). After that, we prove the main result of the paper stating that the solutions of problem (1) converge to the ones of the limit problem (2) in C([τ, τ +T]; L2(O)), for any T > 0 as γ→1−. Finally, in Section 4, we study the upper semicontinuity of the global attractors as the parameter γgoes to 1−in the autonomous situation, that is, when the function h belongs to L2(O). 3 2. Setting of the problem We consider the following fractional reaction-diffusion equation,        ∂u ∂t + (−∆)γu=f(u) + h(t),in O× (τ, ∞), u= 0,on ∂O, u(x, τ) = uτ(x),in O, (3) where (−∆)γ,γ∈(0,1), stands for the fractional Laplace operator, Ois a smooth bounded subset of Rm,τ∈Rand h∈L2 loc(R;L2(O)). Throughout this paper, we assume that f∈C(R) and there exist positive constants Cf,κ,β1,β2and p≥2, such that (f(s)−f(r))(s−r)≤Cf(s−r)2,∀s, r ∈R,(4) −κ−β1|s|p≤f(s)s≤κ−β2|s|p,∀s∈R.(5) From (5), we deduce that there exists a constant β3>0 such that |f(s)| ≤ β3(|s|p−1+ 1),∀s∈R.(6) We fix some µ∈(0, β2) and define the function f(u) = f(u) + µu. Then, the equation in (3) reads as ∂u ∂t + (−∆)γu+µu =f(u) + h(t).(7) It follows from (5), (6) and the Young inequality that f(s)s≤κ−β2|s|p+µs2≤κ−β2|s|p,(8) |f(s)| ≤ β3(|s|p−1+ 1),(9) for some positive constants κ, β2, β3. Let Sdenote the Schwartz space of rapidly decaying C∞functions on Rm. The fractional Laplace operator (−∆)γof u∈ S at point x, for 0 < γ < 1, is defined by, (−∆)γu(x) = −1 2C(m, γ)ZRm u(x+y) + u(x−y)−2u(x) |y|m+2γdy, x ∈Rm,(10) where C(m, γ) is the following positive constant C(m, γ) = γ4γΓ(m+2γ 2) πm 2Γ(1 −γ).(11) It is well known [21] that for any u∈ S, we have (−∆)γu=F−1|ξ|2γFu, where Fis the Fourier transform given by Fu=1 (2π)m/2ZRm e−ix·ξu(x)dx, and F−1is the inverse Fourier transform. For 0 < γ < 1, we define the fractional Sobolev space Wγ,2(Rm) := Hγ(Rm) by: Hγ(Rm) = u∈L2(Rm) : ZRmZRm |u(x)−u(y)|2 |x−y|m+2γdxdy < ∞, endowed with the norm kukHγ(Rm)=ZRm |u(x)|2dx +ZRmZRm |u(x)−u(y)|2 |x−y|m+2γdxdy1 2 . 4 In what follows, we denote by k · kpthe norm in Lp(Rm) for p > 2,whereas by k · k and (·,·), we denote the norm and the inner product of L2(Rm), respectively. By abusing of the notation, (·,·) will be also used for the pairing between Lq(Rm) and Lp(Rm). Moreover, we will consider the Gagliardo semi-norm of Hγ(Rm),denoted by k·k ˙ Hγ(Rm),which is given by, kuk2˙ Hγ(Rm)=ZRmZRm |u(x)−u(y)|2 |x−y|m+2γdxdy, u ∈Hγ(Rm). Therefore, kuk2 Hγ(Rm)=kuk2+kuk2˙ Hγ(Rm)for all u∈Hγ(Rm). Notice that Hγ(Rm) is a Hilbert space with inner product, (u, v)Hγ(Rm)=ZRm u(x)v(x)dx +ZRmZRm (u(x)−u(y))(v(x)−v(y)) |x−y|m+2γdxdy, for u, v ∈Hγ(Rm).Thanks to [21], the norm kukHγ(Rm)is equivalent to kuk2+k(−∆)γ 2uk21 2for u∈Hγ(Rm). Namely, it follows that kuk2 Hγ(Rm)=kuk2+2 C(m, γ)k(−∆)γ 2uk2,∀u∈Hγ(Rm). Since the fractional Laplace operator (−∆)γgiven by (10) is nonlocal, we here interpret the homogeneous Dirichlet boundary problem (3) as follows,        ∂u ∂t + (−∆)γu=f(u) + h(t),in O× (τ, ∞), u= 0,on Rm\O, u(x, τ) = uτ(x),in O. (12) Naturally, the limit problem (2) will be written as        ∂u ∂t −∆u=f(u) + h(t),in O× (τ, ∞), u= 0,on Rm\O, u(x, τ) = uτ(x),in O, (13) so that they keep formally consistent. Thus, we define the spaces Vγ={u∈Hγ(Rm) : u= 0 a.e. on Rm\O}, H={u∈L2(Rm) : u= 0 a.e. on Rm\O}, V={u∈H1(Rm) : u= 0 a.e. on Rm\O}, with their dual spaces denoted by V∗ γ,H∗and V∗, respectively. Moreover, we will use kuk2 Vγ=kuk2+k(−∆)γ 2uk2, to denote the norm of space Vγ. The norm in Vwill be denoted by k·kV. Furthermore, let b:Vγ×Vγ→Rdenote a bilinear form given by, bγ(v1, v2) = µ(v1, v2) + 1 2C(m, γ)ZRmZRm (v1(x)−v1(y))(v2(x)−v2(y)) |x−y|m+2γdxdy, v1, v2∈Vγ,(14) where C(m, γ) is the same as in (11). For convenience, we associate two operators Aγ,Aγ:Vγ→V∗ γ with b, such that < Aγ(v1), v2>(V∗ γ,Vγ)=µ(v1, v2)+ < Aγ(v1), v2>(V∗ γ,Vγ)=bγ(v1, v2),(15) 5 for all v1, v2∈Vγ,where <·,·>(V∗ γ,Vγ)is the duality pairing of V∗ γand Vγ. It is known [21] that the embeddings V⊂Vγ2⊂Vγ1are continuous for 0 < γ1≤γ2<1. Identifying Hwith its dual space, we obtain the chain of continuous embeddings V⊂Vγ⊂H⊂V∗ γ⊂V∗,∀γ∈(0,1) . Pairing between spaces Vand V∗will be denoted by <·,·>(V∗,V ). We consider without loss of generality that h∈L2 loc(R;H) by setting h(t, x) = 0 for x∈Rm\O. Definition 1. Let τ∈R,uτ∈Hand γ∈(0,1). The function u∈C([τ, ∞); H)is said to be a weak solution to problem (12) if u(τ) = uτ,u∈L2 loc(τ, ∞;Vγ)∩Lp loc(τ, ∞;Lp(Rm)),du dt ∈ L2 loc(τ, ∞;V∗ γ) + Lq loc(τ, ∞;Lq(Rm)) and d dt (u, ξ) + 1 2C(m, γ)ZRmZRm (u(t, x)−u(t, y))(ξ(x)−ξ(y)) |x−y|m+2γdxdy =ZO (f(u(t, x)) + h(t, x)) ξ(x)dx, for any ξ∈Vγ∩Lp(Rm), in the sense of scalar distributions in (τ, ∞). In [27, Theorem 2.3], one can see that problem (12) possesses a unique weak solution for any uτ∈Hand γ∈(0,1). In addition, this solution is continuous with respect to the initial datum uτ in H, and satisfies the energy equality, d dt kuk2+C(m, γ)kuk2˙ Hγ(Rm)= 2 ZO (f(u(t, x)) + h(t, x)) u(t, x)dx, for a.a. t≥τ. Definition 2. Let τ∈Rand uτ∈H. The function u∈C([τ, ∞); H)is said to be a weak solution to problem (13) if u(τ) = uτ,u∈L2 loc(τ, ∞;V)∩Lp loc(τ, ∞;Lp(O)),du dt ∈L2 loc(τ, ∞;V∗) + Lq loc(τ, ∞;Lq(O)) and d dt (u, ξ) + ZO ∇u·∇ξdx =ZO (f(u(t, x)) + h(t, x)) ξ(x)dx, for any ξ∈V∩Lp(O), in the sense of scalar distributions in (τ, ∞). Recall that (see, e.g., [5] or [22]) problem (2) possesses a unique weak solution for any uτ∈H, which is continuous with respect to the initial datum uτin H. 3. Convergence of solutions Our aim now is to prove that the solutions of problem (12) converge, as γ→1−, to the unique solution of the limit problem with γ= 1, that is, to the unique solution of the standard reactiondiffusion equation (13). It is well known [21, Proposition 4.4] that if u∈C∞ 0(Rm),then lim γ→1−(−∆)γu(x) = −∆u(x),for all x∈Rm. (16) Let C∞ 0(X) be the space of infinitely differentiable functions u:X→Rwith compact support, where Xis a Banach space. Denote by D0the space of distibutions of D=C∞ 0((τ, τ +T)×Rm) and by <·,·>(D0,D)the duality pairing of D0and D. Denote by BRthe ball in Rmcentered at 0 with radius R. Now, we recall a result proved in [33] which extends the convergence result (16) to a more general one. 6 Lemma 3. (See [33, Lemma 3.1]) For any τ∈R,T > 0and u∈C∞ 0((τ, τ +T)×Rm), the following statement holds, lim γ→1−(−∆)γu=−∆u, strongly in Lp((τ, τ +T)×Rm),∀p≥1. In particular, limγ→1−(−∆)γu=−∆uin the sense of distributions in D0. Now, in a similar way we state the following result. Lemma 4. (See [33, Lemma 3.2]) For any u∈C∞ 0(Rm),limγ→1−(−∆)γu=−∆ustrongly in Lp(Rm)for any p≥1. We also need the following technical lemma which is crucial for our analysis. Lemma 5. (i) For any γ∈1 2,1and v∈Vγ, we have kvkV1 2 ≤r3 2kvkV γ.(17) (ii) There exists a positive constant Ksuch that, for any v∈Vγ,  Aγv V∗≤KkvkV γ, for any γ∈(0,1) .(18) Proof. (i) Since 1 2<1 2γ<1,0<1−1 2γ<1 2. By the Fourier transform and Young inequality, the estimate (17) follows from kvk2 V1 2 =kvk2+k(−∆)1 4vk2=ZRm (1 + |ξ|)|Fv(ξ)|2dξ ≤ZRm2−1 2γ+1 2γ|ξ|2γ|Fv(ξ)|2dξ ≤3 2ZRm1 + |ξ|2γ|Fv(ξ)|2dξ =3 2kvk2 Vγ. (i) For the second statement, let us first prove the existence of positive constant Ksuch that   (−∆)γ 2u  ≤KkukV,∀u∈V. (19) On the one hand, by Fubini’s Theorem, we have ZRmZ|x−y|<1 |u(x)−u(y)|2 |x−y|m+2γdxdy ≤ZRmZB1 |u(y+z)−u(y)|2 |z|2|z|m+2(γ−1) dzdy ≤ZRmZB1 1 |z|m+2(γ−1) Z1 0 |∇u(y+tz)|dt2 dzdy ≤ZB1 1 |z|m+2(γ−1) Z1 0ZRm |∇u(y+tz)|2dydtdz ≤ kuk2 VZB1 1 |z|m+2(γ−1) dz ≤ kuk2 Vωm−1Z1 0 1 ρ2γ−1dρ =ωm−1 2 (1 −γ)kuk2 V. 7 On the other hand, the change of variable implies ZRmZ|x−y|≥1 |u(x)−u(y)|2 |x−y|m+2γdxdy ≤4ZRmZ|x−y|≥1 |u(y)|2 |x−y|m+2γdxdy = 4 kuk2Z|z|≥1 1 |z|m+2γdz ≤4kuk2ωm−1Z∞ 1 1 ρ2γ+1 dρ =2ωm−1 γkuk2 V. Thus, combining the above two estimates, we obtain   (−∆)γ 2u   2=C(m, γ) 2kuk2˙ Hγ(Rm) =C(m, γ) 2ZRmZRm |u(x)−u(y)|2 |x−y|m+2γdxdy ≤ωm−1C(m, γ) 4 (1 −γ)+C(m, γ) γkuk2 V ≤K2kuk2 V, for some constants K > 0, where the last inequality follows from the fact lim γ→0+ C(m, γ) γ= lim γ→0+ C(m, γ) γ(1 −γ)=2 ωm−1 , see [21, Corollary 4.2] for more details. Immediately, by the Young inequality and (19), we find  Aγv V∗= sup u∈V, kukV≤1< Aγ(v), u >(V∗,V ) = sup u∈V, kukV≤1< Aγ(v), u >(V∗ γ,Vγ) = sup u∈V, kukV≤1 1 2C(m, γ)ZRmZRm (v(x)−v(y))(u(x)−u(y)) |x−y|m+2γdxdy ≤sup u∈V, kukV≤1 1 2C(m, γ)kvk˙ Hγ(Rm)kuk˙ Hγ(Rm) = sup u∈V, kukV≤1  (−∆)γ 2v    (−∆)γ 2u   ≤Ksup u∈V, kukV≤1  (−∆)γ 2v  kukV≤K  (−∆)γ 2v  ≤KkvkV γ. We finish the proof of this lemma.  We can now establish the main result of this paper about the convergence of solutions. Theorem 6. Let un τ→uτin H, let un(·)be the solution to problem (12) for γ=γnwith initial value un τ. Then un→uin C([τ, τ +T]; H)as n→ ∞ for any T > 0, where u(·)is the unique solution to problem (13). If un τ→uτweakly in H, then un→uin C([τ+ε, τ +T]; H)as n→ ∞ for any 0< ε < T. 8 Proof. Consider a sequence γn→1−. Let un(·) be the unique solution to problem (12) with initial value un τ. Multiplying equation (7) by un, using the Young inequality and on account of (8), we derive 1 2 d dt kunk2+µkunk2+1 2C(m, γn)kunk2˙ Hγn(Rm) ≤κ−β2kunkp p+kh(t)k kunk ≤κ−β2kunkp p+1 2µkh(t)k2+µ 2kunk2,(20) which is equivalent to d dt kunk2+µkunk2+C(m, γn)kunk2˙ Hγn(Rm)+ 2β2kunkp p≤2κ+1 µkh(t)k2.(21) Multiplying (21) by eµs and integrating it over the interval (τ, t), we deduce kun(t)k2≤ kun τk2e−µ(t−τ)+2κ µ1−e−µ(t−τ)+1 µZt τ e−µ(t−s)kh(s)k2ds. (22) Then for T > 0, we infer that sup t∈[τ,τ+T] kun(t)k2≤MT,(23) where MTdepends on T, and Zτ+T τC(m, γn)kun(s)k2˙ Hγn(Rm)+ 2β2kun(s)kp pds ≤MT.(24) Therefore, the sequence {un}is bounded in L∞(τ, τ +T;H)∩Lp(τ, τ +T;Lp(Rm)). By (23) and (24), we have Zτ+T τ  (−∆)γn 2un(s)   2ds ≤Zτ+T τ C(m, γn)kun(s)k2˙ Hγn(Rm)ds ≤MT,(25) Zτ+T τ kun(s)k2 Vγnds ≤MT+TMT.(26) Hence, in view of (26) and Lemma 5(i), there exists n0>0, such that γn∈[1 2,1) for n≥n0. Then, we have Zτ+T τ kunk2 V1 2 ds ≤3 2MT+TMT, which implies {un}is bounded in L2(τ, τ +T;V1 2). Also, (26) and (18) imply that {Aγn(un)}is bounded in L2(τ, τ +T;V∗). Moreover, by (24) and (9), we infer that {f(un)}is bounded in Lq(τ, τ +T;Lq(Rm)). Therefore, {dun dt }is bounded in Lq(τ, τ +T;Lq(Rm))+L2(τ, τ +T;V∗). Then there exist functions u, χ, ξ such that, up to a subsequence which we relabel the same, we deduce the following convergences, un→uweak-star in L∞(τ, τ +T;H),(27) un→uweakly in L2(τ, τ +T;V1 2),(28) un→uweakly in Lp(τ, τ +T;Lp(Rm)),(29) f(un)→χweakly in Lq(τ, τ +T;Lq(Rm)),(30) Aγn(un)→ξweakly in L2(τ, τ +T;V∗),(31) dun dt →du dt weakly in Lq(τ, τ +T;Lq(Rm)) + L2(τ, τ +T;V∗).(32) 9 Since the embedding V1 2⊂His compact and the embedding H⊂(V∩Lp(Rm))∗is continuous, a standard Compactness Theorem [22, Theorem 8.1] implies that un→ustrongly in L2(τ, τ +T;H),(33) un(t, x)→u(t, x) for a.a. (t, x)∈(τ, τ +T)× O.(34) From the continuity of fand [15, Lemma 1.3], we also obtain that χ=f(u). Further, we need to prove that ξ=−∆u. By Lemma 3, for arbitrary ϕ∈ D, we have < Aγn(un), ϕ >(D0,D)=Zτ+T τ < Aγn(un), ϕ >(V∗,V )dt =Zτ+T τ < un, Aγn(un)>(V,V ∗)dt =Zτ+T τZRm un(−∆)γnϕdxdt γn→1− −→ Zτ+T τZRm u(−∆)ϕdxdt =Zτ+T τ < u, −∆ϕ >(V,V ∗)dt =<−∆u, ϕ >(D0,D), the above convergence holds thanks to the facts that un→uin L2(τ, τ+T;H) and (−∆)γnϕ→ −∆ϕ in L2(τ, τ +T;H). Consequently, Aγn(un)→ −∆uas γn→1−in the distributional sense, which implies that ξ=−∆u. Thus, for any η∈L2(τ, τ +T;V)∩Lp(τ, τ +T;Lp(Rm)), the above convergences yield that 0 = Zτ+T τ <dun dt , η >(V∗+Lq(Rm),V ∩Lp(Rm)) dt +Zτ+T τ < Aγn(un), η >(V∗,V )dt −Zτ+T τZO (f(un) + h)ηdxdt −→ Zτ+T τ <du dt , η >(V∗+Lq(Rm),V ∩Lp(Rm)) dt +Zτ+T τ <−∆u, η >(V∗,V )dt −Zτ+T τZO (f(u) + h)ηdxdt. Moreover, u∈L∞(τ, τ +T;H)∩Lp(τ, τ +T;Lp(Rm)), u∈L2(τ, τ +T;V1 2), du dt ∈Lq(τ, τ +T;Lq(Rm)) + L2(τ, τ +T;V∗), −∆u∈L2(τ, τ +T;V∗). Since −∆u(t)∈V∗, we derive that u(t)∈Vfor a.a. t∈(τ, τ +T). Hence, k∇u(t)k2=<−∆u(t), u(t)>(V∗,V )≤ k∆u(t)kV∗k∇u(t)k, and, consequently, u∈L2(τ, τ +T;V). Therefore, u(·) is the weak solution to problem (13). Since every converging sequence has the same limit, it follows that the convergences (27)-(34) hold for the whole sequence. Also, we deduce from (23) that for tn, t0∈[τ, τ +T] such that tn→t0, we have un(tn)→u(t0) weakly in H. (35)