LONG-TIME BEHAVIOR OF A CLASS OF NONLOCAL PARTIAL DIFFERENTIAL EQUATIONS

. This work is devoted to investigate the well-posedness and long-time behavior of solutions for the following nonlocal nonlinear partial diﬀeren- tial equations in a bounded domain Firstly, due to the lack of an upper growth restriction of the nonlinearity f , we have to utilize a weak compactness approach in an Orlicz space to obtain the well-posedness of weak solutions for the equations. We then establish the existence of ( L 20 (Ω) ,L 20 (Ω))-absorbing sets and ( L 20 (Ω) ,H σ/ 2 0 (Ω))-absorbing sets for the solution semigroup { S ( t ) } t ≥ 0 . Finally, we prove the existence of ( L 20 (Ω) ,L 20 (Ω))-global attractor and ( L 20 (Ω) ,H σ/ 2 0 (Ω))-global attractor by a asymptotic compactness method.


(Communicated by Yuan Lou)
Abstract. This work is devoted to investigate the well-posedness and longtime behavior of solutions for the following nonlocal nonlinear partial differential equations in a bounded domain ut + (−∆) σ/2 u + f (u) = g.
1. Introduction. We consider the existence of global attractors for the following nonlocal nonlinear reaction-diffusion equations: where Ω ⊂ R n with a sufficiently smooth boundary, u 0 ∈ L 2 0 (Ω), g(x) ∈ L 2 0 (Ω), and 0 < σ < 2. We assume that the nonlinearity f satisfies a dissipativity condition for some positive constants C 1 , C 2 . Recently, a lot of interest is devoted to the study of the fractional Laplacian (or nonlocal) operator (−∆) σ/2 , also known as the Laplacian of order σ 2 . It is defined for every function g in the Schwartz class through the Fourier transform: (−∆) σ/2 g(ξ) = |ξ| σ g(ξ).
It can also be represented by a singular integral, where C = 2 σ−1 σΓ((n+σ)/2) π n/2 Γ(1−σ/2) is a normalization constant (see [21]). Furthermore, the fractional Laplacian can further be defined by a σ−harmonic extension which was introduced by Caffarelli and Silvestre [6] in the whole space. This extension is commonly used in the recent literature since it allows nonlocal problems to be written in a local way, which enables the use of variational techniques for these kind of problems (see [2,5,6,22,23,27,31]).
We will consider partial differential equations (PDEs) with the fractional Laplacian operators. It arises in the Fokker-Planck equations (see [10]) for stochastic differential equations with non-Gaussian σ-stable Lévy motion L σ t , for σ ∈ (0, 2). This type of diffusion is nowadays intensively studied both from theoretical and experimental point of views, since it conveniently explains a large number of phenomena in physics, finance, biology, ecology, geophysics, and others. The fractional partial differential equations also appear in the modeling of various complex systems, such as heat transfer processes in fractal and disordered media, and fluid flows and acoustic propagation in porous media (see [19,20,22,23,30]).
Some authors have investigated important properties of fractional PDEs. For example, Ros-Oton and collaborators (see [11,24]) have studied the global regularity of solutions to the fractional elliptic PDEs and heat equations. Lu et. al. (see [18,17]) have obtained the existence of a random attractor for fractional Ginzburg-Landau equation with multiplicative noise. Vázquez et. al. (see [22,23,28]) have investigated the well-posedness and the asymptotic behavior, speed of propagation and many other properties for the fractional porous medium equations.
The main purpose of the present paper is to study the long-time dynamical behavior of solutions for the fractional reaction-diffusion equations (1). We first prove the well-posedness of weak solutions by a Galerkin method. Because of the absence of an upper growth restriction of f , it is impossible to estimate the boundedness of f (u m ) (per Galerkin sequence u m ) to determine its weak limit. In order to overcome this difficulty, we apply the weak compactness theorem in an Orlicz space (see [15]) as it is used in [13]. Then, we examine the existence of absorbing sets in L 2 0 (Ω) and H σ/2 0 (Ω) for the semigroup {S(t)} t≥0 corresponding to the fractional reactiondiffusion equations (1) and the existence of a global attractor in L 2 0 (Ω). Finally, we will obtain the (L 2 0 (Ω), H σ/2 0 (Ω))-asymptotical compactness of the solution semigroup. Utilizing the norm-to-weak continuous semigroup method in [32], we prove the existence of a global attractor in H σ/2 0 (Ω)). Our main results are stated below.
LONG-TIME BEHAVIOR OF NONLOCAL PARTIAL DIFFERENTIAL EQUATIONS 751 Theorem 1.1. Assume that Ω is a bounded smooth domain in R n , f satisfies (2) and 0 < σ < 2. Then for any initial date u 0 ∈ L 2 0 (Ω) and any T > 0, there exists a unique solution and u 0 −→ u(t) is continuous on L 2 0 (Ω). Combining Theorem 1.1 with Lemma 2.6 (next section), we can define a solution semigroup  Notations in these theorems will be explained in the next section. This paper is organized as follows. In the next section, we present some definitions and lemmas used in the sequel. In Section 3, we will prove the well-posedness of weak solutions to the fractional reaction-diffusion equations (1). Finally, in Section 4 we prove the existence of global attractors in L 2 0 (Ω) and in H σ/2 0 (Ω) for the nonlocal system (1).

Preliminaries.
In this section, we introduce several function spaces and recall basic concepts about the global attractors. See [3,29] for more details. We recall L p 0 (Ω) := {u ∈ L p (R n ) : u = 0 a.e. on R n \ Ω} , 1 ≤ p ≤ ∞, and H σ/2 (R n ) := u ∈ L 2 (R n ) : endowed with the natural norm where the term is the so-called Gagliardo (semi)norm of u, and q.e. is the abbreviation for quasieverywhere with respect to Riesz capacity (see [8,12,26]). We also introduce the space

Moreover, we define a linear operator
(Ω) is the classical fractional Sobolev spaces.
and the equations We say that {S(t)} t≥0 is a norm-to weak continuous semigroup on (Y, X), if {S(t)} t≥0 satisfies that (1), (2) and We denote the condition (3) as (Y, X w )-continuous.
n=1 has a convergent subsequence with respect to the topology of X.
Let X, Y be two Banach spaces and X * , Y * be their dual spaces, respectively. Assume that X is a dense subspace of Y , the injection i : X → Y is continuous and its adjoint i * : Y * → X * is densely injective. Under these assumptions, the following results hold.
Lemma 2.6. (See [32]) Let X, Y be two Banach spaces satisfy the assumptions just above, {S(t)} t≥0 be a semigroup on X and Y , respectively, and assume furthermore that {S(t)} t≥0 is continuous or weak continuous on Y . Then {S(t)} t≥0 is a normto-weak continuous semigroup on X if and only if {S(t)} t≥0 maps compact subsets of X × R + into bounded sets of X.
3. Well-posedness. We start with the discussion of existence and uniqueness of solutions by a Galerkin method. Define where e i (x) is eigenfunctions of −∆ with Dirichlet boundary, and {e i } ∞ i=0 denote an orthonormal basis of L 2 (Ω). Hence, it is easy to check that { e i } ∞ i=0 is an orthonormal basis of L 2 0 (Ω), and e i ∈ H σ/2 0 (Ω). We will take f (0) = 0 to simplify the argument in the rest of the paper, but this is not, in fact, necessary.
Proof of Theorem 1.1. Let us consider the approximate solutions where {u mk (t)} m k=1 are the solutions of the following problem: Since the nonlinearity in (4) is locally Lipschitz, there exists a unique solution to (4). Multiplying the equations (4) by the function u mj (t) for each j and adding these equalities for j = 1, ...m, we conclude that Now we can use lemma 2.8 and Cauchy inequality to get By Gronwall inequality, we obtain that Therefore u m L ∞ (0,T ;L 2 0 (Ω)) ≤ C( u 0 L 2 0 (Ω) , g L 2 0 (Ω) ).
Define functions f (s) = f (s) + C 1 s. Because of (5) and (8), we infer that Now, multiplying equation (4), by the function u mj (t), for each j, adding these relations for j = 1, ...m, we deduce that where, F (u) = u 0 f (s)ds. Integrating the last inequality over (s, T ) with respect to the variable t, we conclude that Again integrating the last inequality over (0, T ) with respect to the variable s, we obtain that (Ω) ds + Noting that taking into account (8) and (9), we conclude that (Ω) , T ). Hence, for every ε ∈ (0, T ), we deduce that Thanks to Remark 2 and Aubin-Lions Lemma [25], there exists a subsequence, which we still denote by u m , such that Therefore, applying the diagonalization procedure, we infer that as m → ∞. Hence, we obtain that We now show that f (u m ) ∈ L 1 (0, T ; L 1 0 (Ω)). It is easy to check that T 0 Ω f (u m ) u m dxdt ≤ C, where C m. Let χ Ω1 and χ Ω2 be the characteristic functions of the sets we deduce that f (u m ) ∈ L 1 (0, T ; L 1 0 (Ω)). Furthermore, since by the mean value theorem we infer that Thus we get f (u m )χ Ω2 ∈ L 2 (0, T ; L 2 0 (Ω)). Hence, f (u m ) ∈ L 1 (0, T ; L 1 0 (Ω)) + L 2 (0, T ; L 2 0 (Ω)) ⊆ L 1 (0, T ; L 1 0 (Ω)). We need to show that where u + m = max{u m , 0} and u − m = min{u m , 0}. Define an N-function which has a complementary N-function G as follows, By the definition and (17), we obtain that is the Orlicz space ( [15]). Taking into account u m → u for almost every (x, t) ∈ Ω × [0, T ], continuity of f (·) and the functions max{s, 0} and min{s, 0}, it can be inferred that f ((u + m )) → f (u + ) in measure on Ω×[0, T ]. Hence, we deduce that where E F is the closures of the set of bounded functions in the spaces L * F (Ω×[0, T ]). We obtain that for every w ∈ L ∞ (Ω × (0, T )).
Because u m → u for almost every (x, t) ∈ Ω × [0, T ], we obtain that
We now prove the existence of (L 2 0 (Ω), H  (Ω) < C for every t > T and u 0 ∈ B. Proof. By P oincaré inequality, Hölder inequality and the equality (5), we deduce that d dt u m (t) 2 for some α > 0. Applying the Gronwall inequality, we obtain that By Lemma 4.1, there exists a positive constant T = T (B), such that for every t ≥ T , u m (t) 2 ≤ C, Integrating the equation (10) over (s, t) with respect to the variable t leads to 1 2 Integrating the last inequality over (t−1, t) with respect to the variable s, we obtain that 1 2 Hence, for every t ≥ T 1 2 Taking into account u m * u ∈ L ∞ (T, ∞; H σ/2 0 (Ω)) and passing to the limits when m → ∞, we infer that Integrating the inequality (11) over (T, T + 2) with respect to s and using (8), we conclude that as T large enough, where C is independent of T . Combining (26) with (27), and using the uniform Gronwall lemma, we deduce that Ω |u mt (s)| 2 dx ≤ C.
as s large enough. Therefore, there exists T B such that Ω |u t (s)|dx ≤ lim inf m→∞ Ω |u mt (s)| 2 dx ≤ C for every s ≥ T B .
Thanks to Lemma 2.7, we then obtain Theorem 1.3.