Dynamics and regularity for non-autonomous reaction-diffusion equations with anomalous diffusion

Abstract

This paper is concerned with the long time behavior of solutions for a non-autonomous reaction-diffusion equations with anomalous diffusion. Under suitable assumptions on nonlinearity and external force, the global well-posedness has been studied. Then the pullback attractors in $L^{2} (Ω)$ and $H_{0}^{α} (Ω)$ ( $0 < α < 1$ ) have been achieved with a restriction on the growth order of nonlinearity as $2 ⩽ p ⩽ \frac{2 (n - α)}{n - 2 α}$ . The results presented can be seen as the extension for classical theory of infinite dimensional dynamical system to the fractional diffusion equations.

Keywords

Fractional Laplacian pullback attractor non-compactness measure

1. Introduction

The nonlocal operators appeared in complex systems, such as anomalous diffusion and geophysical flows (see [5,24,25]), the thin obstacle problem [29], finance and stratified materials [12] and [26]. A special but important nonlocal operator is the fractional Laplacian operator arising in non-Gaussian stochastic systems. For a stochastic differential equation with a α-stable Lévy motion (a non-Gaussian stochastic process) $L_{t}^{α}$ for $α \in (0, 1)$ , which can be described as $\begin{matrix} (1.1) & d X_{t} = b (X_{t}) d t + d L_{t}^{α}, X_{0} = x, \end{matrix}$ the corresponding Fokker–Planck equation for the evolution of probability density function of the solution contains fractional Laplacian operator ${(- Δ)}^{α}$ , i.e., when the drift term b in (1.1) depends on the probability distribution, the Fokker–Planck equation becomes a nonlinear, nonlocal partial differential equations (PDEs) (see [1] for more detail). In the past decades, there is a great interest for mathematicians to study the PDEs involving nonlocal operators such as ${(- Δ)}^{α}$ with $0 < α < 1$ , which are called nonlocal PDEs. The well-posedness and regularity of nonlocal PDEs have been investigated in some literatures recently, such as in [3–5,7,15,18,29,30].

In the literatures [19,20], the authors gave probabilistic motivations to study some PDEs with anomalous diffusion, i.e., the Laplacian is replaced by a more general pseudo-differential operator in nonlocal PDEs, which enrich the research of nonlocal PDEs. There is a completely description on a class of time-space fractional semilinear PDEs in [18], the authors gave the well-posedness of problem and decay estimate of solution. To the best of our knowledge, there are some results of evolutionary PDEs with anomalous diffusion, while the long time behavior of solutions is still unknown in many significant cases, which interest us for the past years. In this presented paper, we shall study the dynamics of a non-autonomous reaction-diffusion equations with anomalous diffusion (also nonlocal) that can be written as $\begin{array}{l} (1.2) & \{\begin{matrix} \frac{d u}{d t} + {(- Δ)}^{α} u + f (u) = g (x, t), & in Ω \times [τ, \infty), 0 < α < 1, \\ u (x, t) = 0, & x \in Ω^{c}, t \in [τ, \infty), \\ u (x, τ) = u_{τ}, & x \in Ω, t = τ, \end{matrix} \end{array}$ where Ω is a bounded smooth domain of $R^{n}$ , $Ω^{c} = R^{n} ∖ Ω$ and $u = u (x, t) = (u_{1}, \dots, u_{n})$ , $g = g (x, t) = (g_{1}, \dots, g_{n})$ , $f = f (u) = (f_{1}, \dots, f_{n})$ . Denote the space $H = {(L^{2} (Ω))}^{n}$ and $V = {(H_{0}^{α} (Ω))}^{n}$ with norms $| \cdot |$ and $‖ \cdot ‖$ , respectively.

For mathematical analysis, we assume that the nonlinear term $f = f (u) \in C (R^{n}; R^{n})$ satisfies $\begin{array}{l} (1.3) & \sum_{1}^{n} γ_{i} | u_{i} |^{p_{i}} - C_{1} ⩽ \sum_{i}^{n} f_{i} (u) u_{i} = f (u) u, γ_{i} > 0, \forall u \in R^{n}, \\ (1.4) & \sum_{1}^{n} {| f_{i} (u) |}^{\frac{p_{i}}{p_{i} - 1}} ⩽ C_{2} (\sum_{1}^{n} | u_{i} |^{p_{i}} + 1), \forall u \in R^{n}, \\ p_{1} ⩾ p_{2} ⩾ \dots ⩾ p_{n} ⩾ 2 . \end{array}$ Also let $\begin{matrix} (1.5) & (f (u_{1}) - f (u_{2}), u_{1} - u_{2}) ⩾ - C_{3} | u_{1} - u_{2} |^{2}, \forall u_{1}, u_{2} \in R^{n}, \end{matrix}$ where $C_{i}$ denote special constants, some times for general positive constants that are independent variables, we denote C, maybe different in each line even in the same line.

When α equals to one, the equation (1.2) becomes classical non-autonomous reaction-diffusion equations, which was studied by many mathematicians, here we skip the fruitful results. For the long time behavior of evolutionary PDEs with nonlocal operator, to our best acknowledge, the results are few comparing with that of classical PDEs, here we refer to [2,10,11,21]. Our objective in this paper is to study the pullback dynamics of a non-autonomous reaction-diffusion equation with anomalous diffusion. In particular, the nonlocal operators play an important role in the energy dissipation, a major departure from classical equations. To deal with this technical difficulty, we summarize below some of the highlights of this paper:

The existence and uniqueness of global weak and strong solutions to problem (1.2) are established using the Galerkin technique. The presence of nonlocal operator again requires some special treatments from nonlocal vector calculus developed recently in [16].

A subsequent step after establishing the well-posedness is to study the long time behavior. The non-autonomous external force that we consider here need to satisfy Assumptions A and B, and thus, the forward uniform bounded estimates and invariance might not hold for this non-autonomous system. In this case, the theory of pullback attractors is well suitable for study the long time behavior. In Section 4, the existence of pullback attractors for (1.2) is proved in different functional spaces. We point out that to obtain the results in $L^{2} (Ω)$ , it suffices that external force satisfy Assumption A.

With the help of estimates obtained in $(II)$ , we obtain the existence of pullback attractors in $H_{0}^{α} (Ω)$ ( $0 < α < 1$ ). However, besides assuming that external force satisfy Assumption A, we have to impose a stronger condition Assumption B on the external force. Also we impose a restriction on the growth order of nonlinearity. The main reason is that the solution of problem lack of regularity, so we can’t relax the conditions on the nonlinearity and external force with the help of $L^{p}, p > 2$ estimate of the solution just like [31] does. How to remove the restriction conditions on the nonlinearity and external force let them become more general remains to be an open question.

The main difficulty of mathematical analysis lies in the non-locality of nonlocal operator and fractional Sobolev space.

This paper is organized as follows. In Section 2, we introduce some preliminary facts and assumptions, also give some properties of nonlocal operator, in particular, we prove that it is a sectorial operator. Then in Section 3, we combine the nonlocal calculus and the Galerkin method to prove the existence and uniqueness of weak and strong solutions of our nonlocal model. Finally, in Section 4, after obtaining some energy estimates, we prove the existence of pullback attractors in some different Sobolev spaces. In Section 5, we give some further remarks.

2. Preliminary

This section is devoted to some notations and preliminary results which will be used in the next few sections.

Notations.

We assume that $p_{1} ⩾ p_{2} ⩾ \dots ⩾ p_{n}$ . Let $q_{k}$ be the conjugate of $p_{k}$ , $\frac{1}{q_{k}} + \frac{1}{p_{k}} = 1$ , $k = 1, 2, \dots, n$ . For notation simplicity, denote by $p = (p_{1}, \dots, p_{n})$ , $q = (q_{1}, \dots, q_{n})$ , and $\begin{array}{l} L^{p} (Ω) = L^{p_{1}} (Ω) \times L^{p_{2}} (Ω) \times \dots \times L^{p_{n}} (Ω), \\ L^{p} (0, T; L^{p} (Ω)) = L^{p_{1}} (0, T; L^{p_{1}} (Ω)) \times \dots \times L^{p_{n}} (0, T; L^{p_{n}} (Ω)) . \end{array}$

ℜ be the real part for a complex number.

Λ denote the ${(- Δ)}^{\frac{1}{2}}$ .

$Ω (| u | ⩾ M) = {x \in Ω | | u | ⩾ M}$ .

$m (Ξ)$ denotes the measure of Ξ.

Denote $C_{0}^{\infty} (R^{n})$ by the space of infinitely differentiable functions with compact support in $R^{n}$ .

Denote $S (R^{n})$ as the rapidly decreasing function in $R^{n}$ .

The norm of Banach space E is denoted by $‖ \cdot ‖_{E}$ . The norms of some special spaces are marked with other symbols, but according to the context, there is no misunderstanding.

2.1. The fractional Sobolev space

The generic fractional Sobolev space $H^{α} (Ω) = {(H^{α} (Ω))}^{n}$ is defined as $\begin{matrix} H^{α} (Ω) = {u \in L^{2} (Ω) : \frac{| u (x) - u (y) |}{| x - y |^{\frac{n + 2 α}{2}}} \in L^{2} (Ω \times Ω)} \end{matrix}$ with the Gagliardo norm $\begin{matrix} ‖ u ‖_{H^{α} (Ω)} = ‖ u ‖_{L^{2} (Ω)} + {(\int_{Ω \times Ω} \frac{| u (x) - u (y) |^{2}}{| x - y |^{n + 2 α}} d x d y)}^{\frac{1}{2}}, \end{matrix}$ and $H_{0}^{α} (Ω) = {(H_{0}^{α} (Ω))}^{n}$ for $\begin{matrix} H_{0}^{α} (Ω) = {g \in H^{α} (R^{n}); g = 0 a.e. in Ω^{c}}, \end{matrix}$ where $Ω^{c} = R^{n} ∖ Ω$ . The space $H_{0}^{α} (Ω)$ is not empty ( $C_{0}^{2} (Ω) \subseteq H_{0}^{α} (Ω)$ ) and is a Hilbert space with scalar product (see [27]) $\begin{matrix} {⟨ u, v ⟩}_{H_{0}^{α} (Ω)} = \int_{R^{n} \times R^{n}} \frac{(u (x) - u (y)) (v (x) - v (y))}{| x - y |^{n + 2 α}} d x d y . \end{matrix}$ We use V to represent space $H_{0}^{α} (Ω)$ for convenience. $V^{*}$ represents the dual space of V, and the norm of $V^{*}$ is expressed by $‖ \cdot ‖_{*}$ . For more details of the fractional Sobolev spaces, we refer the reader to [15] and literatures therein.

Lemma 2.1 ([15, Theorem 6.7]).

Let $α \in (0, 1)$ , $n > 2 α$ , Ω be an open bounded set of $R^{n}$ . Then the embedding $J : H_{0}^{α} (Ω) ↪ L^{ν} (Ω)$ is continuous for any $ν \in [1, 2^{*}]$ , while it is compact whenever $ν \in [1, 2^{*})$ , where $2^{*} = \frac{2 n}{n - 2 α}$ is critical Sobolev embedding exponent.

2.2. Properties of nonlocal operator

We first give the definition of nonlocal operator as following.

Definition 2.1 (Fractional Laplacian – [15]).

For $u \in S (R^{n})$ and $α \in (0, 1)$ , we define the nonlocal operator as $\begin{matrix} {(- Δ)}^{α} u (x) = C (n, α) P . V . \int_{R^{n}} \frac{u (x) - u (y)}{| x - y |^{n + 2 α}} d y, x \in R^{n}, \end{matrix}$ where the principle value (P.V.) is taken as the limit of the integral over $Ω ∖ B_{ϵ} (x)$ as $ϵ ⟶ 0^{+}$ , with $B_{ϵ} (x)$ the ball of radius ϵ centered at x, and $\begin{matrix} C (n, α) = {(\int_{R^{n}} \frac{1 - cos ζ_{1}}{| ζ |^{n + 2 α}} d ζ)}^{- 1} . \end{matrix}$

Given the functions $ϱ : R^{n} \times R^{n} ⟶ R^{n}$ with antisymmetric, i.e., $ϱ (x, y) = - ϱ (y, x)$ , $u : R^{n} ⟶ R$ , $z : R^{n} ⟶ R$ and $v : R^{n} \times R^{n} ⟶ R^{n}$ . We will deduce that the fractional Laplacian operator ${(- Δ)}^{α}$ defined above can be expressed in the nonlocal calculus form [16,17].

Definition 2.2.
Let functions u, v, ϱ are defined above. The nonlocal divergence operator can be defined as $\begin{matrix} (2.1) & D (v) (x) : = \int_{R^{n}} (v (x, y) + v (y, x)) \cdot ϱ (x, y) d y for x \in Ω . \end{matrix}$
Definition 2.3.
The action of the interaction operator $N (v) : R^{n} \times R^{n} ⟶ R$ is following $\begin{matrix} (2.2) & N (v) (x) : = - \int_{R^{n}} (v (x, y) + v (y, x)) \cdot ϱ (x, y) d y for x \in Ω^{c} . \end{matrix}$
Definition 2.4 (Nonlocal adjoint operator).

For a nonlocal operator $Q$ with associated interaction operator $P$ , its nonlocal adjoint operator is $Q^{*}$ , such that for any functions u and v, there hold $\begin{matrix} \int_{Ω} v Q (u) d x - \int_{R^{n}} \int_{R^{n}} Q^{*} (v) u d x d y = \int_{Ω^{c}} v P (u) d x . \end{matrix}$

Lemma 2.2.
Suppose that two point antisymmetry function $Ψ \in L^{1} (R^{n} \times R^{n})$ . Then one has the following properties: $\begin{matrix} \int_{R^{n}} \int_{R^{n}} Ψ (x, y) d x d y = 0 . \end{matrix}$
Proof.
Using Fubini’s theorem, we change the order of variables $\begin{array}{l} \int_{R^{n}} \int_{R^{n}} Ψ (x, y) d x d y & = \int_{R^{n}} \int_{R^{n}} Ψ (x, y) d y d x = - \int_{R^{n}} \int_{R^{n}} Ψ (y, x) d y d x \\ = - \int_{R^{n}} \int_{R^{n}} Ψ (x, y) d x d y . \end{array}$ The last line we relabel the variables again. So the desired result hold. □

The adjoint of $D$ is $\begin{matrix} (2.3) & D^{} (u) (x, y) = - (u (y) - u (x)) ϱ (x, y) for x, y \in R^{n} . \end{matrix}$ To see $D$ and $D^{}$ are conjugate, we only need a simple computation according to Definition 2.4. Let v be a vector valued two point function. $\begin{array}{l} \int_{Ω} u D (v) d x - \int_{Ω^{c}} u N (v) d x & = \int_{R^{n}} \int_{R^{n}} u (x) (v (x, y) + v (y, x)) \cdot ϱ (x, y) d y d x \\ = \int_{R^{n}} \int_{R^{n}} (u (x) - u (y)) v (x, y) \cdot ϱ (x, y) d y d x \\ + \int_{R^{n}} \int_{R^{n}} (u (x) v (y, x) + u (y) v (x, y)) \cdot ϱ (x, y) d y d x \\ = \int_{R^{n}} \int_{R^{n}} (u (x) - u (y)) v (x, y) \cdot ϱ (x, y) d y d x \\ = \int_{R^{n}} \int_{R^{n}} (u (x) - u (y)) ϱ (x, y) \cdot v (x, y) d y d x \\ (2.4) & = \int_{R^{n}} \int_{R^{n}} D^{} (u) \cdot v (x, y) d y d x . \end{array}$

The classical Gauss divergence theorem is that for a vector function $f \in C^{1} (\overline{Ω})$ $\begin{matrix} \int_{Ω} \nabla \cdot f d x = \int_{\partial Ω} ν \cdot f d S, \end{matrix}$ where ν is the outward unite normal vector. Analogously, we have a nonlocal version $\begin{array}{l} \int_{Ω} D (v) (x) d x - \int_{Ω^{c}} N (v) (x) d x = \int_{R^{n}} \int_{R^{n}} (v (x, y) + v (y, x)) \cdot ϱ (x, y) d x d y . \end{array}$ The function $(v (x, y) + v (y, x)) \cdot ϱ (x, y)$ is antisymmetry, so by above lemma, the following hold.
Corollary 2.1.
Let* $D$ and $N$ be defined above. Hence for any v, we have $\begin{matrix} \int_{Ω} D (v) (x) d x = \int_{Ω^{c}} N (v) (x) d x . \end{matrix}$

The Green’s first identity in classical calculus is very important, the formula is that for $u, v \in C^{1} (\overline{Ω})$ $\begin{matrix} \int_{Ω} u (\nabla \cdot v) d x + \int_{Ω} (- \nabla u) \cdot v d x = \int_{\partial Ω} u v \cdot ν d S, \end{matrix}$ where ν is the outward unite normal vector. In the following, we will give an analogous nonlocal version.
Theorem 2.1.
Let $D$ and $N$ are nonlocal divergence and interaction operator respectively, u and z are scalar one point value functions. Then we have $\begin{matrix} (2.5) & \int_{Ω} u D (D^{} z) d x - \int_{R^{n}} \int_{R^{n}} D^{} u \cdot D^{} z d y d x = \int_{Ω^{c}} u N (D^{} z) d x . \end{matrix}$
Proof.
We put $v = D^{} z$ in (2.4) to get $\begin{array}{l} (2.6) & \int_{Ω} u D (D^{} z) d x - \int_{Ω^{c}} u N (D^{} z) d x = \int_{R^{n}} \int_{R^{n}} D^{} u \cdot D^{} z d y d x, \end{array}$ which is the desired result. □

The nonlocal operator ${(- Δ)}^{α}$ can be rewritten as $\begin{matrix} (2.7) & {(- Δ)}^{α} = D D^{}, \end{matrix}$ with $\frac{C (n, s)}{2 | x - y |^{n + 2 α}} = ‖ ϱ ‖_{R^{n}}^{2}$ .

We can also define the other nonlocal operators in the sense of principle value, just as [16,17] does, and get the similar results, such as nonlocal Green’s second identity, the relationship between nonlocal operators and classical differential operators.

In [16,17], the authors also make a comparison between the nonlocal and the corresponding local operators. For example, if we choose $ϱ (x, y) = - \nabla_{y} δ (y - x)$ , where $δ (\cdot)$ denotes the Dirac delta function, $\nabla_{y}$ denotes the differential gradient with respect to y in distribution sense, we have $\begin{array}{l} \int_{R^{n}} D^{} (u) d y & = - \int_{R^{n}} (u (x) - u (y)) \nabla_{y} δ (y - x) d y \\ = \int_{R^{n}} δ (y - x) \nabla_{y} (u (x) - u (y)) d y = - \nabla_{y} u (y) |_{y = x} = - \nabla u (x) . \end{array}$

Next, we give some properties of the nonlocal operator ${(- Δ)}^{α}$ .
Proposition 2.1.
The operator* ${(- Δ)}^{α}$ has the following properties:
If u is a constant, then ${(- Δ)}^{α} u = 0$ .

If $u \in H_{0}^{α} (Ω)$ , then ${(- Δ)}^{α}$ is a positive semidefinite operator, i.e., $({(- Δ)}^{α} u, u) ⩾ 0$ in the sense of $L^{2} (Ω)$ inner product.

Proof.
By definition of nonlocal operator ${(- Δ)}^{α}$ , $(i)$ is trivial.

Next, let $\frac{C (n, s)}{2 | x - y |^{n + 2 α}} = ‖ ϱ ‖_{R^{n}}^{2}$ , then by Theorem 2.1 it yields $\begin{array}{l} ({(- Δ)}^{α} u, u) & = (D D^{} u, u) = \int_{R^{n}} \int_{R^{n}} D^{} (u) \cdot D^{} (u) d y d x + \int_{Ω^{c}} u N (D^{} (u)) d x \\ = \frac{C (n, α)}{2} \int_{R^{n}} \int_{R^{n}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{n + 2 α}} d y d x ⩾ 0, \end{array}$ which shows that property $(ii)$ holds. We complete the proof. □

∙ Nonlocal Poincaré’s Inequality.

We use the following nonlocal counterpart of the Poincaré’s inequality in the proof: $\begin{matrix} ‖ u ‖_{L^{2} (Ω)} ⩽ C ‖ u ‖_{H_{0}^{α} (Ω)}, \end{matrix}$ see [15,28] for more details.

∙ Eigenvalues of ${(- Δ)}^{α}$ .

Consider the following eigenvalue problem of ${(- Δ)}^{α}$ : $\begin{array}{l} (2.8) & \{\begin{matrix} {(- Δ)}^{α} u = λ u, & in Ω, \\ u = 0, & in Ω^{c} . \end{matrix} \end{array}$

The weak formulation of problem (2.8) is following $\begin{array}{l} (2.9) & \{\begin{matrix} C \int_{R^{n}} \int_{R^{n}} \frac{(u (x) - u (y)) (φ (x) - φ (y))}{| x - y |^{n + 2 α}} d x d y = λ \int_{Ω} u (x) φ (x) d x, φ (x) \in H_{0}^{α} (Ω), \\ u \in H_{0}^{α} (Ω) . \end{matrix} \end{array}$ We say $λ \in R$ is an eigenvalue of problem (2.8), if there exists a non-trivial solution $u \in H_{0}^{α} (Ω)$ of problem (2.8) satisfying the weak formulation. Any such solution will be called an eigenfunction corresponding to the eigenvalue λ.
Lemma 2.3 ([28]).

Let $α \in (0, 1)$ , $n > 2 α$ , Ω be an open bounded set of $R^{n}$ . Then

Problem ( 2.8 ) admits an eigenvalue $λ_{1} > 0$ that is simple and has an associated nonnegative eigenfunction $e_{1} \in H_{0}^{α} (Ω)$ with $| e_{1} | = 1$ ,

the set of the eigenvalues of problem ( 2.8 ) consists of a sequence ${λ_{j}}_{j \in N}$ with $\begin{array}{l} (2.10) & 0 < λ_{1} < λ_{2} ⩽ \dots ⩽ λ_{j} ⩽ \cdot \cdot \cdot, λ_{j} \to + \infty as j \to \infty, \end{array}$

the sequence ${e_{j}}_{j \in N}$ of eigenfunctions corresponding to $λ_{j}$ is an orthonormal basis of $L^{2} (Ω)$ and an orthogonal basis of $H_{0}^{α} (Ω)$ .

∙ Sectorial operator.

Definition 2.5.
A linear operator A is said to be a sectorial operator, if A is a densely defined closed operator, and for $M ⩾ 1$ , there exist real numbers $ϕ \in (0, \frac{π}{2})$ , such that $S_{a, ϕ} = {λ | ϕ ⩽ | arg (λ - a) | ⩽ π, λ \neq a} \subset ρ (A)$ for $a \in R$ , where $ρ (A)$ is the resolvent of A and $‖ {(λ I - A)}^{- 1} ‖ ⩽ M / | λ - a |$ .

In the space $L^{2} (Ω)$ , we have the following results for ${(- Δ)}^{α}$ with zero boundary condition in $Ω^{c}$ . Lemma 2.4.
The nonlocal Laplacian operator $A_{α} = {(- Δ)}^{α}$ is sectorial and satisfies the following estimates in $L^{2} (Ω)$ : $\begin{matrix} (2.11) & {‖ e^{- t A_{α}} ‖}_{L^{2} (Ω)} ⩽ C e^{- δ t}, {‖ A_{α} e^{- t A_{α}} ‖}_{L^{2} (Ω)} ⩽ \frac{C}{t} e^{δ t}, \end{matrix}$ where $C, δ > 0$ are constants independent of t.
Proof.
By the nonlocal calculus, we have $\begin{matrix} (2.12) & (A_{α} u, v) = \int_{R^{n}} \int_{R^{n}} D^{} (u) \cdot D^{} (v) d y d x = (u, A_{α} v), \end{matrix}$ for any $v \in D (A_{α})$ . An integration by parts argument gives that $\begin{matrix} D (A_{α}) = {u \in H_{0}^{α} (Ω), A_{α} u \in L^{2} (Ω)} . \end{matrix}$ Since $C_{0}^{\infty} (R^{n}) \subset D (A_{α}) \subset L^{2} (Ω)$ is a dense subset of $L^{2} (Ω)$ , the operator $A_{α}$ is densely defined and is a self-adjoint operator in Hilbert space $L^{2} (Ω)$ . Moreover $\begin{matrix} (2.13) & (A_{α} u, u) = ‖ u ‖_{H_{0}^{α} (Ω)}^{2} . \end{matrix}$ By nonlocal Poincaré’s Inequality, there exist $m \in R$ , we have $(A_{α} u, u) ⩾ m | u |^{2}$ , hence using Lax–Milgram theorem, for every λ satisfying $λ > λ_{0}$ and every $f \in L^{2} (Ω)$ , there exists a unique solution $u \in H_{0}^{α} (Ω)$ satisfying the equation $\begin{array}{l} \{\begin{matrix} {(- Δ)}^{α} u + λ u = f, & x \in Ω, 0 < α < 1 . \\ u = 0, & x \in Ω^{c} . \end{matrix} \end{array}$

By above preparation, $A_{α}$ is a sectorial operator ([14, Theorem 5.1.2]) and for $t > 0$ $\begin{matrix} e^{- A_{α} t} = \frac{1}{2 π i} \int_{Γ} e^{λ t} {(λ I - A_{α})}^{- 1} d λ \end{matrix}$ with $e^{- A_{α} t} = I (Identity)$ at $t = 0$ , where Γ is a contour in the resolvent set $ρ (- A_{α})$ .

Set $μ = λ t$ , $λ > 0$ , $\begin{array}{l} (2.14) & {‖ e^{- t A_{α}} ‖}_{L^{2} (Ω)} = {‖ \frac{1}{2 π i} \int_{Γ} e^{μ} {(\frac{μ}{t} - A_{α})}^{- 1} \frac{d μ}{t} ‖}_{L^{2} (Ω)} ⩽ \frac{M}{2 π} \int_{Γ} | e^{μ} | \frac{| d μ |}{| μ |} ⩽ C e^{- δ t} . \\ (2.15) & \begin{matrix} {‖ A_{α} e^{- t A_{α}} ‖}_{L^{2} (Ω)} & = ‖ \frac{d}{d t} e^{- t A_{α}} ‖ \\ = \frac{1}{2 π i} \int_{Γ} λ e^{λ t} {(λ I - A_{α})}^{- 1} d λ \\ ⩽ \frac{1}{2 π} \frac{M}{δ} \int_{Γ} | e^{μ} | \frac{| d μ |}{| μ |} \frac{1}{t} \\ ⩽ \frac{C}{t} e^{- δ t} . \end{matrix} \end{array}$ We complete the proof. □

3. Global well-posedness

In this section, the global well-posedness of problem (1.2)–(1.4) will be stated. Without loss of generality, we assume that $C (n, α) = 1$ in Definition 2.1.

We first state the definition of weak solution.

Definition 3.1.
A function $u (x, t)$ is a weak solution of equation (1.2) if $\begin{matrix} u \in L^{p} (τ, T; L^{p} (Ω)) \cap L^{2} (τ, T; V), \frac{d u}{d t} \in L^{2} (τ, T; V^{*}) \end{matrix}$ and u satisfies equation (1.2) in the distribution sense.

If $u \in L^{\infty} (τ, T; H)$ , then $u \in C_{w} (τ, T; H)$ [9, II Theorem 1.7]. Therefore, the initial value $u_{τ}$ we given in H is meaningful.
Theorem 3.1 (Weak solution).

Suppose $g (t)$ belongs to $L^{2} (τ, T; V^{*})$ and f meets the conditions ( 1.3 ) and ( 1.4 ). Then the problem ( 1.2 )–( 1.4 ) has an unique weak solution: for any $T > 0$ , given $u_{τ} \in H$ , there exists a solution $\begin{matrix} u \in L^{2} (τ, T; V) \cap L^{p} (τ, T; L^{p} (Ω)), u \in C (τ, T; H), \end{matrix}$ and $u_{τ} ⟼ u (t)$ is continuous on H. Equation ( 1.2 ) holds as an equality in $L^{q} (τ, T; V^{*})$ , where q is conjugate to p.

Proof.
By Lemma 2.3, we project the equation (1.2) onto the span of the first n eigenfunctions of nonlocal operator ${(- Δ)}^{α}$ . Let $P_{n} : L^{2} (Ω) ⟶ span {w_{1}, \dots, w_{n}}$ be the orthogonal projector and $u_{n} = P_{n} u$ . We consider the ordinary differential equations $\begin{matrix} (3.1) & \frac{d u_{n}}{d t} + {(- Δ)}^{α} u_{n} + P_{n} f (u_{n}) = P_{n} g (t), u_{n} (τ) = P_{n} u_{τ} . \end{matrix}$

Since the nonlinearity in equation (3.1) is locally Lipschitz continuous, hence classical ordinary differential equation theory indicate that equation (3.1) has an unique solution on some finite time interval. In what follows, we will show that the solutions of the above equations are bounded in time and uniformly bounded in n.

Taking inner product the equation (3.1) with $u_{n}$ , we get $\begin{matrix} (3.2) & (\frac{d u_{n}}{d t}, u_{n}) + ({(- Δ)}^{α} u_{n}, u_{n}) + (P_{n} f (u_{n}), u_{n}) = (P_{n} g (t), u_{n}) . \end{matrix}$ And $\begin{matrix} (3.3) & \frac{1}{2} \frac{d | u_{n} |^{2}}{d t} + \int_{R^{n}} {| {(- Δ)}^{\frac{α}{2}} u_{n} |}^{2} d x + \int_{Ω} f (u_{n}) u_{n} d x = \int_{Ω} g (t) u_{n} d x . \end{matrix}$ Thanks to condition (1.3), we deduce that $\begin{matrix} (3.4) & \sum_{1}^{n} γ_{i} ‖ u_{n} ‖_{p_{i}}^{p_{i}} - C m (Ω) ⩽ \int_{Ω} f (u_{n}) u_{n} d x . \end{matrix}$ It implies by Young’s and nonlocal Poincare’s inequality that $\begin{matrix} (3.5) & | (P_{n} g (t), u_{n}) | ⩽ \frac{C | g (t) |^{2}}{2} + \frac{‖ u_{n} ‖^{2}}{2} . \end{matrix}$ Substituting (3.4) and (3.5) into (3.3), the following estimate hold $\begin{matrix} (3.6) & \frac{d | u_{n} |^{2}}{d t} + ‖ u_{n} ‖^{2} + C ‖ u_{n} ‖_{p}^{p} ⩽ C + C {| g (t) |}^{2} ⩽ C {| g (t) |}^{2} . \end{matrix}$ Integrating both sides of above inequality between τ and T gives $\begin{matrix} (3.7) & {| u_{n} (T) |}^{2} + \int_{τ}^{T} ‖ u_{n} ‖^{2} d t + C \int_{τ}^{T} ‖ u_{n} ‖_{p}^{p} d t ⩽ {| u_{n} (τ) |}^{2} + C \int_{τ}^{T} {| g (t) |}^{2} d t . \end{matrix}$ It follows from inequality (3.7) that there exists some positive constant M such that $\begin{array}{rcl} (3.8) & sup_{t \in [τ, T]} {| u_{n} (t) |}^{2} ⩽ M, \int_{τ}^{T} ‖ u_{n} ‖^{2} d t ⩽ M, \int_{τ}^{T} ‖ u_{n} ‖_{p}^{p} d t ⩽ M, \end{array}$ hold for the initial data in H. One can write these as $\begin{array}{l} {u_{n}} is uniformly bounded in L^{\infty} (τ, T; H), \\ (3.9) & {u_{n}} is uniformly bounded in L^{2} (τ, T; V), \\ {u_{n}} is uniformly bounded in L^{p} (τ, T; L^{p} (Ω)) . \end{array}$ Now, we want to obtain the bound of $f (u_{n})$ , by condition (1.4), we have $\begin{array}{l} \sum_{1}^{n} \int_{τ}^{T} \int_{Ω} {| f_{i} (u_{n}) |}^{q_{i}} d x d t & ⩽ C_{2} \int_{τ}^{T} (\sum_{1}^{n} \int_{Ω} {| u_{n}^{i} |}^{p_{i}} d x + m (Ω)) d t \\ = C_{2} \int_{τ}^{T} \int_{Ω} | u_{n} |^{p} d x d t + C_{2} (T - τ) . \end{array}$ Hence, we conclude that $\begin{matrix} f (u_{n}) is uniformly bounded in L^{q} (τ, T; L^{q} (Ω)) . \end{matrix}$

Finally, we shall give estimate on the derivative of $u_{n}$ .

Noting that $L^{2} (τ, T; V^{})$ and $L^{q} (τ, T; L^{q} (Ω))$ are both continuously included in $L^{q} (τ, T; V^{} (Ω))$ , by using the equation $\begin{matrix} \frac{d u_{n}}{d t} = - {(- Δ)}^{α} u_{n} - P_{n} f (u_{n}) + P_{n} g (t), \end{matrix}$ hence we have $\begin{matrix} \frac{d u_{n}}{d t} is uniformly bounded in L^{q} (τ, T; V^{} (Ω)) . \end{matrix}$

Now, by functional analysis theory, we can extract a weakly convergent subsequence, still we denoted by $u_{n}$ , with $\begin{array}{l} u_{n} weakly convergent to u in L^{2} (τ, T; V) . \\ u_{n} weakly convergent to u in L^{p} (τ, T; L^{p} (Ω)) . \\ f (u_{n}) weakly convergent to χ in L^{q} (τ, T; L^{q} (Ω)) . \\ \frac{d u_{n}}{d t} weakly convergent to \frac{d u}{d t} in L^{q} (τ, T; V^{} (Ω)) . \end{array}$ In fact, we obtain $P_{n} f (u_{n})$ weakly convergent to χ in $L^{q} (τ, T; L^{q} (Ω))$ . If we take limit in equation (3.1), then all the terms convergent in the dual space of $L^{2} (τ, T; V) \cap L^{p} (τ, T; L^{p} (Ω))$ , and the equality $\begin{matrix} \frac{d u}{d t} + {(- Δ)}^{α} u + χ = g (t) \end{matrix}$ holds in $L^{2} (τ, T; V^{}) + L^{q} (τ, T; L^{q} (Ω)) \subset L^{q} (τ, T; V^{} (Ω))$ .

In order to prove our results, we need to prove $χ = f (u)$ . Since $u_{n}$ converges to u in $L^{2} (τ, T; H)$ strongly ([9, II Theorem 1.4]), hence it guarantees that $u_{n}$ converges to u for almost every $(x, t) \in Ω \times (τ, T)$ . Using the continuity of f, it follows that $f (u_{n})$ converges to $f (u)$ for almost every $(x, t) \in Ω \times (τ, T)$ . According to the above proof process, $f (u_{n})$ is uniformly bounded in $L^{q} (τ, T; L^{q} (Ω))$ , this implies that $f (u_{n})$ weakly convergent to $f (u)$ in $L^{q} (τ, T; L^{q} (Ω))$ . By the uniqueness of weak limits, we obtain that $χ = f (u)$ .

Subsequently, we prove the continuity of the solution from $u (t)$ into H and $u (τ) = u_{τ}$ . Denote that $u \in L^{2} (τ, T; V) \cap L^{p} (τ, T; L^{p} (Ω))$ and $\frac{d u}{d t} \in L^{2} (τ, T; V^{})$ , use the theorem ([9, II Theorem 1.8]), we can obtain $u \in C (τ, T; H)$ . Considering the limiting equation of equation (3.1), we have $\begin{matrix} (3.10) & (\frac{d u}{d t}, φ) + ({(- Δ)}^{\frac{α}{2}} u, {(- Δ)}^{\frac{α}{2}} φ) + (f (u), φ) = (g (t), φ), \end{matrix}$ where we choose $φ \in C^{1} (τ, T; V \cap L^{p} (Ω))$ with $φ (T) = 0$ . Applying integration by part gives $\begin{array}{l} - \int_{τ}^{T} (u, \frac{d φ}{d t}) d s + \int_{τ}^{T} ({(- Δ)}^{\frac{α}{2}} u, {(- Δ)}^{\frac{α}{2}} φ) d s + \int_{τ}^{T} (f (u), φ) d s \\ (3.11) & = \int_{τ}^{T} (g (s), φ) d s + (u (τ), φ (τ)) . \end{array}$ Similarly, for the Galerkin approximation, $\begin{array}{l} - \int_{τ}^{T} (u_{n}, \frac{d φ}{d t}) d s + \int_{τ}^{T} ({(- Δ)}^{\frac{α}{2}} u_{n}, {(- Δ)}^{\frac{α}{2}} φ) d t + \int_{τ}^{T} (P_{n} f (u_{n}), φ) d s \\ (3.12) & = \int_{τ}^{T} (g (s), φ) d s + (u_{n} (τ), φ (τ)) . \end{array}$ Hence taking limits as $n \to \infty$ yields $\begin{array}{l} - \int_{τ}^{T} (u, \frac{d φ}{d t}) d s + \int_{τ}^{T} ({(- Δ)}^{\frac{α}{2}} u, {(- Δ)}^{\frac{α}{2}} φ) d t + \int_{τ}^{T} (f (u), φ) d s \\ (3.13) & = \int_{τ}^{T} (g (s), φ) d s + (u_{τ}, φ (τ)), \end{array}$ where we use $u_{n} (τ) = P_{n} u_{τ} ⟶ u_{τ}$ as $n ⟶ \infty$ . Comparing with (3.11) and (3.13), we have $u (τ) = u_{τ}$ .

Finally, we prove the uniqueness of solution and the continuous dependence of the solution on the initial value. Let $u_{τ}$ and $v_{τ}$ be two initial values in H, also let $w (t) = u (t) - v (t)$ , where $u (t)$ and $v (t)$ are solutions of equation (1.2) corresponding to initial datum $u_{τ}$ and $v_{τ}$ respectively. Hence $w (t)$ satisfies $\begin{matrix} (3.14) & \frac{d w}{d t} + {(- Δ)}^{α} w + f (u) - f (v) = 0 . \end{matrix}$

Taking inner product above equation with w in H, we have $\begin{matrix} (3.15) & \frac{1}{2} \frac{d | w |^{2}}{d t} + ‖ w ‖^{2} + (f (u) - f (v), w) = 0 . \end{matrix}$

By condition (1.5), we obtain $\begin{matrix} (3.16) & \frac{1}{2} \frac{d | w |^{2}}{d t} ⩽ \frac{1}{2} \frac{d | w |^{2}}{d t} + ‖ w ‖^{2} ⩽ C_{3} | w |^{2} . \end{matrix}$ Using the uniform Gronwall’s inequality, this gives $\begin{matrix} | u (t) - v (t) | ⩽ e^{2 C_{3} t} | u_{τ} - v_{τ} |, \end{matrix}$ which implies our results. We complete the proof. □

Next, we apply the same method to prove the existence of strong solution. Definition 3.2.
If we increase the regularity of g and $u_{τ}$ , then the solution becomes more smoothness, namely, if $g \in L^{2} (0, T; H)$ and $u_{τ} \in V$ , then the solutions are continuously embedded into V and are elements of H for almost every t. We call it a strong solution.
Theorem 3.2 (Strong solution).

If $u_{τ} \in V \cap L^{p} (Ω)$ , then there exists an unique strong solution $\begin{matrix} u (t) \in C (τ, T; V) \cap L^{\infty} (τ, T; L^{p} (Ω)) \cap L^{2} (τ, T; D ({(- Δ)}^{α})), \end{matrix}$ where $D (\cdot)$ denotes the domain of operator.

Proof.
Taking inner product equation (3.1) with ${(- Δ)}^{α} u_{n}$ implies that $\begin{matrix} (3.17) & \frac{1}{2} \frac{d ‖ u_{n} ‖^{2}}{d t} + {| {(- Δ)}^{α} u_{n} |}^{2} = - \int_{Ω} f (u_{n}) {(- Δ)}^{α} u_{n} d x + \int_{Ω} g (t) {(- Δ)}^{α} u_{n} d x . \end{matrix}$ Using condition (1.5) of nonlinear term and nonlocal calculus, we have $\begin{array}{l} (3.18) & \begin{matrix} - \int_{Ω} f (u_{n}) {(- Δ)}^{α} u_{n} d x & = - \frac{1}{2} \int_{R^{n}} \int_{R^{n}} \frac{(u_{n} (x) - u_{n} (y)) (f (u_{n} (x)) - f (u_{n} (y)))}{| x - y |^{n + 2 α}} d x d y \\ ⩽ C \int_{R^{n}} \int_{R^{n}} \frac{{(u_{n} (x) - u_{n} (y))}^{2}}{| x - y |^{n + 2 α}} d x d y \\ = C ‖ u_{n} ‖^{2} . \end{matrix} \\ (3.19) & \int_{Ω} g (t) {(- Δ)}^{α} u_{n} d x ⩽ | \int_{Ω} g (t) {(- Δ)}^{α} u_{n} d x | ⩽ \frac{| g (t) |^{2}}{2} + \frac{| {(- Δ)}^{α} u_{n} |^{2}}{2} . \end{array}$ Substituting (3.18) and (3.19) into (3.17), we have $\begin{matrix} (3.20) & \frac{d ‖ u_{n} ‖^{2}}{d t} + {| {(- Δ)}^{α} u_{n} |}^{2} ⩽ C ‖ u_{n} ‖^{2} + {| g (t) |}^{2} . \end{matrix}$ Integrating both sides from τ to T gives $\begin{matrix} (3.21) & {‖ u_{n} (T) ‖}^{2} + \int_{τ}^{T} {| {(- Δ)}^{α} u_{n} |}^{2} d s ⩽ C \int_{τ}^{T} ‖ u_{n} ‖^{2} d t + {‖ u_{n} (τ) ‖}^{2}, \end{matrix}$ which implies that $\begin{array}{l} u_{n} is uniformly bounded in L^{2} (τ, T; D ({(- Δ)}^{α})), \\ u_{n} is uniformly bounded in L^{\infty} (τ, T; V), \end{array}$ here we also use the fact that $u \in L^{2} (τ, T; V)$ from the proof of Theorem 3.1.

Next, we give an estimates of $\frac{d u_{n}}{d t}$ . Taking scalar product of equation (3.1) with $\frac{d u_{n}}{d t}$ , we have $\begin{matrix} (3.22) & {| \frac{d u_{n}}{d t} |}^{2} + ({(- Δ)}^{α} u_{n}, \frac{d u_{n}}{d t}) + (P_{n} f (u_{n}), \frac{d u_{n}}{d t}) = (g (t), \frac{d u_{n}}{d t}) . \end{matrix}$ Setting $F (u) = \int_{0}^{u} f (ξ) d ξ$ . Using nonlocal Green’s first identity, we get $\begin{matrix} (3.23) & {| \frac{d u_{n}}{d t} |}^{2} + \frac{d}{d t} (‖ u_{n} ‖^{2} + 2 \int_{Ω} F (u_{n}) d x) ⩽ {| g (t) |}^{2} . \end{matrix}$ Integrating above inequality from τ to T gives $\begin{matrix} (3.24) & \begin{matrix} \int_{τ}^{T} {| \frac{d u_{n}}{d t} |}^{2} d s + {‖ u_{n} (T) ‖}^{2} + 2 \int_{Ω} F (u_{n} (T)) d x \\ ⩽ {‖ u_{n} (τ) ‖}^{2} + 2 \int_{Ω} F (u_{n} (τ)) d x + \int_{τ}^{T} {| g (s) |}^{2} d s . \end{matrix} \end{matrix}$ Taking into account condition (1.3), we have $\begin{matrix} (3.25) & - C + C | u |^{p} ⩽ F (u) ⩽ C + C | u |^{p} . \end{matrix}$ Combining with (3.24) and (3.25), the following estimate holds $\begin{matrix} (3.26) & \begin{matrix} \int_{τ}^{T} {| \frac{d u_{n}}{d t} |}^{2} d s + {‖ u_{n} (T) ‖}^{2} + C \int_{Ω} {| u_{n} (T) |}^{p} d x \\ ⩽ C + {‖ u_{n} (τ) ‖}^{2} + C \int_{Ω} {| u_{n} (τ) |}^{p} d x + \int_{τ}^{T} {| g (s) |}^{2} d s . \end{matrix} \end{matrix}$ It follows from (3.26) that $\begin{matrix} \frac{d u_{n}}{d t} is uniformly bounded in L^{2} (τ, T; H), \end{matrix}$ and $\begin{matrix} u_{n} is uniformly bounded in L^{\infty} (τ, T; L^{p} (Ω)) . \end{matrix}$

By extracting a subsequence, it implies that $\begin{matrix} u \in L^{\infty} (τ, T; L^{p} (Ω)) \cap L^{2} (τ, T; D ({(- Δ)}^{α})) and \frac{d u}{d t} \in L^{2} (τ, T; H) . \end{matrix}$ According to theorem ([9, II Theorem 1.5]), we obtain that $C (τ, T; V)$ .

The uniqueness of strong solution follows from Theorem 3.1 by similar technique, here we skip the details. We complete the proof. □
Remark 3.1.
Similar to the classical reaction-diffusion equation with normal diffusion, if without further restriction on order p of the nonlinear term, then we cannot prove the map $u_{τ} ⟶ u (t)$ is continuous in V, this is due to the embedding (Lemma 2.3).

4. Pullback dynamics for problem (1.2)–(1.4)

4.1. The abstract theory of pullback attractors

In this section, we recall the abstract theory of the pullback attractor for non-autonomous systems, for more details, see [6,8,32].

Let E be a Banach space with norm $‖ \cdot ‖_{E}$ . The collection of the bounded sets of E represented by $B (E)$ and ${dist}_{E} (A, B)$ denotes the semi-distance between two sets $A \subset E$ and $B \subset E$ . ${D (t)} = {D (t)}_{t \in R}$ also represents a family of bounded sets in E. A two-parameter family of mappings ${U (t, τ)} = {U (t, τ) | t ⩾ τ, τ \in R}$ , $U (t, τ) : E ⟶ E$ is considered.

Definition 4.1.
A family ${U (t, τ)} = {U (t, τ) | t ⩾ τ, τ \in R}$ is a process in E if $\begin{array}{l} (4.1) & U (t, s) \circ U (s, τ) = U (t, τ), \forall t ⩾ s ⩾ τ . \\ (4.2) & U (τ, τ) = Id (identity), \forall τ \in R . \end{array}$

We say ${U (t, τ)}$ is continuous in E if $(t, τ, x) \mapsto U (t, τ) x$ is a continuous map from ${(t, τ) \in R^{2} | t ⩾ τ} \times E$ to E. If $U (t, τ) x_{n}$ weakly convergent to $U (t, τ) x$ in E while $x_{n}$ strongly convergent to x in E as $n ⟶ \infty$ , then we say ${U (t, τ)}$ is norm-to-weak continuous.

Let ${U (t, τ)}$ be a family of processes defined in Banach space E.
Definition 4.2.
A family of sets ${B (t)}$ is said to be a family of pullback absorbing sets with respect to the process $U (t, τ)$ if there exists a time $T = T (B, t) ⩽ t$ for all $t, τ \in R$ , $t ⩾ τ$ , and each $B \in B (E)$ , such that $\begin{matrix} U (t, τ) B \subseteq B (t), \end{matrix}$ for all $τ ⩽ T$ .
Definition 4.3.
A family of compact sets $A = {A (t)}$ is the pullback attractor for ${U (t, τ)}$ in E, if it satisfies the following conditions:
Invariance: $U (t, τ) A (τ) = A (t)$ for all $t ⩾ τ$ ,

Pullback attraction: given $t \in R$ , ${lim}_{τ \to - \infty} {dist}_{E} (U (t, τ) B, A (t)) = 0$ for any $B \in B (E)$ .

Minimality: $A (t) \subset C (t)$ for all $t \in R$ if a family of closed sets ${C (t)}$ exists, such that pullback attracts bounded sets of E.

The notion of a pullback asymptotically compact process is naturally associated with processes which posses a pullback attractor.
Definition 4.4.
${U (t, τ)}$ is said to be pullback asymptotically compact in E if for each $t \in R$ , sequence ${τ_{n}}$ , ${τ_{n}} ⟶ - \infty$ as $n ⟶ \infty$ and bounded sequence ${x_{n}} \subset E$ , $U (t, τ_{n}) x_{n}$ has a convergent subsequence.
Theorem 4.1 ([6,8]).

Let ${U (t, τ)}$ be a family of processes in E. If ${U (t, τ)}$ is pullback asymptotically compact and continuous (or norm-to-weak continuous), then ${U (t, τ)}$ possesses a pullback attractor $A = {A (t)}$ and $\begin{matrix} A (t) = \overline{⋃ {ω (B, t) : B \in B (E)}}, \end{matrix}$ where $ω (B, t)$ is pullback ω-limit set of B at time t, which defined by $\begin{matrix} ω (B, t) : = ⋂_{s ⩽ t} \overline{⋃_{τ ⩽ s} U (t, τ) B}, \end{matrix}$ or equivalently $\begin{array}{l} ω (B, t) & = {x \in E | there are sequences {s_{k}}_{1}^{\infty} \subset (- \infty, t], s_{k} ⟶ - \infty, as k ⟶ \infty \\ and {x_{k}}_{1}^{\infty} \subset B, such that lim_{k ⟶ \infty} U (t, s_{k}) x_{k} = x} . \end{array}$

According to Theorem 4.1, how to verify the process ${U (t, τ)}$ pullback asymptotically compact is crucial for prove the existence of pullback attractor. A very useful method that was presented in papers [23,32] use the non-compactness measure, which is called pullback condition- $(C)$ .

Definition 4.5.
The family of processes ${U (t, τ)}$ in Banach space E verifies the pullback condition-(C) if for any $t \in R$ , $B \in B (E)$ and $ε > 0$ there exists a $τ_{1} = τ_{1} (t, B, ε)$ such that
the set $⋃_{τ ⩽ τ_{1}} U (t, τ) B$ is bounded, and

there exists a finite dimensional space $E_{n}$ and a bounded projection $P_{n} : E ⟶ E_{n}$ such that $‖ (I - P_{n}) ⋃_{τ ⩽ τ_{1}} U (t, τ) B ‖_{E} ⩽ ε$ .

Theorem 4.2 ([22,23,31,32]).

Assume that the family of processes ${U (t, τ)}$ possesses a family of pullback absorbing set in E and it is continuous or norm-to-weak continuous. If ${U (t, τ)}$ satisfies pullback condition-(C), then it have a family of pullback attractor in E, if the space E is a uniformly convex Banach space, then the converse is also true.

4.2. Pullback attractors for problem (1.2)

In this subsection we consider the long time behavior of solutions of problem (1.2)-(1.4). We have obtained existence and uniqueness of solutions and their continuous dependence on initial datum. So by Theorems 3.1, 3.2, we can define processes ${U (t, τ) | t ⩾ τ, τ \in R}$ in spaces H and V by $\begin{matrix} U (t, τ) : H ⟶ H, u_{τ} ⟶ u (t) \end{matrix}$ and $\begin{matrix} U (t, τ) : V ⟶ V, u_{τ} ⟶ u (t), \end{matrix}$ respectively, which only continuous in H, $u (t)$ is the solution of problem with initial data $u_{τ}$ .

Assumption A.
The external force $g = g (x, t)$ belonging to $L_{loc}^{2} (R; H)$ and satisfy $\begin{matrix} (4.3) & \int_{- \infty}^{t} e^{σ s} {| g (s) |}^{2} d s < \infty \end{matrix}$ for $t \in R$ , here σ is a positive constant which will be determined later.
Assumption B.
We suppose that for any $t \in R$ , there exist constants $M ⩾ 0$ and $κ ⩾ 0$ that determined latter, such that $\begin{matrix} {| g (t) |}^{2} ⩽ M e^{κ | t |} . \end{matrix}$
Lemma 4.1.
Assume that g belongs to $L_{loc}^{2} (R, H)$ , f satisfies conditions ( 1.3 ) and ( 1.4 ). Then the following estimates hold $\begin{array}{l} (4.4) & \frac{d | u |^{2}}{d t} + ‖ u ‖^{2} ⩽ C {| g (t) |}^{2} . \\ (4.5) & \frac{d ‖ u ‖^{2}}{d t} + {| {(- Δ)}^{α} u |}^{2} ⩽ C ‖ u ‖^{2} + {| g (t) |}^{2} . \end{array}$
Proof.
Multiplying equation (1.2) by u and ${(- Δ)}^{α} u$ . Then integrating over Ω, also using the same estimates as proving Theorems 3.1 and 3.2 (in Galerkin’s approximation sense), we have inequalities (4.4) and (4.5) respectively. We complete the proof. □

Because of the embedding $H_{0}^{α} (Ω) \subset L^{2} (Ω)$ is compact (Lemma 2.1), so by Theorem 4.1, we have the following result.
Theorem 4.3.
( $L^{2}$ pullback attractor) Let $α \in (0, 1)$ , $n > 2 α$ , Ω be an open bounded set of $R^{n}$ . Assume that g belongs to $L_{loc}^{2} (R, H)$ and satisfies $\int_{- \infty}^{t} e^{σ s} | g (s) |^{2} d s < \infty$ , where σ is a positive constant. It is also assumed that f satisfies conditions ( 1.3 )–( 1.5 ). Then the process associated to problem ( 1.2 ) has a family of pullback attractors ${A_{H} (t)}$ in $L^{2} (Ω)$ .
Proof.
Combining the compact embedding and existence of pullback absorbing set, also the process is continuous in H, we can derive the theorem easily. □

According to Theorem 4.1, some continuity of process is necessary for the existence of pullback attractor, however, strong continuity of process is not available in space V (see Remark 3.1). Fortunately, the process is norm-to-weak continuous ([23, Theorem 3.8]), since by Lemma 4.1, the process maps a compact subset of V to a bounded set of V.

In order to prove the existence of pullback attractor in V, we shall use a decomposition of an element that belongs to V by using the eigenfunctions of nonlocal operator ${(- Δ)}^{α}$ . By Lemma 2.3, let $V_{m} = span {e_{1}, e_{2}, \dots, e_{m}}$ in V and $P_{m} : V ⟶ V_{m}$ be the orthogonal projector. Then for any $u \in V$ , u has a unique decomposition: $u = u_{1} + u_{2}$ , $u_{1} = P_{m} u$ , $u_{2} = (I - P_{m}) u$ . Therefore we are ready to state the following theorem. Theorem 4.4 ( $H_{0}^{α}$ pullback attractor).

Let $α \in (0, 1)$ , $n > 2 α$ , $2 ⩽ p ⩽ \frac{2 (n - α)}{n - 2 α}$ , Ω be an open bounded set of $R^{n}$ . Assume that g satisfy Assumptions A and B . Then, the process associated to the problem ( 1.2 ) has a family of pullback attractors ${A_{V} (t)}$ in V.

Proof.
According to Lemma 4.1 and Theorem 4.2, we only need to verify pullback condition- $(C)$ . The validity of first condition of Definition 4.5 is obviously. We only need to prove the second condition of Definition 4.5 in sequel.

We decompose equation (1.2) as $\begin{matrix} (4.6) & \frac{d u_{2} (t)}{d t} + {(- Δ)}^{α} u_{2} (t) + f (u) - P_{m} f (u) = (I - P_{m}) g (t) \end{matrix}$ and $\begin{matrix} (4.7) & \frac{d u_{1} (t)}{d t} + {(- Δ)}^{α} u_{1} (t) + P_{m} f (u) = P_{m} g (t) . \end{matrix}$ Taking inner product of equation (4.6) with ${(- Δ)}^{α} u_{2} (t)$ in H, we have $\begin{matrix} (4.8) & \frac{1}{2} \frac{d (‖ u_{2} ‖^{2})}{d t} + {| {(- Δ)}^{α} u_{2} |}^{2} = - (f (u), {(- Δ)}^{α} u_{2}) + (g (t), {(- Δ)}^{α} u_{2}) . \end{matrix}$

We estimate each terms of right hand side of equation (4.8). Due to conditions (1.3) and (1.4), we have $| f (u) | ⩽ C (| u |^{p - 1} + 1)$ , also thanks to Young’s inequality and Lemma 2.1, the following estimate holds $\begin{array}{l} | (f (u), {(- Δ)}^{α} u_{2}) | & ⩽ | f (u) | | {(- Δ)}^{α} u_{2} | ⩽ \frac{1}{4} {| {(- Δ)}^{α} u_{2} |}^{2} + {| f (u) |}^{2} \\ ⩽ \frac{1}{4} {| {(- Δ)}^{α} u_{2} |}^{2} + C \int_{Ω} {(| u |^{p - 1} + 1)}^{2} d x \\ ⩽ \frac{1}{4} {| {(- Δ)}^{α} u_{2} |}^{2} + C + C ‖ u ‖_{2 p - 2}^{2 p - 2} \\ (4.9) & ⩽ \frac{1}{4} {| {(- Δ)}^{α} u_{2} |}^{2} + C + C ‖ u ‖^{2 p - 2} . \end{array}$ Using Young’s inequality, we have $\begin{array}{l} (4.10) & | (g (t), {(- Δ)}^{α} u_{2}) | ⩽ | g (t) | | {(- Δ)}^{α} u_{2} | ⩽ \frac{1}{4} {| {(- Δ)}^{α} u_{2} |}^{2} + {| g (t) |}^{2} . \end{array}$ Substituting (4.9) and (4.10) into (4.8), it holds $\begin{array}{l} (4.11) & \frac{d (‖ u_{2} ‖^{2})}{d t} + λ_{m + 1} ‖ u_{2} ‖^{2} ⩽ C + C ‖ u ‖^{2 p - 2} + 2 {| g (t) |}^{2} . \end{array}$ Thanks to uniform Gronwall’s inequality, we deduce from the above inequality that $\begin{array}{l} ‖ u_{2} ‖^{2} & ⩽ e^{- λ_{m + 1} (t - τ)} ‖ u_{τ} ‖^{2} + \int_{τ}^{t} e^{- λ_{m + 1} (t - s)} (C + 2 {| g (s) |}^{2}) d s \\ (4.12) & + C \int_{τ}^{t} e^{- λ_{m + 1} (t - s)} ‖ u ‖^{2 p - 2} d s . \end{array}$

At last, we shall estimate each term on the right hand side of above inequality, which makes the process is asymptotically compact by pullback condition- $(C)$ technique.

Thanks to (4.4) and nonlocal Poincaré’s inequality, we have $\begin{matrix} (4.13) & | u |^{2} ⩽ e^{- C_{4} (t - τ)} | u_{τ} |^{2} + C \int_{τ}^{t} e^{C_{4} (s - t)} {| g (s) |}^{2} d s . \end{matrix}$ Multiply above equality by $e^{C_{4} t}$ and integrating τ to t, we get $\begin{matrix} (4.14) & \int_{τ}^{t} e^{C_{4} s} {| u (s) |}^{2} d s ⩽ (t - τ) e^{C_{4} τ} | u_{τ} |^{2} + C \int_{τ}^{t} \int_{τ}^{r} e^{C_{4} s} {| g (s) |}^{2} d s d r . \end{matrix}$ By (4.5), we get $\begin{array}{l} \frac{d ((t - τ) e^{C_{4} t} ‖ u (t) ‖^{2})}{d t} & ⩽ e^{C_{4} t} {‖ u (t) ‖}^{2} + C (t - τ) e^{C_{4} t} {‖ u (t) ‖}^{2} + (t - τ) e^{C_{4} t} \frac{d ‖ u (t) ‖^{2}}{d t} \\ ⩽ e^{C_{4} t} {‖ u (t) ‖}^{2} + C (t - τ) e^{C_{4} t} {‖ u (t) ‖}^{2} + C (t - τ) e^{C_{4} t} {‖ u (t) ‖}^{2} \\ + (t - τ) e^{C_{4} t} {| g (t) |}^{2} \\ (4.15) & = C (t - τ) e^{C_{4} t} {‖ u (t) ‖}^{2} + (t - τ) e^{C_{4} t} {| g (t) |}^{2} . \end{array}$ Integrating above inequality from τ to t, we deduce that $\begin{matrix} (4.16) & {‖ u (t) ‖}^{2} ⩽ C e^{- C_{4} t} \int_{τ}^{t} e^{C_{4} s} {‖ u (s) ‖}^{2} d s + C e^{- C_{4} t} \int_{τ}^{t} e^{C_{4} s} {| g (s) |}^{2} d s . \end{matrix}$

We compute by (4.4) obtain that $\begin{matrix} (4.17) & \frac{d (e^{C_{4} t} | u (t) |^{2})}{d t} + e^{C_{4} t} {‖ u (t) ‖}^{2} ⩽ C e^{C_{4} t} {| u (t) |}^{2} + C e^{C_{4} t} {| g (t) |}^{2} . \end{matrix}$ Integrating above inequality from τ to t, and thanks to (4.14), we get $\begin{array}{l} \int_{τ}^{t} e^{C_{4} s} {‖ u (s) ‖}^{2} d s & ⩽ e^{C_{4} τ} {| u (τ) |}^{2} + C \int_{τ}^{t} e^{C_{4} s} {| u (s) |}^{2} d s + C \int_{τ}^{t} e^{C_{4} s} {| g (s) |}^{2} d s \\ ⩽ e^{C_{4} τ} {| u (τ) |}^{2} + C (t - τ) e^{C_{4} τ} | u_{τ} |^{2} + C \int_{τ}^{t} \int_{τ}^{r} e^{C_{4} s} {| g (s) |}^{2} d s d r \\ (4.18) & + C \int_{τ}^{t} e^{C_{4} s} {| g (s) |}^{2} d s . \end{array}$

Combining (4.16) and (4.18), it implies that there exists some $τ_{1}$ such that when $τ ⩽ τ_{1}$ $\begin{matrix} (4.19) & {‖ u (t) ‖}^{2} ⩽ C e^{- C_{4} t} \int_{τ}^{t} \int_{τ}^{r} e^{C_{4} s} {| g (s) |}^{2} d s d r + C e^{- C_{4} t} \int_{τ}^{t} e^{C_{4} s} {| g (s) |}^{2} d s . \end{matrix}$ By Assumption B, we choose $κ < C_{4}$ , all the terms in the right hand side of above inequality are finite, so we let $\begin{matrix} r (t) = C e^{- C_{4} t} \int_{τ}^{t} \int_{τ}^{r} e^{C_{4} s} {| g (s) |}^{2} d s d r + C e^{- C_{4} t} \int_{τ}^{t} e^{C_{4} s} {| g (s) |}^{2} d s . \end{matrix}$

For arbitrary $ε > 0$ , there exists a $τ_{2}$ such that $\begin{matrix} (4.20) & e^{- λ_{m + 1} (t - τ)} ‖ u_{τ} ‖^{2} < ε, \end{matrix}$ if $τ ⩽ τ_{2}$ , and $\begin{matrix} (4.21) & C \int_{τ}^{t} e^{- λ_{m + 1} (t - s)} d s = C \frac{1 - e^{- λ_{m + 1} (t - τ)}}{λ_{m + 1}} < ε, \end{matrix}$ for some $m ⩾ m_{1}$ .

By Assumption A, we choose $σ = λ_{m + 1}$ . For arbitrary $ε > 0$ , there exists some $m_{2}$ such that $\begin{array}{l} (4.22) & 2 \int_{τ}^{t} e^{- λ_{m + 1} (t - s)} {| g (s) |}^{2} d s ⩽ 2 M e^{- λ_{m} t} \int_{- \infty}^{t} e^{λ_{m + 1} s} e^{α | s |} d s < ε, \end{array}$ for $m ⩾ m_{2}$ .

By estimate (4.19), there exists $m_{3}$ such that when $m ⩾ m_{3}$ , we have $\begin{array}{l} (4.23) & C \int_{τ}^{t} e^{- λ_{m + 1} (t - s)} ‖ u ‖^{2 p - 2} d s = C e^{- λ_{m + 1} t} \int_{τ}^{t} e^{λ_{m + 1} s} {(r (ξ))}^{p - 1} d ξ < ε . \end{array}$

Choosing $τ_{0} = min {τ_{1}, τ_{2}}$ , $m_{0} = max {m_{1}, m_{2}, m_{3}}$ , choose $κ < C_{4}$ , then we conclude that $\begin{matrix} (4.24) & ‖ u_{2} ‖^{2} < ε, \end{matrix}$ for $τ ⩽ τ_{0}$ and $m > m_{0}$ , which implies the pullback asymptotically compactness for process, i.e., our result hold. □
Remark 4.1.
If the external force satisfies the pullback tempered condition, a weaker hypothesis as in [33], the existence of a family of pullback attractors still holds.

5. Conclusion and further research

In the above sections, we have presented the well-posedness and pullback dynamics for a non-autonomous reaction-diffusion equations with anomalous diffusion. In [18], the authors considered a class of semilinear fractional kinetic equation like $\begin{matrix} (5.1) & \frac{d^{c} u}{d t^{c}} = A u + f (x, t, u), \end{matrix}$ with suitable boundary, initial condition and growth condition on f, where A is a class of diffusion operators that include fractional Laplacian operator we considered in this paper, $\frac{d^{c} u}{d t^{c}}$ is the time derivative of u in the sense of Caputo. In final chapter, there are several remarks and interesting problems presented by authors, one of problems is to investigate the long-term behavior of (5.1) in terms of global attractors and ω-limits set. Our model considered in this paper is a particular case when $c = 1$ in (5.1), so it is worth going to study the equation (5.1) in the point view of infinite dimensional dynamical system.

To obtain the results in this paper, we impose some restriction conditions on nonlinearity and external force. We can remove that restriction conditions along the ideas in [31]. The key is that we need the $L^{p}$ estimate of solution. However, the following result

Lemma 5.1 (Positivity Lemma ([4,13])).

Suppose that $α \in (0, 1)$ , u and ${(- Δ)}^{α} u \in L^{p} (R^{n})$ , where $p ⩾ 2$ . Then $\begin{matrix} \int_{Ω} | u |^{p - 2} u {(- Δ)}^{α} u d x ⩾ \frac{2}{p} \int_{Ω} {({(- Δ)}^{\frac{α}{2}} | u |^{\frac{p}{2}})}^{2} d x \end{matrix}$

isn’t hold, because of loss of regularity. We will do it in forthcoming paper.

Footnotes

Acknowledgements

Xingjie Yan was partially supported by the Nature Science Foundation of Jiangsu Province (No. BK20130170). Xin-Guang Yang was partially supported by the Fund of Young Backbone Teachers in Henan Province (No. 2018GGJS039), Incubation Fund Project of Henan Normal University (No. 2020PL17) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003).

Xingjie Yan would like to thank Professors Jinqiao Duan, Meihua Yang and Dr. Jinchun He for helpful discussion when he was visiting the Center for Mathematical Sciences, Huazhong University of Science and Technology.

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