Regularity of Global Attractor for the Reaction-Diffusion Equation
By using an iteration procedure, regularity estimates for the linear semigroups, and a classical existence theorem of global attractor, we prove that the reaction-diffusion equation possesses a global attractor in Sobolev space for all , which attracts any bounded subset of () in the -norm.
This paper is concerned with the following initial-boundary problem of reaction-diffusion systems involving an unknown function : where is a given constant. is the Laplace operator. denotes an open bounded set of with smooth boundary . is a polynomial on , which is given by where should be an odd number, that is, , and
The reaction-diffusion systems (1.1) have been extensively studied during the last decades, one of the motivations being that such systems could account for phenomena occurring in living organisms. In 1952, Turing  proposed that a combination of chemical reaction and diffusion produces spatial patterns of chemical concentration, under certain conditions. Such patterns are of interest because they give a possible explanation for the development of pattern and form in developmental biology [2–4] and experimental chemical systems . Schneider et al. [6–9] have studied existence of periodic travelling wave solutions and positive periodic solution of reaction-diffusion systems. In [10–17], asymptotic behaviour of the nonlinear reaction-diffusion equation, such as global attractors, inertial manifolds, and approximate inertial manifolds, has been studied.
The global asymptotical behaviors of solutions and existence of global attractors are important for the study of the dynamical properties of general nonlinear dissipative dynamical systems. So, many authors are interested in the existence of global attractors such as [12–19]. As for the reaction-diffusion equation (1.1), the existence of global solutions and global attractors in has been proved by Temam , Marion [18, 19], and Zhong et al. . For convenience, we introduce the main results as follows.
Lemma 1.1. Under the conditions (1.2) and (1.3) for , the following three claims hold.(1)Equation (1.1) has a unique global weak solution , for ;(2)Equation (1.1) has a unique strong solution for any ;(3)Equation (1.1) has a global attractor , which attracts any bounded set of in the -norm.
Here the spaces and are defined as follows:
In this paper, we shall use the regularity estimates for the linear semigroups, combining with the classical existence theorem of global attractors, to prove that the reaction-diffusion equation possesses, in any th differentiable function spaces , a global attractor, which attracts any bounded set of in norm. The basic idea is an iteration procedure, which is from recent books and papers [20–24].
Let and be two Banach spaces and a compact and dense inclusion. Consider the abstract nonlinear evolution equation defined on , given by where is an unknown function, a linear operator, and a nonlinear operator.
A family of operators is called a semigroup generated by (2.1) provided that satisfies the properties:(1) is a continuous mapping for any ;(2) the identity; (3), and the solution of (2.1) can be expressed as
Next, we introduce the concepts and definitions of invariant sets, global attractors, limit sets for the semigroup .
Definition 2.1. Let be a semigroup defined on . A set is called an invariant set of if . An invariant set is an attractor of if is compact, and there exists a neighborhood of such that for any :
In this case, we say that attracts . Especially, if attracts any bounded set of , is called a global attractor of .
For a set , we define the limit set of as follows: where the closure is taken in the -norm. The following Lemma 2.2 is the classical existence theorem of global attractor by Temam .
Lemma 2.2. Let be the semigroup generated by (2.1). Assume that the following conditions hold:(1) has a bounded absorbing set , that is, for any bounded set there exists a time such that for all and ;(2) is uniformly compact, that is, for any bounded set and some sufficiently large, the set is compact in .
Then, that the limit set of is a global attractor of (2.1), and is connected providing B is connected.
Note that we used to assume that the linear operator in (2.1) is a sectorial operator which generates an analytic semigroup . It is known that there exists a constant such that generates the fractional power operators and fractional order spaces for , where . Without loss of generality, we assume that generates the fractional power operators and fractional order spaces as follows: where is the domain of . By the semigroup theory of linear operators (Pazy ), we know that is a compact inclusion for any .
Thus, Lemma 2.2 can be equivalently expressed in the following Lemma 2.3 .
Lemma 2.3. Let be a solution of (2.1) and the semigroup generated by (2.1). Let be the fractional order space generated by . Assume(1)for some , there is a bounded set ; for any , there exists such that
(2)there is a , for any bounded set there are and such that
Then (2.1) has a global attractor , which attracts any bounded set of in the -norm.
For sectorial operators, we also have the following properties, which can be found in .
Lemma 2.4. Let be a sectorial operator which generates an analytic semigroup . If all eigenvalues of satisfy for some real number , then for we have (1) is bounded for all and ;(2) for all ;(3) for each is bounded, and where some is a constant only depending on ;(4) the norm can be defined by (5) if is symmetric, for any , we have
3. Main Results
Let and be the spaces defined as in (1.4). We define the operators and by where is the same as one of (1.2). Thus, the reaction-diffusion equation (1.1) can be written in the abstract form (2.1). It is well known that the linear operator given by (3.1) is a sectorial operator and . The space is the same as (1.4), is given by closure of in , and for .
The main result in this paper is given by the following theorem, which provides the existence of global attractors of the reaction-diffusion equation (1.1) in any th order space .
Theorem 3.1. Let the function be a polynomial of order
with leading coefficient
Assume for . Then, for any (1.1) has a global attractor in , and attracts any bounded set of in the -norm.
Proof. From Lemma 1.1, we know that the solution of system (1.1) is a weak solution for any . Hence, the solution of system (1.1) can be written as
By (3.1), we rewrite (3.4) as
Next, according to Lemma 2.3, we prove Theorem 3.1 in the following six steps.
Step 1. We prove that for any bounded set there is a constant such that the solution of system (1.1) is uniformly bounded by the constant for any and . To do that, we firstly check that system (1.1) has a global Lyapunov function as follows: where
In fact, if is a weak solution of system (1.1), we have
By (3.1) and (3.6), we get
Hence, it follows from (3.8) and (3.9) that which implies that (3.6) is a Lyapunov function.
Integrating (3.10) from 0 to gives
Using (1.2) and (3.6), we have Choosing such that , and noting that is an odd number, that is, , we get Combining with (3.11) yields which implies where , and are positive constants. only depends on .
Step 2. We prove that for any bounded set there exists such that By , we have which implies that is bounded.
Hence, it follows from (2.9) and (3.5) that where . Hence, (3.16) holds.
Step 3. We prove that for any bounded set there exists such that In fact, by the embedding theorems of fractional order spaces : we have which implies Therefore, it follows from (3.16) and (3.22) that
Then, by using same method as that in Step 2, we get from (3.23) that where . Hence, (3.19) holds.
Step 4. We prove that for any bounded set there exists such that
In fact, by the embedding theorems of fractional order spaces :
we have which implies that
Therefore, it follows from (3.19) and (3.28) that
Then, we get from (3.29) that where . Hence, (3.25) holds.
Step 5. We prove that for any bounded set there exists such that In fact, by the embedding theorems of fractional order spaces : we have which implies that
Therefore, it follows from (3.25) and (3.34) that
Then, we get from (3.35) that where . Hence, (3.31) holds for .
By doing the same procedures as Steps 1–4, we can prove that (3.31) holds for all .
Step 6. We show that for any , system (1.1) has a bounded absorbing set in . We first consider the case of .
It is well known that the reaction-diffusion equation possesses a global attractor in space, and the global attractor of this equation consists of equilibria with their stable and unstable manifolds. Thus, each trajectory has to converge to a critical point. From (3.31) and (3.10), we deduce that for any the solution of system (1.1) converges to a critical point of . Hence, we only need to prove the following two properties: (1); (2)the set is bounded.
Property (1) is obviously true, we now prove property (2) in the following. It is easy to check if , is a solution of the following equation: where is given by (1.2). Taking the scalar product of (3.37) with , then we derive that
By (1.3) and (3.38), we have
Using Hölder inequality and the above inequality, we have where is a constant. Thus, property (2) is proved.
Now, we show that system (1.1) has a bounded absorbing set in for any , that is, for any bounded set there are and a constant independent of such that
From the above discussion, we know that (3.41) holds as . By (3.5), we have
Let be the bounded absorbing set of system (1.1), and such that
It is well known that where is the first eigenvalue of the equation: Hence, for any given and , we have From (3.42)-(3.43) and Lemma 2.4 for any , we get that where is a constant independent of .
Then, we infer from (3.46) and (3.47) that (3.41) holds for all . By the iteration method, we have that (3.41) holds for all .
Finally, this theorem follows from (3.31)–(3.41) and Lemma 2.3. The proof is completed.
This work is funded by the National Natural Science Foundation of China (no. 11071177) and the NSF of Sichuan Education Department of China (no. 11ZA1102).
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