Based on the recent theory of time-dependent global attractors in the works of Conti et al. (2013) and di Plinio et al. (2011), we prove the existence of time-dependent global attractors as well as the regularity of the time-dependent global attractor for a class of nonclassical parabolic equations.
Let
The nonlinearity
The classical reaction diffusion equation has a long history in mathematical physics and appears in many mathematical models. It arises in several bead mark problems of hydrodynamics and heat transfer theory, such as heat transfer, as well as in solid-fluid in Hradionconfigurations and, of course, in standard situations mass diffusion and flow through porous media [
In the case when
The paper is organized as follows. In Section
In this section, we introduce some notations and definitions, along with a lemma.
We set
Note that the spaces
According to (
The following inequalities hold for some
For any
Using the Galerkin approximation method, we can obtain the following result concerning the existence and uniqueness of solutions; see, for example, [
Under the assumptions of (
According to Lemma
A time-dependent absorbing set for the process
Under the assumptions of (
Multiplying (
Under the assumptions of (
From the proof of Lemma
We can assume that the time-dependent absorbing set
As introduced in [
One calls a time-dependent global attractor the smallest element of
If
According to Definition
For the nonlinearity
Noting that
Under assumptions of (
Multiplying (
From Lemmas
Under the assumptions of (
Multiplying (
From Lemma
If
If the process
Under the assumptions of (
According to Lemma
We fix
Under assumptions of (
Multiplying (
Denoting by
This completes the proof.
Therefore, we have the following regularity result.
Under the assumptions of (
In fact, we define
From Theorem
Hence,
The author declares that there is no conflict of interests regarding the publication of this paper.
The author expresses her sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. She also thanks the editors for their kind help.