Continuous Dependence on a Parameter of Exponential Attractors for Nonclassical Diffusion Equations

Abstract

In this paper, a new abstract result is given to verify the continuity of exponential attractors with respect to a parameter for the underlying semigroup. We do not impose any compact embedding on the main assumptions in the abstract result which is different from the corresponding result established by Efendiev et al. in 2004. Consequently, it can be used for equations whose solutions have no higher regularity. As an application, we prove the continuity of exponential attractors in H01 for a class of nonclassical diffusion equations with initial datum in H01.

Strong pullback attractors for a nonclassical diffusion equation

“…Under a Sobolev type growth condition of the nonlinearity g, Lee and Toi [18] investigated the existence and upper semi-continuity of global attractors of nonclassical diffusion equation in a smooth bounded domain Ω with dynamic boundary condition. Wang and Hu [37] studied the continuity of exponential attractors in H 1 0 (Ω) when the nonlinearity g ∈ C 1 satisfies the some suitable assumptions. To be more precise, our motivation in this paper is to use the method of [30] to prove the global existence and uniqueness of strong solutions in H 2 (Ω) ∩ H 1 0 (Ω) to the non-autonomous nonclassical diffusion equations, then we prove the existence of pullback attractors in H 2 (Ω) ∩ H 1 0 (Ω) by applying asymptotic a priori estimate method [44], which is still an open problem before the present paper solved it.…”