Upper semicontinuity of uniform attractors for nonclassical diffusion equations

Abstract

We study the upper semicontinuity of a uniform attractor for a nonautonomous nonclassical diffusion equation with critical nonlinearity. In particular, we prove that the uniform (with respect to (w.r.t.) g ∈ ) attractor A ε (ε ≥ 0) for equation (1.1) satisfies lim ε→ε 0 dist H 1 0 ( ) (A ε , A ε 0 ) = 0 for any ε 0 ≥ 0. MSC: 37L05; 35B40; 35B41

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

“…And the long-time behavior of solutions of Eq. (1.1) has been considered by some researchers (see, e.g., [20,21] and the references therein).…”

“…Therefore, it is natural to examine the limiting behavior of solutions to Eq. (1.1) as ν → 0 (see, e.g., [20][21][22][23][24] and the references therein). For example, in [24], the authors studied the existence of global attractors in D(A) (generated by strong solutions) and their upper semicontinuity in H 1 0 ( ) for Eq.…”

“…(1.1) with subcritical nonlinearity was considered. In [20], upper semicontinuity of uniform attractors for Eq. (1.1) was considered, and the nonlinearity satisfies critical exponential growth conditions.…”

“…(1.1) with memory [3] or in unbounded domain [1,2,23]. Particularly, the nonlinearity f is assumed to satisfy the polynomial growth of arbitrary order instead of critical nonlinearity (see, e.g., [20,23]).…”

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

“…We assume that g ∈ L 2 (Ω) and the nonlinearity f ∈ C 1 (R, R) satisfies the following (see, e.g., [1]): Nonclassical diffusion equations appear in fluid mechanics, soil mechanics, and heat conduction theory (see, e.g., [2]). e long-time behavior of solutions to nonclassical diffusion equations has been extensively studied by many authors for both autonomous and nonautonomous cases [3][4][5][6][7][8][9]. e global attractor plays an important role in the study of long-time behavior of infinite dimension systems arising from physics and mechanics.…”