Upper semicontinuity of uniform attractors for nonclassical diffusion equations

Abstract

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 ∈ Σ $g\in \Sigma $ ) attractor A Σ ε $\mathcal {A}^{\varepsilon }_{\Sigma }$ ( ε ⩾ 0 $\varepsilon \geqslant 0$ ) for equation (1.1) satisfies lim ε → ε 0 dist H 0 1 ( Ω ) ( A Σ ε , A Σ ε 0 ) = 0 $\lim_{\varepsilon \to \varepsilon _{0}}\operatorname{dist}_{H_{0}^{1}(\Omega )}(\mathcal {A}^{\varepsilon } _{\Sigma },\mathcal {A}^{\varepsilon _{0}}_{\Sigma })=0$ for any ε 0 ⩾ 0 $\varepsilon _{0}\geqslant 0$ .

Keywords

• nonautonomous
• nonclassical diffusion equation
• uniform attractor
• upper semicontinuity