Existence and upper semicontinuity of uniform attractors in $H^1(\mathbb {R}^N)$ for nonautonomous nonclassical diffusion equations
Volume 111 / 2014
Annales Polonici Mathematici 111 (2014), 271-295
MSC: Primary 35B41; Secondary 35K70, 35D30.
DOI: 10.4064/ap111-3-5
Abstract
We prove the existence of uniform attractors $\mathcal A_{\varepsilon }$ in the space $H^1(\mathbb {R}^N)$ for the nonautonomous nonclassical diffusion equation $$ u_t - \varepsilon \varDelta u_t - \varDelta u + f(x,u)+\lambda u = g(x,t),\hskip 1em \varepsilon \in [0,1]. $$ The upper semicontinuity of the uniform attractors $\{\mathcal A_{\varepsilon }\}_{\varepsilon \in [0,1]}$ at $\varepsilon = 0$ is also studied.
