Parameter dependent quasi-linear parabolic equations
The second-order nonlinear partial differential equation: \[ \begin{gathered} {d\over dt} u^\varepsilon(t)- \text{div}(a^\varepsilon|\nabla u^\varepsilon(t)|^{p- 2}\nabla u^\varepsilon(t))= Bu^\varepsilon(t),\quad t> 0,\\ u^\varepsilon(0)= u_0\in W^{1,p}_0(\Omega)\end{gathered}\tag{1} \] is considered. There \(\Omega\subset\mathbb{R}^n\) is a bounded domain with smooth boundary, \(p> \max\{2,{n\over 2}\}\), \(a_\varepsilon\in L^\infty(\Omega)\), \(a^\varepsilon(x)\geq \delta> 0\) almost everywhere in \(\Omega\) for every \(\varepsilon\in[0,\varepsilon_0]\) and \(a^\varepsilon\) converges to \(a\) in \(L^\infty(\Omega)\) as \(\varepsilon\) goes to \(0\). Following the paper of M. Otani some conditions are assumed on the perturbation \(B\). They imply global existence of solutions of the problem (1) in \(W^{1,p}_0(\Omega)\) and the existence of global attractors \(A_\varepsilon\) in \(W^{1,p}_0(\Omega)\) for every \(\varepsilon\in [0,\varepsilon_0]\). In this paper the upper semicontinuity of the global attractors \(A_\varepsilon\) at \(\varepsilon= 0\) on the \(W^{1,p}_0(\Omega)\) topology is shown.
scientific article; zbMATH DE number 2136557