Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems
The author studies the existence and global boundedness of the solutions to the double obstacle problem: Find \(u\in C([s,+\infty);L^2(\Omega))\) and \(q\in L^2_{\text{loc}} ((s,+\infty);L^2(\Omega))\) such that \[ \begin{gathered} lu'(t)-\text{div}(| \nabla u(t)| ^{p-2}\nabla u(t))-g(u(t))=q(t,x)\quad \text{in }Q_s=[s,+\infty)\times \Omega,\\ 0\leq q(t,x)\leq h(t,u(t,x)),\quad \text{a.e. on }(s,+\infty)\times \Omega,\\ u(t)=l(t),\quad \text{a.e. on }(s,+\infty)\times\Gamma,\\ u(s)=u_0 \quad \text{in }\Omega, \end{gathered}\leqno(P_s) \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\Gamma\). He proves the asymptotic stability for \((P_s)\), by constructing global attractors for the multivalued dynamical systems associated with \((P_s)\) and with the corresponding limiting autonomous problem.
scientific article; zbMATH DE number 2046270