On the convergence of solutions of multivalued parabolic equations and applications
The authors are investigating sequence of parabolic inclusions of the form \[ \partial u_k/\partial_t- \text{div }a_k(x, Du_k))\ni f(t,x)\tag{1} \] in \((0,T)\times \Omega\), where \(\Omega\subset \mathbb{R}^N\) is a bounded domain, \(0< T< \infty\), while the multivalued nonlinear maps \(a_k: \Omega\times \mathbb{R}^N\to 2^{\mathbb{R}^N}\) are measurable in \(x\), with \(a_k(x,\xi)\) maximal monotone in \(\xi\). It is also assumed that certain growth and coercivity conditions are satisfied. Under appropriate hypotheses on data, the authors establish two basic results. The first is an existence result for (1), with an initial condition on \(\Omega\) and with zero condition on the lateral boundary of \((0,T)\times\Omega\). The second result regards the convergence of solutions of (1), as \(k\to\infty\), to the solution of a limit equation, of the same form as (1). An application to an optimal control problem is also discussed.
scientific article; zbMATH DE number 1957311