Extremal solutions of quasilinear parabolic subdifferential inclusions.
The authors consider the following initial boundary value problem for the quasilinear parabolic inclusion: \[ \frac{\partial u}{\partial t}+Au+\beta(\cdot,u)\ni Fu+h \quad {\text{in}} \;Q, \qquad u(x,0)=0\;{\text{in}}\;\Omega \quad{\text{and}}\;u=0\;{\text{on}}\;\Gamma, \] where \(\Omega\subseteq\mathbb{R}^N\) is a bounded domain with Lipschitz boundary \(\partial\Omega\), \(Q=\Omega\times(0,\tau)\), \(\Gamma=\partial\Omega\times(0,\tau)\), \(\tau>0\), \(A\) is a second order quasilinear differential operator in divergence form, \(F\) is the Nemytskii operator corresponding to the Carathéodory function \(f\colon Q\times\mathbb{R}\rightarrow\mathbb{R}\), \(\beta(x,t,\cdot):\mathbb{R}\rightarrow 2^{\mathbb{R}}\setminus\{\emptyset\}\) is a maximal monotone multifunction. Under an appropriate set of assumptions and using regularization methods, comparison and truncation techniques and special test function techniques, the existence of extremal solutions within a sector formed by upper and lower solutions is shown.
scientific article; zbMATH DE number 2004899