Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient.
Let \(\Omega \subset \mathbb{R}^N\) be a bounded domain with Lipschitz boundary \(\partial \Omega\); \(Q = \Omega \times (0,\tau)\) and \(\Gamma = \partial \Omega \times (0,\tau)\). The authors consider the following initial boundary value problem: \[ \begin{aligned} \frac{\partial u}{\partial t} + Au + \partial g(\cdot,\cdot,u) \ni Fu + h &\text{in }Q, \cr u(\cdot,0) = 0 &\text{in }\Omega \cr u = 0 &\text{on }\Gamma \end{aligned} \] where \(A\) is a second-order quasilinear differential operator in divergence form of Leray-Lions type, \(F\) is a Nemytski operator associated with a Carathéodory function \(f:Q \times \mathbb{R} \rightarrow \mathbb{R}\), and the function \(g:Q \times \mathbb{R} \rightarrow \mathbb{R}\) is locally Lipschitz in the last variable. The notation \(\partial g\) stands for the generalized Clarke gradient w.r.t. the third variable. It is proved that the problem admits extremal solutions within the order interval formed by given lower and upper solutions.
scientific article; zbMATH DE number 1915672