Extremal solutions of quasilinear parabolic inclusions with generalized Clarke’s gradient. (English) Zbl 1042.35092

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.


35R70 PDEs with multivalued right-hand sides
49J52 Nonsmooth analysis
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K30 Initial value problems for higher-order parabolic equations
35K35 Initial-boundary value problems for higher-order parabolic equations
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