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

Carl, S.; Motreanu, D.

J. Differ. Equations 191, No. 1, 206-233 (2003).

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.

Reviewer: Valerii V. Obukhovskij (Voronezh)

MSC:

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

Keywords:

quasilinear parabolic inclusion; initial boundary value problem; extremal solution; upper solution; lower solution; pseudo-monotone operator; generalized gradient; comparison

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