## 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.

### 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
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### References:

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