×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Berkovits, J.; Mustonen, V., Monotonicity methods for nonlinear evolution equations, Nonlinear anal., 27, 1397-1405, (1996) · Zbl 0894.34055
[2] Brezis, H., Analyse fonctionelle. theorie et applications, (1983), Masson Paris
[3] Carl, S., Extremal solutions of parabolic hemivariational inequalities, Nonlinear anal., 47, 5077-5088, (2001) · Zbl 1042.35585
[4] Carl, S., A survey of recent results on the enclosure and extremality of solutions for quasilinear hemivariational inequalities, (), 15-28 · Zbl 1058.35134
[5] Carl, S., Existence of extremal solutions of boundary hemivariational inequalities, J. differential equations, 171, 370-396, (2001) · Zbl 1180.35255
[6] Carl, S.; Heikkilä, S., Nonlinear differential equations in ordered spaces, (2001), Chapman & Hall/CRC Boca Raton · Zbl 0974.35062
[7] Carl, S.; Motreanu, D., Extremal solutions of quasilinear parabolic subdifferential inclusions, Differential integral equations, 16, 241-255, (2003) · Zbl 1036.35112
[8] Chipot, M.; Rodrigues, J.F., Comparison and stability of solutions to a class of quasilinear parabolic problems, Proc. royal soc. Edinburgh, 110 A, 275-285, (1988) · Zbl 0669.35052
[9] Clarke, F.H., Optimization and nonsmooth analysis, (1983), Wiley New York · Zbl 0727.90045
[10] Naniewicz, Z.; Panagiotopoulos, P.D., Mathematical theory of hemivariational inequalities and applications, (1995), Marcel Dekker New York · Zbl 0968.49008
[11] Panagiotopoulos, P.D., Hemivariational inequalities and applications in mechanics and engineering, (1993), Springer New York · Zbl 0826.73002
[12] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vols. II A/B, Springer, Berlin, 1990. · Zbl 0684.47029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.