Evolution equations generated by subdifferentials in the dual space of $(H^1(\Omega))$

A. Damlamian; Nobuyuki Kenmochi

doi:10.3934/dcds.1999.5.269

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1999, Volume 5, Issue 2: 269-278. Doi: 10.3934/dcds.1999.5.269

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Evolution equations generated by subdifferentials in the dual space of $(H^1(\Omega))$

A. Damlamian^1, and
Nobuyuki Kenmochi^2,

1.
Mathématiques, UFR des Sciences et de la Technologie, Universite, Paris 12-Val de Mauve, 94000 Creteil
2.
Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-chō, Inage-ku, Chiba, 263–8522

Received: October 31, 1996

Revised: March 31, 1998

Published: March 1999

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Abstract

This paper is concerned with the subdifferential operator approach to nonlinear (possibly degenerate and singular) parabolic PDE's of the form $u_t-\Delta \beta(u) \ni f$ formulated in the dual space of $H^1(\Omega)$, where $\beta$ is a maximal monotone graph in $\mathbf R\times \mathbf R$. In the set-up considered so far [8], some coerciveness condition has been required for $\beta$, corresponding at least to the fact that it is onto $\mathbf R$. In the present paper, we show that the subdifferential operator approach is possible for any maximal monotone graph $\beta$ without any growth condition.

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Mathematics Subject Classification: 80A22, 35A15, 35K05, 35K85, 35R35.

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Evolution equations generated by subdifferentials in the dual space of $(H^1(\Omega))$

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