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

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April 1999, 5(2): 269-278. doi: 10.3934/dcds.1999.5.269

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, France
2.	Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-chō, Inage-ku, Chiba, 263–8522

Received November 1996 Revised April 1998 Published January 1999

Abstract
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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.

Keywords: subdifferential, Nonlinear semigroups, degenerate Stefan problem..

Mathematics Subject Classification: 80A22, 35A15, 35K05, 35K85, 35R3.

Citation: A. Damlamian, Nobuyuki Kenmochi. Evolution equations generated by subdifferentials in the dual space of $(H^1(\Omega))$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 269-278. doi: 10.3934/dcds.1999.5.269

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