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Existence of bounded solutions of some nonlinear parabolic equations

Published online by Cambridge University Press:  14 November 2011

A. Mokrane
Département de Mathematiques, Ecole Normale Superieure, Vieux-Kouba, Alger, Algeria


This paper proves the existence of (at least) one solution of the following equation:

Here, is an elliptic operator of Leray-Lions type acting from into Lp′(0, T; W−1.p′ (Ω)), (1/p + 1/p′ = 1) and |F(u, ∇u)| ≧b(|u|)(l + |≧u|P). There are no smoothness assumptions on the bounded open set Ω; the operator and the nonlinearity F(u, ∇u) are denned in terms of Carathéodory functions. These points are the most characteristic features of this paper.

Assuming the existence of upper and lower solutions allows us to obtain L(Q)-estimates. An estimate is then proved. The final step is to prove the strong convergence in of the approximations. This proof relies on the method introduced by L. Boccardo, F. Murat and J. P. Puel for elliptic and parabolic problems of this type.

Research Article
Copyright © Royal Society of Edinburgh 1987

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