Electron. J. Diff. Eqns., Vol. 2005(2005), No. 36, pp. 1-20.

An Lp-approach for the study of degenerate parabolic equations

Rabah Labbas, Ahmed Medeghri, Boubaker-Khaled Sadallah

Abstract:
We give regularity results for solutions of a parabolic equation in non-rectangular domains $U=\cup_{t\in ] 0,1[}\{ t\} \times I_{t}$ with $I_{t}=\{x:0<x<\varphi (t)\}$. The optimal regularity is obtained in the framework of the space $L^{p}$ with $p$ greater than 3/2 by considering the following cases: (1) When $\varphi (t)=t^{\alpha }$, $\alpha$ greater than 1/2 with a regular right-hand side belonging to a subspace of $L^{p}(U)$ and under assumption $p$ greater than $1+\alpha $. We use Labbas-Terreni results [11]. (2) When $\varphi (t)=t^{1/2}$ with a right-hand side taken only in $L^{p}(U)$. Our approach make use of the celebrated Dore-Venni results [2].

Submitted July 16, 2004. Published March 29, 2005.
Math Subject Classifications: 35K05, 35K65, 35K90
Key Words: Sum of linear operators, diffusion equation, non rectangular domain,bounded imaginary powers of operators

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Rabah Labbas
Laboratoire de Mathématiques
Université du Havre, U.F.R. Sciences et Techniques
b.p. 540, 76058 Le Havre, France
email: labbas@univ-lehavre.fr
  Ahmed Medeghri
Université de Mostaganem
b.p. 188, 27000, Mostaganem, Algeria
email: medeghri@univ-mosta.dz
  Boubaker-Khaled Sadallah
Ecole Normale Supérieure, Département
de Maths, 16050 Kouba, Alger, Algeria
email: sadallah@wissal.dz

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