An Lp-approach for the study of degenerate parabolic equations

Abstract

We give regularity results for solutions of a parabolic equation in non-rectangular domains $U=cup_{tin ] 0,1[}{ t} imes I_{t}$ 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].

Keywords

• Sum of linear operators
• diffusion equation
• non rectangular domain
• bounded imaginary powers of operators