MMN-3192

# L^{p}-regularity results for 2m-th order parabolic equations in time-varying domains

*Arezki Kheloufi*;

*Abderrahmane Ikassoulene*;

## Abstract

This paper is devoted to the analysis of the following linear $2m$-th order parabolic equation
$
\partial _{t}u+(-1)^{m}\sum_{k=1}^{N}\partial _{x_{k}}^{2m}u=f,
$
subject to Dirichlet type condition
$
\partial_{\nu}^{l}u=0,\;l=0,1,...,m-1,
$
on the lateral boundary, where $m$ is a positive integer. The right-hand side $f$ of the equation is taken in the Lebesgue space $L^{p}.$ The problem is set in a domain of the form
$
Q=\left\{ \left( t,x_{1},...,x_{N}\right) \in\mathbb{R}^{N+1}:0\leq \sqrt{x_{1}^{2}+...+x_{N}^{2}}<t^{\alpha} \right\}
$
with $\alpha >1/2m.$ We use Labbas-Terreni results [17] on the operator's sum theory in the non-commutative case. This work is an extension of the Hilbertian case studied in [6].

Vol. 23 (2022), No. 1, pp. 211-230

DOI: 10.18514/MMN.2022.3192