MMN-3192

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

• Arezki Kheloufi, Department of Technology, Faculty of Technology, Lab. of Applied Mathematics, University of Bejaia, 06000 Bejaia, Algeria, arezkinet2000@yahoo.fr
• Abderrahmane Ikassoulene, Department of Mathematics, Faculty of Exact Sciences, University of Bejaia, 06000 Bejaia, Algeria, ikabdset@gmail.com

## 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