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Study of a 2m-th order parabolic equation in a non-regular type of prism of ℝN+1

Arezki Kheloufi

Abstract

This paper deals with the existence, uniqueness and maximal regularity of a solution to the multidimensional parabolic equation of order 2m (with m*)

tu+(-1)mk=1Nxk2mu=f,

defined on a type of the prism

Q=(t,x1)2:0<t<T,φ1(t)<x1<φ2(t)×i=1N-1]0,bi[

of ℝN+1, which is a nonstandard domain since its base shrinks at t = 0 (φ1(0)=φ2(0)). The equation is associated with Cauchy–Dirichlet boundary conditions and the right-hand side term f is taken in a weighted Lebesgue space. More precisely, we look for minimal conditions on the functions φ1 and φ2 under which the solution is regular. For this purpose, the domain decomposition method is employed. This work is an extension of the one space variable case studied in [19] and of the case m = 1 studied in [11].

MSC: 35K05; 35K55

The author wants to thank the anonymous referee for a careful reading of the manuscript and for his/her helpful suggestions.

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Received: 2014-4-13
Revised: 2014-11-13
Accepted: 2014-11-28
Published Online: 2016-3-5
Published in Print: 2016-6-1

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