Research article

Moroccan Journal of Pure and Applied Analysis

, 1:6

First online:

Open Access

# On a Fourth Order Parabolic Equation with Mixed Type Boundary Conditions in a Nonrectangular Domain

• Arezki KheloufiAffiliated withFaculté des Sciences Exactes, Université de Bejaia Email author

## Abstract

This paper is devoted to the study of the following fourth order parabolic equation $${\partial _t}u + \partial _x^4u = f$$ in the non-necessarily rectangular domain

$$Q=\left\{ (t,x) \right.\in {{\mathbb{R}}^{2}}:0<t<T,{{\varphi }_{1}}(t)<x<{{\varphi }_{2}}(t)\}.$$

The equation is subject to mixed type conditions $${\partial _x}u = \partial _x^3u + \beta u = 0$$, on the lateral boundary of Q. The right-hand side term f of the equation lies in $$L_\omega^2(Q)$$ the space of square-integrable functions on Q with the measure ωdtdx. Our aim is to find sufficient conditions on the coefficient β and on the functions φ i; i = 1; 2 and on the weight ω such that the solution of this equation belongs to the anisotropic Sobolev space

$$H_\omega ^{1,4}(Q) = \{ u \in L_\omega ^2(Q):{\partial _t}u,\partial _x^ju \in L_\omega ^2(Q),j = 1,2,3,4\} .$$

The analysis is performed by using the domain decomposition method.

### Key words and phrases

Higher-order parabolic equations Nonregular domains Anisotropic weighted Sobolev spaces Mixed type conditions