Electron. J. Diff. Equ., Vol. 2010(2010), No. 25, pp. 1-14.

Parabolic equations with Robin type boundary conditions in a non-rectangular domain

Arezki Kheloufi, Boubaker-Khaled Sadallah

Abstract:
In this article, we study the parabolic equation $\partial_{t}u-c^2(t)\partial_x^2u=f$ in the non-necessarily rectangular domain
$$ \Omega =\{ (t,x)\in\mathbb{R}^2:0<t<T,\, \varphi_1(t)<x<\varphi_2(t)\}. $$
The boundary conditions are of Robin type, while the right-hand side lies in the Lebesgue space $L^2(\Omega )$. Our aim is to find conditions on $c$ and the functions $(\varphi_i)_{i=1,2}$ such that the solution belongs to the anisotropic Sobolev space $H^{1,2}(\Omega )=\{u\in L^2(\Omega ):\partial_{t}u, \partial_xu,\partial_x^2u\in L^2(\Omega )\} $. For goal we use the method of approximation of domains.

Submitted July 29, 2009. Published February 10, 2010.
Math Subject Classifications: 35K05, 35K20.
Key Words: Parabolic equation; non-rectangular domains; Robin condition; anisotropic Sobolev space.

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Arezki Kheloufi
Department of Sciences and Techniques, Faculty of Technology
Béjaia University, 6000. Béjaia, Algeria
email: arezkinet2000@yahoo.fr
Boubaker-Khaled Sadallah
Department of Mathematics, E.N.S.
16050 Kouba. Algiers, Algeria
email: sadallah@ens-kouba.dz

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