Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 25, pp. 1-14.

Title: Parabolic equations with Robin type boundary conditions
       in a non-rectangular domain
       
Authors: Arezki Kheloufi (Bejaia Univ., Algeria)
         Boubaker-Khaled Sadallah (E.N.S., Algiers, Algeria)
	 
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.