Abstract and Applied Analysis
Volume 2003 (2003), Issue 10, Pages 573-589
doi:10.1155/S1085337503210010

On the weak solution of a three-point boundary value problem for a class of parabolic equations with energy specification

Abdelfatah Bouziani1,2

1Department of Mathematics, The Larbi Ben M'hidi University Centre, Oum El Bouagui 04000, Algeria
2The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Trieste 34100, Italy

Received 3 September 2002

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with weak solution in weighted Sobolev spaces, of three-point boundary value problems which combine Dirichlet and integral conditions, for linear and quasilinear parabolic equations in a domain with curved lateral boundaries. We, firstly, prove the existence, uniqueness, and continuous dependence of the solution for the linear equation. Next, analogous results are established for the quasilinear problem, using an iterative process based on results obtained for the linear problem.