Table of Contents
ISRN Applied Mathematics
Volume 2012, Article ID 138308, 9 pages
http://dx.doi.org/10.5402/2012/138308
Research Article

An Energy Inequality and Its Applications of Nonlocal Boundary Conditions of Mixed Problem for Singular Parabolic Equations in Nonclassical Function Spaces

1Department of Mathematics, University of Lomé, P.O. Box 1515, Lomé, Togo
2Faculty of Mechanics and Mathematics, Belarusian State University, Minsk 220050, Belarus

Received 23 July 2012; Accepted 14 August 2012

Academic Editors: C. Lu and E. Skubalska-Rafajlowicz

Copyright © 2012 Moussa Zakari Djibibe et al. 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

The aim of this paper is to establish a priori estimates of the following nonlocal boundary conditions mixed problem for parabolic equation: πœ•π‘£/πœ•π‘‘βˆ’(π‘Ž(𝑑)/π‘₯2)(πœ•/πœ•π‘₯)(π‘₯2πœ•π‘£/πœ•π‘₯)+𝑏(π‘₯,𝑑)𝑣=𝑔(π‘₯,𝑑), 𝑣(π‘₯,0)=πœ“(π‘₯), 0≀π‘₯≀ℓ, 𝑣(β„“,𝑑)=𝐸(𝑑), 0≀𝑑≀𝑇, βˆ«β„“0π‘₯3𝑣(π‘₯,𝑑)𝑑π‘₯=𝐺(𝑑), 0≀𝑑≀ℓ. It is important to know that a priori estimates established in nonclassical function spaces is a necessary tool to prove the uniqueness of a strong solution of the studied problems.