A Boundary Value Problem with Multivariables Integral Type Condition for Parabolic Equations

A L Marhoune,F Lakhal

doi:10.1155/2009/975601

Abstract

We study a boundary value problem with multivariables integral type condition for a class of parabolic equations. We prove the existence, uniqueness, and continuous dependence of the solution upon the data in the functional wieghted Sobolev spaces. Results are obtained by using a functional analysis method based on two-sided a priori estimates and on the density of the range of the linear operator generated by the considered problem.

Highlights

Certain problems of modern physics and technology can be effectively described in terms of nonlocal problems with integral conditions for partial differential equations
We consider in the rectangular domain Ω 0, 1 × 0, T, the following nonclassical boundary value problem of finding a solution u x, t such that
Mathematical modelling of different phenomena leads to problems with nonlocal or integral boundary conditions

Summary

Introduction

Certain problems of modern physics and technology can be effectively described in terms of nonlocal problems with integral conditions for partial differential equations. Mathematical modelling of different phenomena leads to problems with nonlocal or integral boundary conditions Such a condition occurs in the case when one measures an averaged value of some parameter inside the domain. This amounts to the specification of the energy or mass contained in a portion of the conductor or porous medium as a function of time. This problems arise in plasma physics, heat conduction, biology and demography, as well as modelling of technological process, see, for example, 1–5. We prove that the operator L is a linear homeomorphism between the spaces E and F

Two-Sided A Priori Estimates

Re τ 0

Solvability of the Problem

Re exp