A Mixed Problem with an Integral Two-Space-Variables Condition for Parabolic Equation with The Bessel Operator

Bouziani Abdelfatah,Oussaeif Taki-Eddine,Ben Aoua Leila

doi:10.1155/2013/457631

Bouziani Abdelfatah, Oussaeif Taki-Eddine + Show 1 more

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https://doi.org/10.1155/2013/457631

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Abstract

We study a mixed problem with an integral two-space-variables condition for parabolic equation with the Bessel operator. The existence and uniqueness of the solution in functional weighted Sobolev space are proved. The proof is based on a priori estimate “energy inequality” and the density of the range of the operator generated by the problem considered.

Highlights

The importance of boundary value problems with integral boundary condition has been pointed out by Samarskiı [1] and problems with integral conditions for parabolic equations were treated by Kamynin [2], Ionkin [3], Yurchuk [4], Benouar and Yurchuk [5], Bouziani [6], Bouziani and Benouar [7, 8], and Mesloub and Bouziani [9]
The presence of integral terms in boundary condition can, in general, greatly complicate the application of standard functional or numerical techniques, specially the integral two-space-variables condition. To avoid this difficulty, we introduce a technique for transfering this problem to another classically less complicated one that does not contain integral conditions
The method used here is one of the most efficient functional analysis methods in solving partial differential equations with integral conditions, the so-called a priori estimate method or the energy-integral method. This method is essentially based on the construction of multiplicators for each specific given problem, which provides the a priori estimate from which it is possible to establish the solvability of the posed problem

Summary

Introduction

The importance of boundary value problems with integral boundary condition has been pointed out by Samarskiı [1] and problems with integral conditions for parabolic equations were treated by Kamynin [2], Ionkin [3], Yurchuk [4], Benouar and Yurchuk [5], Bouziani [6], Bouziani and Benouar [7, 8], and Mesloub and Bouziani [9]. Other parabolic problems arise in plasma physics by Samarskiı [1], heat conduction by Cannon [10], Ionkin [3], dynamics of ground waters by Nakhushev [11], Vodakhova [12], Kartynnik [13], and Lin [14]. Regular case of this problem is studied in [15]. We study a mixed problem with an integral two-space-variables condition for parabolic equation with the Bessel operator

Setting of the Problem

A Priori Estimate

Existence of Solution