Mixed problem with a weighted integral condition for a parabolic equation with the Bessel operator

Abstract

In this paper, we prove the existence, uniqueness and continuous dependence on the data of a solution of a mixed problem with a weighted integral condition for a parabolic equation with the Bessel operator. The proof uses a functional analysis method based on an a priori estimate and on the density of the range of the operator generated by the considered problem.Singular Parabolic Equations, Weighted Integral Condition, A

Global Existence, Decay, and Blow up of Solutions of a Singular Nonlocal Viscoelastic Problem

“…In fact, most of the research was devoted to the classical solutions. Later, mixed problems with integral conditions for both parabolic and hyperbolic equations were studied by Ionkin [13], Ionkin and Moiseev [14], Kamynin [15], Pulkina [32,33], Volkodavov and Zhukov [36], Yurchuk [37], Kartynnik [16], Muravei and Philinovskii [31], Bouziani [4], Mesloub [19], Mesloub and Bouziani [20][21][22], Mesloub and Messaoudi [24][25][26], Mesloub and Lekrine [23]. We should mention here that the presence of the integral term in the boundary condition can greatly complicate the application of standard functional techniques (see [24,25]).…”

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

“…The importance of boundary value problems with integral boundary condition has been pointed out by Samarskiȋ [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 also arise in plasma physics by Samarskiȋ [1], heat conduction by Cannon [10], Ionkin [3], dynamics of ground waters by Nakhushev [11], Vodakhova [12], Kartynnik [13], and Lin [14].…”

On a certain class of a nonlinear evolution partial differential equations

“…We start by applying a functional analysis method [12][13][14]18] for obtaining a priori estimate for the solution to problems (1)-(3) stated below. The technique of deriving such a priori estimate is based on constructing a suitable multiplicator.…”