A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation

Abstract

We study a nonlocal mixed problem for a nonlinear pseudoparabolic equation, which can, for example, model the heat conduction involving a certain thermodynamic temperature and a conductive temperature. We prove the existence, uniqueness and continuous dependence of a strong solution of the posed problem. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplicator. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.  2005 Elsevier Inc. All rights reserved.

Solvability of a Mixed Nonlocal Problem for a Nonlinear Singular Viscoelastic Equation

“…The proofs for the nonlinear problem are mainly based on the application of some iterative processes. This work can be considered as a further elaboration of that in Bouziani [1] and Mesloub [14].…”

“…For the hyperbolic case, we cite Mesloub and Bouziani [18], Muravei and Philinovskii [22], Pulkina [23]. For the pseudoparabolic case, we cite Bouziani [1] and Mesloub [14]. For the case of initial and boundary value problems for linear and nonlinear viscoelastic equations with classical conditions we can cite for example, Cavalcanti et al [5] established an exponential rate of decay for the equation…”

Solving nonlinear pseudoparabolic equations with nonlocal boundary conditions in reproducing kernel space

“…And the reproducing kernel r y (x) can be presented by r y (x) = a 1 + a 2 x + a 3 x 2 + · · · + a 6 x 5 + cx 6 6! , b 1 + b 2 x + b 3 x 2 + · · · + b 6 x 5 + cx 6 6!…”

“…Some qualitative theories for the nonlinear parabolic equations are established in [5][6][7][8][9][10][11][12][13][14][15], but there are few references about their numerical solutions. In this paper, we provide a very simple numerical algorithm for the approximations of problem (1.1)-(1.4) based on the reproducing kernel space.…”

On the higher dimension Boussinesq equation with nonclassical condition

“…The nonlocal condition appearing in this mathematical model represents the total mass of impurities in the lamina. For some hyperbolic nonlocal mixed problems, the reader should consult the works done by Beilin [5], Mesloub and Lekrine [6], Mesloub and Messaoudi [7,8], Mesloub and Bouziani [9], Muravei and Philinovskii [10], Nukushev [11], and Pulkina [12]. Recent works dealing with nonlinear nonlocal mixed problems can be found in Mesloub [13--15].…”