On a Nonlinear Mixed Problem for a Parabolic Equation with a Nonlocal Condition

Abstract

The aim of this work is to prove the well-posedness of some linear and nonlinear mixed problems with integral conditions defined only on two parts of the considered boundary. First, we establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense using a functional analysis method. 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.

On the Multi-Dimensional Sumudu-Generalized Laplace Decomposition Method and Generalized Pseudo-Parabolic Equations

“…The authors in [5] applied two different method schemes to solve nonlinear pseudo-parabolic equations. The authors in [6] have proved the well-posedness of some linear and nonlinear mixed problems with integral conditions. The solution of the pseudo-parabolic equation has been studied in diverse articles, including those that concern exact and approximation methods, for example, the three-dimensional Laplace Adomian decomposition method [7], the double Sumudu-generalized Laplace decomposition method [8], the double Laplace decomposition method [9,10], the three-layer difference method [11], and two approximation methods [12].…”