The Existence and Uniqueness Theorem for Initial-Boundary Value Problem of the Same Class of Integro-Differential PDEs

Abstract

The second initial-boundary value problem for a class of nonlinear PDEs of the second order and an integral operator of a given form is considered. Dependence of its solution with the solution of the standard second linear initial-boundary value problem for a second order hyperbolic equation is shown. The proof of energy inequality presents at the beginning for an auxiliary linear problem and then for a nonlinear problem. The existence and uniqueness theorem of the corresponding initial-boundary value problem is proved with its help. For better understanding of the problems under consideration as particular representatives of the studied class of Integro-differential equations, the examples of Integro-differential equations for various integral operators of this type are given at the final of the article. The author believes that, despite the existing purely theoretical interest, this class of Integro-differential equations will attract great attention for further research. The solution of the nonlinear problem has a curious nontrivial property, and the existence and uniqueness theorem of the original problem will help to justify the application of various applications to her.