Study of existence and uniqueness of the solution of a class of linear and semi-linear equations

Abstract

This memoir aims to study two cases of general mixed problems with integral con- ditions, both classical and non-local.

The memoir begins with an introduction providing background and interest in the addressed topic, along with mentioning some preliminary concepts useful later on. In the second chapter, a problem involving a non-local boundary condition for a parabolic differential equation is addressed. We employ functional analysis method to prove the existence and uniqueness of the strong solution to the problem. In the third chapter, our focus shifts to exploring the weak solution of the problem and proving its existence and uniqueness related to the specific issue, relying on the a priori estimation method used in the previous chapter for proving uniqueness. As for the existence of the solution, the Galerkin method is utilized. Finally, a fixed point argument is used to prove the existence of the local solution to the problem.