Nonlocal complement value problem for a global in time parabolic equation

TL;DR

This paper studies the existence and uniqueness of solutions for a complex semilinear parabolic equation with nonlocal interactions in space and time, relevant to biological nano-sensors and polymer dynamics.

Contribution

It introduces a novel analysis of a semilinear parabolic equation with double nonlocality, proving existence and uniqueness of solutions under specific conditions.

Findings

Existence and uniqueness of weak solutions for small time.

Application of Galerkin approximation method.

Conditions on the interaction potential for solvability.

Abstract

The overreaching goal of this paper is to investigate the existence and uniqueness of weak solution of a semilinear parabolic equation with double nonlocality in space and in time variables that naturally arises while modeling a biological nano-sensor in the chaotic dynamics of a polymer chain. In fact, the problem under consideration involves a symmetric integrodifferential operator of L\'{e}vy type and a term called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. Owing to the Galerkin approximation, the existence and uniqueness of a weak solution of the nonlocal complement value problem is proven for small time under fair conditions on the interaction potential.