Strong Solutions to a Nonlocal-in-Time Semilinear Heat Equation

TL;DR

This paper proves the existence of strong solutions for a nonlocal semilinear heat equation where the nonlocal term depends on the entire solution history, using Schauder's fixed point theorem and semigroup theory.

Contribution

It introduces a novel approach to establish strong solutions for a nonlocal-in-time heat equation with a history-dependent term.

Findings

Existence of strong solutions is established.

The method employs Schauder's fixed point theorem.

Semigroup theory is used to handle the nonlocal term.

Abstract

Existence of strong solutions to a nonlocal semilinear heat equation is shown. The main feature of the equation is that the nonlocal term depends on the unknown on the whole time interval of existence, the latter being given a priori. The proof relies on Schauder's fixed point theorem and semigroup theory.