A Remark on a Nonlocal-in-Time Heat Equation

TL;DR

This paper investigates a nonlocal-in-time heat equation with a nonlinear term depending on the integral of the solution over time, using Schauder's fixed point theorem to establish the existence of solutions.

Contribution

It introduces a novel approach to proving existence for a heat equation with a nonlocal temporal nonlinear term using fixed point theory.

Findings

Existence of solutions established for the nonlocal-in-time heat equation.

Application of Schauder's fixed point theorem to a new class of semilinear equations.

Insights into the behavior of solutions with nonlocal temporal dependencies.

Abstract

Schauder's fixed point theorem is used to derive the existence of solutions to a semilinear heat equation. The equation features a nonlinear term that depends on the time-integral of the unknown on the whole, a priori given, interval of existence.