Unbounded solutions of the nonlocal heat equation

TL;DR

This paper investigates the nonlocal heat equation with a symmetric probability density, providing a comprehensive analysis of unbounded solutions, including their initial trace, existence, uniqueness, and explicit polynomial solutions.

Contribution

It offers new insights into unbounded solutions of the nonlocal heat equation, including optimal class estimates and explicit polynomial solutions, extending previous understanding.

Findings

Characterization of initial traces for unbounded solutions

Existence and uniqueness results in specific classes

Explicit construction of unbounded polynomial solutions

Abstract

We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: u_t = J * u - u, where J is a symmetric continuous probability density. Depending on the tail of J, we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a suitable class; (iii) giving explicit unbounded polynomial solutions.