Decay/growth rates for inhomogeneous heat equations with memory. The case of large dimensions

TL;DR

This paper investigates the decay and growth rates of solutions to an inhomogeneous nonlocal heat equation with memory effects in high-dimensional spaces, revealing how these rates depend on space-time scales and forcing terms.

Contribution

It provides new insights into the decay/growth behavior of solutions to inhomogeneous heat equations with memory in large dimensions, extending existing theories.

Findings

Decay rates depend on space-time scale and forcing term behavior

Growth rates are characterized for large dimensions

Results apply to equations with Caputo derivatives and fractional Laplacians

Abstract

We study the decay/growth rates in all $L^p$ norms of solutions to an inhomogeneous nonlocal heat equation in $\mathbb{R}^N$ involving a Caputo $\alpha$-time derivative and a power $\beta$ of the Laplacian when the dimension is large, $N> 4\beta$. Rates depend strongly on the space-time scale and on the time behavior of the spatial $L^1$ norm of the forcing term.