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

TL;DR

This paper investigates the decay and growth rates of solutions to an inhomogeneous nonlocal heat equation with memory effects in low-dimensional spaces, extending existing results to dimensions 1 through 4.

Contribution

It provides a comprehensive analysis of decay/growth rates for small spatial dimensions, considering various norms, scales, and forcing term behaviors, filling a gap in the existing literature.

Findings

Decay rates depend on $p$, space-time scale, and forcing term behavior.

Results complete the understanding for dimensions $1 o 4eta$, previously known for larger dimensions.

The study characterizes how solutions grow or decay in different $L^p$ norms over time.

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 spatial dimension is small, $1\le N\le 4\beta$, thus completing the already available results for large spatial dimensions. Rates depend not only on $p$, but also on the space-time scale and on the time behavior of the spatial $L^1$ norm of the forcing term.