Asymptotic profiles for inhomogeneous heat equations with memory

TL;DR

This paper investigates the long-term behavior of solutions to an inhomogeneous nonlocal heat equation with memory effects, revealing how asymptotic profiles depend on space-time scales and the forcing term's properties.

Contribution

It provides a detailed analysis of the asymptotic profiles for solutions to a nonlocal heat equation with memory, considering large dimensions and various forcing terms.

Findings

Asymptotic profiles vary with space-time scale and forcing term behavior.

Large dimension ($N>4eta$) influences the solution's decay and profile.

Results extend understanding of heat equations with memory effects in high dimensions.

Abstract

We study the large-time behavior 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, $N> 4\beta$. The asymptotic profiles depend strongly on the space-time scale and on the time behavior of the spatial $L^1$ norm of the forcing term.