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

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.

16 pages, 1 figure