Paper detail

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

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 $α$-time derivative and a power $β$ of the Laplacian when the spatial dimension is small, $1\le N\le 4β$, 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.