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.

Keywords

• heat equation with nonlocal time derivative
• caputo derivative
• fully nonlocal heat equations
• fractional laplacian
• large-time behavior