Asymptotic behavior of solutions to a space fractional diffusion equation

Abstract

We improve the time decay estimates of solutions to the one-dimensional fractional diffusion equation involving the Caputo derivative. The equation is considered on the half-line. Depending on the boundary condition, we show that solutions converge in $L^p$, $p>1$ to a multiple of the self-similar solutions or decay to zero. The convergence rate is provided.