Weighted mixed-norm estimates for fractional parabolic equations with space-time nonlocal operators

Abstract

We establish weighted mixed-norm estimates for fractional parabolic equations \begin{equation*} \partial_t^αu=Lu-λu+f \text{ in } (0,T)\times\mathbb{R}^d, \end{equation*} with nonlocal operators in both time and space. Here, ${\partial }_t^{\alpha }$ is the Caputo derivative of order $\alpha \in (0,1)$, and $L$ is a spatially nonlocal operator of order $\sigma \in (0,2)$ whose kernel is merely measurable in time. We also obtain the corresponding estimates in the odd mixed-norm spaces, where the inner integration is taken in time and the outer one in space. The estimates are robust in the limit $\alpha \to 1$ and $\sigma \to 2$. We also establish unique solvability when either $T<\infty$ or $\lambda >0$. The proof is based on a direction-by-direction extension argument.

46 pages