Sobolev estimates for fractional parabolic equations with space-time non-local operators

Abstract

We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $${\partial }_t^{\alpha }u-Lu+\lambda u=f\mathrm{in}(0,T)\times {\mathbb{R}}^d,$$ where ${\partial }_t^{\alpha }u$ is the Caputo fractional derivative of order $\alpha \in (0,1]$, $\lambda \ge 0$, $T\in (0,\infty )$, and $$Lu(t,x):={\int }_{{\mathbb{R}}^d}(u(t,x+y)-u(t,x)-y\cdot {\nabla }_{x}u(t,x){\chi }^{(\sigma )}(y))K(t,x,y)dy$$ is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel $K$ with respect to $t$ and $y$. We also derive a weighted mixed-norm estimate for the equations with operators that are local in time, i.e., $\alpha =1$, which extend the previous results by using a quite different method.

Updated on 28th Dec, 2021

References in corpus (1)

• Weighted mixed norm estimates for fractional wave equations with VMO coefficients

Cited by in corpus (2)

• Boundedness of non-local operators with spatially dependent coefficients and $L_p$-estimates for non-local equations
• An $L_q(L_p)$-theory for time-fractional diffusion equations with nonlocal operators generated by Lévy processes with low intensity of small jumps