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 $$\begin{aligned} \partial _t^\alpha u - Lu + \lambda u= f \quad \text {in } (0,T) \times {\mathbb {R}}^d , \end{aligned}$$ where $$\partial _t^\alpha u$$ is the Caputo fractional derivative of order $$\alpha \in (0,1]$$ , $$T\in (0,\infty )$$ , and $$\begin{aligned} Lu(t,x) {:}{=} \int _{{\mathbb {R}}^d} \bigg ( u(t,x+y)-u(t,x) - y\cdot \nabla _xu(t,x)\chi ^{(\sigma )}(y)\bigg )K(t,x,y)\,dy \end{aligned}$$ 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 in Mikulevičius and Pragarauskas (J Differ Equ 256(4):1581–1626, 2014) and Zhang (Annales l’IHPAnalyse Nonlinéaire 30:573–614, 2013) by using a quite different method.