An $L_{q}(L_{p})$-regularity theory for parabolic equations with integro-differential operators having low intensity kernels

Abstract

In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators $$ \partial_{t}u(t,x) = \mathcal{L}^{a}u(t,x) + f(t,x), \quad t>0 $$ in $L_{q}(L_{p})$ spaces. Our spatial operator $\mathcal{L}^{a}$ is an integro-differential operator of the form $$ \int_{\mathbb{R}^{d}} \left( u(x+y)-u(x) -\nabla u(x) \cdot y \mathrm{1}_{|y|\leq 1} \right) a(t,y) j_{d}(|y|)dy. $$ Here, $a(t,y)$ is a merely bounded measurable coefficient, and we employed the theory of additive process to handle it. We investigate conditions on $j_{d}(r)$ which yield $L_{q}(L_{p})$-regularity of solutions. Our assumptions on $j_d$ are general so that $j_d(r)$ may be comparable to $r^{-d}\ell(r^{-1})$ for a function $\ell$ which is slowly varying at infinity. For example, we can take $\ell(r)=\log{(1+r^{\alpha})}$ or $\ell(r) = \min{\{r^{\alpha},1\}}$ ($\alpha\in(0,2)$). Indeed, our result covers the operators whose Fourier multiplier $\psi(\xi)$ does not have any scaling condition for $|\xi|\geq 1$. Furthermore, we give some examples of operators, which cannot be covered by previous results where smoothness or scaling conditions on $\psi$ are considered.