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 $${\partial }_{t}u(t,x)={\mathcal{L}}^{a}u(t,x)+f(t,x),t>0$$ in $L_q(L_p)$ spaces. Our spatial operator ${\mathcal{L}}^a$ is an integro-differential operator of the form $${\int }_{{\mathbb{R}}^d}\left(u(x+y)-u(x)-\nabla u(x)\cdot y{1}_{|y|\le 1}\right)a(t,y)j_d(|y|)dy.$$ 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 $|\xi |\ge 1$. Furthermore, we give some examples of operators, which cannot be covered by previous results where smoothness or scaling conditions on $\psi$ are considered.