An Lq(Lp)-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${\partial }_{t}u(t,x)=L^{a}u(t,x)+f(t,x),t>0$ in $L_q\left(L_p\right)$ spaces. Our spatial operator $L^a$ is an integro-differential operator of the form$\underset{R^d}{\int }\left(u\right(x+y)-u(x)-\nabla u(x)\cdot y{1}_{\left|y\right|\le 1})a(t,y)j_d\left(\right|y\left|\right)dy.$ 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 $j_d\left(r\right)$ which yield $L_q\left(L_p\right)$-regularity of solutions. Our assumptions on $j_d$ are general so that $j_d\left(r\right)$ may be comparable to $r^{-d}\ell \left(r^{-1}\right)$ for a function ℓ which is slowly varying at infinity. For example, we can take $\ell \left(r\right)=\log (1+r^{\alpha })$ or $\ell \left(r\right)=\min \{r^{\alpha },1\}$ ($\alpha \in (0,2)$). Indeed, our result covers the operators whose Fourier multiplier $\psi \left(\xi \right)$ does not have any scaling condition for $\left|\xi \right|\ge 1$. Furthermore, we give some examples of operators, which cannot be covered by previous results where smoothness or scaling conditions on ψ are considered.