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∂tu(t,x)=Lau(t,x)+f(t,x),t>0 in Lq(Lp) spaces. Our spatial operator La is an integro-differential operator of the form∫Rd(u(x+y)−u(x)−∇u(x)⋅y1|y|≤1)a(t,y)jd(|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 jd(r) which yield Lq(Lp)-regularity of solutions. Our assumptions on jd are general so that jd(r) may be comparable to r−dℓ(r−1) for a function ℓ which is slowly varying at infinity. For example, we can take ℓ(r)=log⁡(1+rα) or ℓ(r)=min⁡{rα,1} (α∈(0,2)). Indeed, our result covers the operators whose Fourier multiplier ψ(ξ) does not have any scaling condition for |ξ|≥1. Furthermore, we give some examples of operators, which cannot be covered by previous results where smoothness or scaling conditions on ψ are considered.