An Lq(Lp)-regularity theory for parabolic equations with integro-differential operators having low intensitykernels

초록

In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators partial derivative(t)u(t,x)=L(a)u(t,x) + f(t,x), t > 0 in L-q(L-p) spaces. Our spatial operator L-a is an integro-differential operator of the form integral(Rd)(u(x + y) - u(x) - del u(x) center dot y1(vertical bar y vertical bar <= 1))a(t, y)j(d)(vertical bar y vertical bar)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)l(r(-1)) for a function l which is slowly varying at infinity. For example, we can take l(r)=log (1+r(alpha)) or l(r)=min {r(alpha), 1} (alpha is an element of(0,2)). Indeed, our result covers the operators whose Fourier multiplier psi(xi) does not have any scaling condition for vertical bar xi vertical bar >= 1. Furthermore, we give some examples of operators, which cannot be covered by previous results where smoothness or scaling conditions on psi are considered. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

키워드