An L-q(L-p)-theory for diffusion equations with space-time nonlocal operators

Abstract

We present an Lq(Lp)-theory for the equation a(t)(alpha) u = phi(Delta)u + f, t > 0, x is an element of R-d ; u(0, center dot) = u(0). Here p, q > 1, alpha is an element of(0, 1), partial derivative(alpha)(t) atis the Caputo fractional derivative of order alpha, and phi is a Bernstein function satisfying the following: there exists delta(0) is an element of(0, 1] and c > 0such that c(R/r)(delta 0) <= phi(R)/phi(r), 0 < < R < infinity. (0.1) We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove parallel to vertical bar partial derivative(alpha)(t) u vertical bar + vertical bar u vertical bar + vertical bar phi(Delta)u vertical bar parallel to(Lq([0,T];Lp)) <= N(parallel to f parallel to(Lq([0,T]; Lp)) + parallel to u(0)parallel to B-p,B-q phi,2-2/alpha q), where B-p,q(phi,2-2/alpha q) is a modified Besov space on R-d related to phi. Our approach is based on BMO estimate for p = q and vector-valued Calderon-Zygmund theorem for p not equal q. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory. (c) 2021 Elsevier Inc. All rights reserved.