An Lq(Lp)-theory for diffusion equations with space-time nonlocal operators

Kyeong Hun Kim, Daehan Park, Junhee Ryu

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


We present an Lq(Lp)-theory for the equation ∂tαu=ϕ(Δ)u+f,t>0,x∈Rd;u(0,⋅)=u0. Here p,q>1, α∈(0,1), ∂tα is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying the following: ∃δ0∈(0,1] and c>0 such that [Formula presented] We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove ‖|∂tαu|+|u|+|ϕ(Δ)u|‖Lq([0,T];Lp)≤N(‖f‖Lq([0,T];Lp)+‖u0Bp,qϕ,2−2/αq), where Bp,qϕ,2−2/αq is a modified Besov space on Rd related to ϕ. Our approach is based on BMO estimate for p=q and vector-valued Calderón-Zygmund theorem for p≠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.

Original languageEnglish
Pages (from-to)376-427
Number of pages52
JournalJournal of Differential Equations
Publication statusPublished - 2021 Jun 25


  • Caputo fractional derivative
  • Integro-differential operator
  • L(L)-theory
  • Space-time nonlocal equations

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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