## Abstract

We present an L_{q}(L_{p})-theory for the equation ∂_{t}^{α}u=ϕ(Δ)u+f,t>0,x∈R^{d};u(0,⋅)=u_{0}. Here p,q>1, α∈(0,1), ∂_{t}^{α} is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying the following: ∃δ_{0}∈(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 ‖|∂_{t}^{α}u|+|u|+|ϕ(Δ)u|‖_{Lq([0,T];Lp)}≤N(‖f‖_{Lq([0,T];Lp)}+‖u_{0}‖_{Bp,qϕ,2−2/αq}), where B_{p,q}^{ϕ,2−2/αq} is a modified Besov space on R^{d} 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 language | English |
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Pages (from-to) | 376-427 |

Number of pages | 52 |

Journal | Journal of Differential Equations |

Volume | 287 |

DOIs | |

Publication status | Published - 2021 Jun 25 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics