## Abstract

We introduce an L_{q}(L_{p})-theory for the semilinear fractional equations of the type∂_{t} ^{α}u(t,x)=a^{ij}(t,x)u_{xixj }(t,x)+f(t,x,u),t>0,x∈R^{d}. Here, α∈(0,2), p,q>1, and ∂_{t} ^{α} is the Caupto fractional derivative of order α. Uniqueness, existence, and L_{q}(L_{p})-estimates of solutions are obtained. The leading coefficients a^{ij}(t,x) are assumed to be piecewise continuous in t and uniformly continuous in x. In particular a^{ij}(t,x) are allowed to be discontinuous with respect to the time variable. Our approach is based on classical tools in PDE theories such as the Marcinkiewicz interpolation theorem, the Calderon–Zygmund theorem, and perturbation arguments.

Original language | English |
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Pages (from-to) | 123-176 |

Number of pages | 54 |

Journal | Advances in Mathematics |

Volume | 306 |

DOIs | |

Publication status | Published - 2017 Jan 14 |

## Keywords

- Caputo fractional derivative
- Fractional diffusion-wave equation
- L(L)-theory
- L-theory
- Variable coefficients

## ASJC Scopus subject areas

- Mathematics(all)