An Lq(Lp)-theory for the time fractional evolution equations with variable coefficients

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We introduce an Lq(Lp)-theory for the semilinear fractional equations of the type∂t αu(t,x)=aij(t,x)uxixj (t,x)+f(t,x,u),t>0,x∈Rd. Here, α∈(0,2), p,q>1, and ∂t α is the Caupto fractional derivative of order α. Uniqueness, existence, and Lq(Lp)-estimates of solutions are obtained. The leading coefficients aij(t,x) are assumed to be piecewise continuous in t and uniformly continuous in x. In particular aij(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 languageEnglish
Pages (from-to)123-176
Number of pages54
JournalAdvances in Mathematics
Publication statusPublished - 2017 Jan 14

Bibliographical note

Publisher Copyright:
© 2016 Elsevier Inc.


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

ASJC Scopus subject areas

  • General Mathematics


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