An Lp-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators

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2 Citations (Scopus)


In this article we prove the existence and uniqueness of a (weak) solution u in Lp ((0, T);Λγ+m) to the Cauchy problem (equetion presented) where d ∈ ℕ, p ∈ (1,∞], γ, m ∈ (0,∞), Λγ+m is the Lipschitz space on Rd whose order is γ + m, f ∈ Lp ((0, T),Λγ), and ψ (t, iΔ) is a time measurable pseudo-differential operator whose symbol is ψ(t, ξ), (equetion presented) with the assumptions (equetion presented) and (equetion presented): Furthermore, we show (equetion presented) where N is a positive constant depending only on d, p, γ, ν, m, and T, The unique solvability of equation (1) in Lp-Hölder space is also considered. More precisely, for any f ∈ Lp((0, T),Cn+α), there exists a unique solution u ∈ Lp((0, T),Cγ+n+α(Rd)) to equation (1) and for this solution u, (equetion presented) where n ∈ ℤ+, α ∈ (0, 1), and γ + α ∉ ℤ+.

Original languageEnglish
Pages (from-to)2751-2771
Number of pages21
JournalCommunications on Pure and Applied Analysis
Issue number6
Publication statusPublished - 2018 Nov

Bibliographical note

Funding Information:
2000 Mathematics Subject Classification. 35K99, 47G30, 26A16. Key words and phrases. Time measurable pseudo-differential operator, Lp-Lipschitz estimate, Cauchy problem. The author was supported by the TJ Park Science Fellowship of POSCO TJ Park Foundation.

Publisher Copyright:
© 2018 American Institute of Mathematical Sciences. All rights reserved.


  • Cauchy problem
  • L-Lipschitz estimate
  • Time measurable pseudo-differential operator

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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