TY - JOUR

T1 - An Lp-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators

AU - Kim, Ildoo

N1 - 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.

PY - 2018/11

Y1 - 2018/11

N2 - 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+.

AB - 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+.

KW - Cauchy problem

KW - L-Lipschitz estimate

KW - Time measurable pseudo-differential operator

UR - http://www.scopus.com/inward/record.url?scp=85056889402&partnerID=8YFLogxK

U2 - 10.3934/cpaa.2018130

DO - 10.3934/cpaa.2018130

M3 - Article

AN - SCOPUS:85056889402

VL - 17

SP - 2751

EP - 2771

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

SN - 1534-0392

IS - 6

ER -