### Abstract

In this article we prove the existence and uniqueness of a (weak) solution u in L_{p} ((0, T);Λ_{γ+m}) to the Cauchy problem (equetion presented) where d ∈ ℕ, p ∈ (1,∞], γ, m ∈ (0,∞), Λ_{γ+m} is the Lipschitz space on R^{d} whose order is γ + m, f ∈ L_{p} ((0, T),Λ_{γ}), 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 L_{p}-Hölder space is also considered. More precisely, for any f ∈ L_{p}((0, T),C^{n+α}), there exists a unique solution u ∈ L_{p}((0, T),C^{γ+n+α}(R^{d})) to equation (1) and for this solution u, (equetion presented) where n ∈ ℤ_{+}, α ∈ (0, 1), and γ + α ∉ ℤ_{+}.

Original language | English |
---|---|

Pages (from-to) | 2751-2771 |

Number of pages | 21 |

Journal | Communications on Pure and Applied Analysis |

Volume | 17 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2018 Nov 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**An L _{p}-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators.** / Kim, Ildoo.

Research output: Contribution to journal › Article

}

TY - JOUR

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

AU - Kim, Ildoo

PY - 2018/11/1

Y1 - 2018/11/1

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

UR - http://www.scopus.com/inward/citedby.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 -