Remark on a weakly coupled system of nonlinear damped wave equations

Nakao Hayashi; Pavel I. Naumkin; Masayo Tominaga

doi:10.1016/j.jmaa.2015.03.008

Remark on a weakly coupled system of nonlinear damped wave equations

Nakao Hayashi, Pavel I. Naumkin, Masayo Tominaga

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

We study global existence of small solutions to the Cauchy problem for a weakly coupled nonlinear damped wave equation{(∂²_t+∂_t-δ)u=N₁(v), (∂²_t+∂_t-δ)v=N₂(u),x∈Rⁿ, t>0 u(0,x)=εu₀(x), ∂_tu(0,x)=εu₁(x), v(0,x)=εv₀(x), ∂_tv(0,x)=εv₁(x), x∈Rⁿ, with super-critical nonlinearities N_k(ϕ)=|ϕ|^ρk, k= 1, 2, where ε>0, the space dimension n≥4. Our purpose is to remove the exponential decay condition on the data and the lower bound for ρ₁ which was assumed in [4] when proving the global existence of solutions in the case of higher space dimensions.

Original language	English
Pages (from-to)	490-501
Number of pages	12
Journal	Journal of Mathematical Analysis and Applications
Volume	428
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2015.03.008
Publication status	Published - 2015
Externally published	Yes

Keywords

A weakly coupled system
Global existence
Super-critical case
System of damped wave equations
Time decay

ASJC Scopus subject areas

Analysis
Applied Mathematics

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title = "Remark on a weakly coupled system of nonlinear damped wave equations",

abstract = "We study global existence of small solutions to the Cauchy problem for a weakly coupled nonlinear damped wave equation{(∂2t+∂t-δ)u=N1(v), (∂2t+∂t-δ)v=N2(u),x∈Rn, t>0 u(0,x)=εu0(x), ∂tu(0,x)=εu1(x), v(0,x)=εv0(x), ∂tv(0,x)=εv1(x), x∈Rn, with super-critical nonlinearities Nk(ϕ)=|ϕ|ρk, k= 1, 2, where ε>0, the space dimension n≥4. Our purpose is to remove the exponential decay condition on the data and the lower bound for ρ1 which was assumed in [4] when proving the global existence of solutions in the case of higher space dimensions.",

keywords = "A weakly coupled system, Global existence, Super-critical case, System of damped wave equations, Time decay",

author = "Nakao Hayashi and Naumkin, {Pavel I.} and Masayo Tominaga",

note = "Funding Information: We would like to thank the unknown referee for useful comments on the first draft. The work of N.H. is partially supported by JSPS KAKENHI Grant Number 25220702 . The work of P.I.N. is partially supported by CONACYT and PAPIIT project IN100113 . Publisher Copyright: {\textcopyright} 2015 Elsevier Inc.",

year = "2015",

doi = "10.1016/j.jmaa.2015.03.008",

language = "English",

volume = "428",

pages = "490--501",

journal = "Journal of Mathematical Analysis and Applications",

issn = "0022-247X",

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AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

AU - Tominaga, Masayo

N1 - Funding Information: We would like to thank the unknown referee for useful comments on the first draft. The work of N.H. is partially supported by JSPS KAKENHI Grant Number 25220702 . The work of P.I.N. is partially supported by CONACYT and PAPIIT project IN100113 . Publisher Copyright: © 2015 Elsevier Inc.

PY - 2015

Y1 - 2015

N2 - We study global existence of small solutions to the Cauchy problem for a weakly coupled nonlinear damped wave equation{(∂2t+∂t-δ)u=N1(v), (∂2t+∂t-δ)v=N2(u),x∈Rn, t>0 u(0,x)=εu0(x), ∂tu(0,x)=εu1(x), v(0,x)=εv0(x), ∂tv(0,x)=εv1(x), x∈Rn, with super-critical nonlinearities Nk(ϕ)=|ϕ|ρk, k= 1, 2, where ε>0, the space dimension n≥4. Our purpose is to remove the exponential decay condition on the data and the lower bound for ρ1 which was assumed in [4] when proving the global existence of solutions in the case of higher space dimensions.

AB - We study global existence of small solutions to the Cauchy problem for a weakly coupled nonlinear damped wave equation{(∂2t+∂t-δ)u=N1(v), (∂2t+∂t-δ)v=N2(u),x∈Rn, t>0 u(0,x)=εu0(x), ∂tu(0,x)=εu1(x), v(0,x)=εv0(x), ∂tv(0,x)=εv1(x), x∈Rn, with super-critical nonlinearities Nk(ϕ)=|ϕ|ρk, k= 1, 2, where ε>0, the space dimension n≥4. Our purpose is to remove the exponential decay condition on the data and the lower bound for ρ1 which was assumed in [4] when proving the global existence of solutions in the case of higher space dimensions.

KW - A weakly coupled system

KW - Global existence

KW - Super-critical case

KW - System of damped wave equations

KW - Time decay

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JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

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Remark on a weakly coupled system of nonlinear damped wave equations

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