### Abstract

We consider the Cauchy problem for the wave equation with time-dependent damping and absorbing semilinear term u_{tt}-Δu+b(t)u _{t}+|u|^{ρ-1}u=0, (t,x)∈R_{+}×R ^{N}, (u,u_{t})(0,x)=(u_{0},u_{1})(x), x∈R^{N}. When b(t)=b_{0}(t+1)^{-β} with -1<β<1 and b_{0}>0, we want to seek for the asymptotic profile as t→∞ of the solution u to in the supercritical case ρ>ρ_{F}(N):=1+2/N. By the weighted energy method we can show the basic decay rates of u, which are almost the same as those to the corresponding linear parabolic equation φ_{t}-1/b(t)Δφ=0, (t,x)∈R_{+}×R^{N}. When N=1, the decay rates of higher order derivatives of u are obtained by the energy method, so that the solution u can be regarded as that of with source term -1/b(t)(u _{tt}+|u|^{ρ-1}u). Thus, we will show θ _{0}G_{B}(t,x) (θ_{0}: suitable constant) to be an asymptotic profile of u, where G_{B}(t,x) is the fundamental solution of.

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

Pages (from-to) | 185-205 |

Number of pages | 21 |

Journal | Asymptotic Analysis |

Volume | 71 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2011 |

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### Keywords

- asymptotic profile
- supercritical exponent
- time-dependent damping
- wave equation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Asymptotic Analysis*,

*71*(4), 185-205. https://doi.org/10.3233/ASY-2010-1018

**Asymptotic profile of solutions for 1-D wave equation with time-dependent damping and absorbing semilinear term.** / Nishihara, Kenji.

Research output: Contribution to journal › Article

*Asymptotic Analysis*, vol. 71, no. 4, pp. 185-205. https://doi.org/10.3233/ASY-2010-1018

}

TY - JOUR

T1 - Asymptotic profile of solutions for 1-D wave equation with time-dependent damping and absorbing semilinear term

AU - Nishihara, Kenji

PY - 2011

Y1 - 2011

N2 - We consider the Cauchy problem for the wave equation with time-dependent damping and absorbing semilinear term utt-Δu+b(t)u t+|u|ρ-1u=0, (t,x)∈R+×R N, (u,ut)(0,x)=(u0,u1)(x), x∈RN. When b(t)=b0(t+1)-β with -1<β<1 and b0>0, we want to seek for the asymptotic profile as t→∞ of the solution u to in the supercritical case ρ>ρF(N):=1+2/N. By the weighted energy method we can show the basic decay rates of u, which are almost the same as those to the corresponding linear parabolic equation φt-1/b(t)Δφ=0, (t,x)∈R+×RN. When N=1, the decay rates of higher order derivatives of u are obtained by the energy method, so that the solution u can be regarded as that of with source term -1/b(t)(u tt+|u|ρ-1u). Thus, we will show θ 0GB(t,x) (θ0: suitable constant) to be an asymptotic profile of u, where GB(t,x) is the fundamental solution of.

AB - We consider the Cauchy problem for the wave equation with time-dependent damping and absorbing semilinear term utt-Δu+b(t)u t+|u|ρ-1u=0, (t,x)∈R+×R N, (u,ut)(0,x)=(u0,u1)(x), x∈RN. When b(t)=b0(t+1)-β with -1<β<1 and b0>0, we want to seek for the asymptotic profile as t→∞ of the solution u to in the supercritical case ρ>ρF(N):=1+2/N. By the weighted energy method we can show the basic decay rates of u, which are almost the same as those to the corresponding linear parabolic equation φt-1/b(t)Δφ=0, (t,x)∈R+×RN. When N=1, the decay rates of higher order derivatives of u are obtained by the energy method, so that the solution u can be regarded as that of with source term -1/b(t)(u tt+|u|ρ-1u). Thus, we will show θ 0GB(t,x) (θ0: suitable constant) to be an asymptotic profile of u, where GB(t,x) is the fundamental solution of.

KW - asymptotic profile

KW - supercritical exponent

KW - time-dependent damping

KW - wave equation

UR - http://www.scopus.com/inward/record.url?scp=79953328977&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953328977&partnerID=8YFLogxK

U2 - 10.3233/ASY-2010-1018

DO - 10.3233/ASY-2010-1018

M3 - Article

AN - SCOPUS:79953328977

VL - 71

SP - 185

EP - 205

JO - Asymptotic Analysis

JF - Asymptotic Analysis

SN - 0921-7134

IS - 4

ER -