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

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U2 - 10.3233/ASY-2010-1018

DO - 10.3233/ASY-2010-1018

M3 - Article

AN - SCOPUS:79953328977

SN - 0921-7134

VL - 71

SP - 185

EP - 205

JO - Asymptotic Analysis

JF - Asymptotic Analysis

IS - 4

ER -