Decay property of solutions for damped wave equations with space-time dependent damping term

Jiayun Lin, Kenji Nishihara, Jian Zhai

    Research output: Contribution to journalArticle

    12 Citations (Scopus)

    Abstract

    We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t,x) and absorbing semilinear term |u|ρ-1u. Here, b(t,x)=b0(1+|x|2)α-2(1+t)-β with b0>0, α,β≥0 and α+β∈[0,1). Using the weighted energy method, we can obtain the L2 decay rate of the solution, which is almost optimal in the case ρ>ρc(N,α,β):=1+2/(N-α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L2-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρc(N,α,β) is a critical exponent. Note that when α=β=0, ρc(N,alpha;,beta;) coincides to the Fujita exponent ρF(N):=1+2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.

    Original languageEnglish
    Pages (from-to)602-614
    Number of pages13
    JournalJournal of Mathematical Analysis and Applications
    Volume374
    Issue number2
    DOIs
    Publication statusPublished - 2011 Feb 15

    Fingerprint

    Damped Wave Equation
    Damping Term
    Wave equations
    Decay Rate
    Damping
    Space-time
    Exponent
    Decay
    Dependent
    Energy Method
    Absorbing
    Semilinear
    Estimate
    Critical Exponents
    Cauchy Problem
    Differential equations
    Differential equation
    Term

    Keywords

    • Damped wave equation
    • Decay rate
    • Supercritical case
    • The weighted energy method

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    Decay property of solutions for damped wave equations with space-time dependent damping term. / Lin, Jiayun; Nishihara, Kenji; Zhai, Jian.

    In: Journal of Mathematical Analysis and Applications, Vol. 374, No. 2, 15.02.2011, p. 602-614.

    Research output: Contribution to journalArticle

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    N2 - We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t,x) and absorbing semilinear term |u|ρ-1u. Here, b(t,x)=b0(1+|x|2)α-2(1+t)-β with b0>0, α,β≥0 and α+β∈[0,1). Using the weighted energy method, we can obtain the L2 decay rate of the solution, which is almost optimal in the case ρ>ρc(N,α,β):=1+2/(N-α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L2-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρc(N,α,β) is a critical exponent. Note that when α=β=0, ρc(N,alpha;,beta;) coincides to the Fujita exponent ρF(N):=1+2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.

    AB - We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t,x) and absorbing semilinear term |u|ρ-1u. Here, b(t,x)=b0(1+|x|2)α-2(1+t)-β with b0>0, α,β≥0 and α+β∈[0,1). Using the weighted energy method, we can obtain the L2 decay rate of the solution, which is almost optimal in the case ρ>ρc(N,α,β):=1+2/(N-α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L2-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρc(N,α,β) is a critical exponent. Note that when α=β=0, ρc(N,alpha;,beta;) coincides to the Fujita exponent ρF(N):=1+2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.

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