Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term

Kenji Nishihara

    研究成果: Article

    22 引用 (Scopus)

    抄録

    We consider the Cauchy problem for the damped wave equation with space dependent potential V(x)ut and absorbed semilinear term {pipe}u{pipe}ρ-1u in RN. Our assumption on V(x) ~ (1 + {pipe}x{pipr}2)-α/2 (0 ≤ α < 1) still implies the diffusion phenomena and the decay rates of solutions are expected to be the same as the corresponding parabolic problem. In this paper we obtain two kinds of decay rates of the solution effective for ρ > ρc(N, α):=1+2/(N - α) and for ρ < ρc(N, α). We believe that in the "supercritical" exponent the decay rates obtained are almost the same as those for the linear parabolic problem, while, in the "subcritical" exponent the solution decays faster than that of linear equation, thanks to the absorbed semilinear term. So we believe that ρc(N, α) is a critical exponent. Note that ρc(N, α) with α = 0 coincides to the Fujita exponent ρF(N):=1+2/N.

    元の言語English
    ページ(範囲)1402-1418
    ページ数17
    ジャーナルCommunications in Partial Differential Equations
    35
    発行部数8
    DOI
    出版物ステータスPublished - 2010 8

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    Damped Wave Equation
    Wave equations
    Semilinear
    Exponent
    Pipe
    Decay
    Dependent
    Term
    Parabolic Problems
    Linear equations
    Decay Rate
    Critical Exponents
    Linear equation
    Cauchy Problem

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    これを引用

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    N2 - We consider the Cauchy problem for the damped wave equation with space dependent potential V(x)ut and absorbed semilinear term {pipe}u{pipe}ρ-1u in RN. Our assumption on V(x) ~ (1 + {pipe}x{pipr}2)-α/2 (0 ≤ α < 1) still implies the diffusion phenomena and the decay rates of solutions are expected to be the same as the corresponding parabolic problem. In this paper we obtain two kinds of decay rates of the solution effective for ρ > ρc(N, α):=1+2/(N - α) and for ρ < ρc(N, α). We believe that in the "supercritical" exponent the decay rates obtained are almost the same as those for the linear parabolic problem, while, in the "subcritical" exponent the solution decays faster than that of linear equation, thanks to the absorbed semilinear term. So we believe that ρc(N, α) is a critical exponent. Note that ρc(N, α) with α = 0 coincides to the Fujita exponent ρF(N):=1+2/N.

    AB - We consider the Cauchy problem for the damped wave equation with space dependent potential V(x)ut and absorbed semilinear term {pipe}u{pipe}ρ-1u in RN. Our assumption on V(x) ~ (1 + {pipe}x{pipr}2)-α/2 (0 ≤ α < 1) still implies the diffusion phenomena and the decay rates of solutions are expected to be the same as the corresponding parabolic problem. In this paper we obtain two kinds of decay rates of the solution effective for ρ > ρc(N, α):=1+2/(N - α) and for ρ < ρc(N, α). We believe that in the "supercritical" exponent the decay rates obtained are almost the same as those for the linear parabolic problem, while, in the "subcritical" exponent the solution decays faster than that of linear equation, thanks to the absorbed semilinear term. So we believe that ρc(N, α) is a critical exponent. Note that ρc(N, α) with α = 0 coincides to the Fujita exponent ρF(N):=1+2/N.

    KW - Absorbed semilinear term

    KW - Damped wave equation

    KW - Space dependent potential

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