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

Kenji Nishihara

    Research output: Contribution to journalArticle

    24 Citations (Scopus)

    Abstract

    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.

    Original languageEnglish
    Pages (from-to)1402-1418
    Number of pages17
    JournalCommunications in Partial Differential Equations
    Volume35
    Issue number8
    DOIs
    Publication statusPublished - 2010 Aug

    Keywords

    • Absorbed semilinear term
    • Damped wave equation
    • Space dependent potential

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

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