Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption

Kenji Nishihara, Huijiang Zhao

    Research output: Contribution to journalArticle

    37 Citations (Scopus)

    Abstract

    We consider the Cauchy problem for the damped wave equation with absorption utt - Δu + ut + u p-1u = 0, (t,x) ∈ R+ × RN. The behavior of u as t → ∞ is expected to be the same as that for the corresponding heat equation φt - Δφ + φ p-1φ = 0, (t,x) ∈ R+ × RN, which has the similarity solution wa(t,x) with the form t-1/(p-1)f(x/√t) depending on a = lim x →∞ x 2/(p-1) f(x) ≥ 0 provided that p is less than the Fujita exponent pc(N) := 1+2/N. In this paper, as a first step, if 1 < p < pc(N) and the data (u0, u1) (x) decays exponentially as x → ∞ without smallness condition, the solution is shown to decay with rates as t → ∞, (∥u(t)∥L2, ∥u(t)∥Lp+1, ∥∇u(t)∥L2) = O (t-1/p-1+N/4, t-1/p-1+N/2(p+1), t-1/p-1-1/2+N/4), (*) those of which seem to be reasonable, because the similarity solution wa (t,x) has the same decay rates as (*). For the proof, the weighted L2-energy method will be employed with suitable weight, similar to that in Todorova and Yordanov [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

    Original languageEnglish
    Pages (from-to)598-610
    Number of pages13
    JournalJournal of Mathematical Analysis and Applications
    Volume313
    Issue number2
    DOIs
    Publication statusPublished - 2006 Jan 15

    Fingerprint

    Damped Wave Equation
    Similarity Solution
    Wave equations
    Cauchy Problem
    Absorption
    Decay
    Nonlinear Wave Equation
    Energy Method
    Decay Rate
    Heat Equation
    Critical Exponents
    Damping
    Differential equations
    Exponent
    Differential equation
    Form
    Hot Temperature

    Keywords

    • Decay rate
    • Similarity solution
    • Subcritical
    • Weighted energy method

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption. / Nishihara, Kenji; Zhao, Huijiang.

    In: Journal of Mathematical Analysis and Applications, Vol. 313, No. 2, 15.01.2006, p. 598-610.

    Research output: Contribution to journalArticle

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    abstract = "We consider the Cauchy problem for the damped wave equation with absorption utt - Δu + ut + u p-1u = 0, (t,x) ∈ R+ × RN. The behavior of u as t → ∞ is expected to be the same as that for the corresponding heat equation φt - Δφ + φ p-1φ = 0, (t,x) ∈ R+ × RN, which has the similarity solution wa(t,x) with the form t-1/(p-1)f(x/√t) depending on a = lim x →∞ x 2/(p-1) f(x) ≥ 0 provided that p is less than the Fujita exponent pc(N) := 1+2/N. In this paper, as a first step, if 1 < p < pc(N) and the data (u0, u1) (x) decays exponentially as x → ∞ without smallness condition, the solution is shown to decay with rates as t → ∞, (∥u(t)∥L2, ∥u(t)∥Lp+1, ∥∇u(t)∥L2) = O (t-1/p-1+N/4, t-1/p-1+N/2(p+1), t-1/p-1-1/2+N/4), (*) those of which seem to be reasonable, because the similarity solution wa (t,x) has the same decay rates as (*). For the proof, the weighted L2-energy method will be employed with suitable weight, similar to that in Todorova and Yordanov [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].",
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    N2 - We consider the Cauchy problem for the damped wave equation with absorption utt - Δu + ut + u p-1u = 0, (t,x) ∈ R+ × RN. The behavior of u as t → ∞ is expected to be the same as that for the corresponding heat equation φt - Δφ + φ p-1φ = 0, (t,x) ∈ R+ × RN, which has the similarity solution wa(t,x) with the form t-1/(p-1)f(x/√t) depending on a = lim x →∞ x 2/(p-1) f(x) ≥ 0 provided that p is less than the Fujita exponent pc(N) := 1+2/N. In this paper, as a first step, if 1 < p < pc(N) and the data (u0, u1) (x) decays exponentially as x → ∞ without smallness condition, the solution is shown to decay with rates as t → ∞, (∥u(t)∥L2, ∥u(t)∥Lp+1, ∥∇u(t)∥L2) = O (t-1/p-1+N/4, t-1/p-1+N/2(p+1), t-1/p-1-1/2+N/4), (*) those of which seem to be reasonable, because the similarity solution wa (t,x) has the same decay rates as (*). For the proof, the weighted L2-energy method will be employed with suitable weight, similar to that in Todorova and Yordanov [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

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