The LP - Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media

Pierangelo Marcati, Kenji Nishihara

    Research output: Contribution to journalArticle

    98 Citations (Scopus)

    Abstract

    We first obtain the Lp-Lq estimates of solutions to the Cauchy problem for one-dimensional damped wave equation Vtt, - Vxx + Vt = 0 (V, Vt t=0 = (V0, V1)(x), ( x, t)∈R × R+, corresponding to that for the parabolic equation φt - φxx = 0 φ t=0 = (V0 + V1)(x). The estimates are shown by An equation is presented etc. for 1 ≤q≤p≤ ∞. To show (*), the explicit formula of the damped wave equation will be used. To apply the estimates to nonlinear problems is the second aim. We will treat the system of a compressible flow through porous media. The solution is expected to behave as the diffusion wave, which is the solution to the porous media equation due to the Darcy law. When the initial data has the same constant state at ± ∞, a sharp Lp-convergence rate for p ≥ 2 has been recently obtained by Nishihara (Proc. Roy. Soc. Edinburgh, Sect. A, 133A (2003), 1-20) by choosing a suitably located diffusion wave. We will show the L1 convergence, applying (*).

    Original languageEnglish
    Pages (from-to)445-469
    Number of pages25
    JournalJournal of Differential Equations
    Volume191
    Issue number2
    DOIs
    Publication statusPublished - 2003 Jul 1

    Fingerprint

    Lp Estimates
    Damped Wave Equation
    Compressible flow
    Compressible Flow
    Wave equations
    Porous Media
    Porous materials
    L1-convergence
    Darcy's Law
    Porous Medium Equation
    Estimate
    Parabolic Equation
    Nonlinear Problem
    Convergence Rate
    Explicit Formula
    Cauchy Problem

    ASJC Scopus subject areas

    • Analysis

    Cite this

    The LP - Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media. / Marcati, Pierangelo; Nishihara, Kenji.

    In: Journal of Differential Equations, Vol. 191, No. 2, 01.07.2003, p. 445-469.

    Research output: Contribution to journalArticle

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