The \(L^{p}\)--\(L^{q}\) estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media
type: Item
instance of: scholarly article
title: The \(L^{p}\)--\(L^{q}\) estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media
zbMATH Open document ID: 1031.35031
DOI: 10.1016/S0022-0396(03)00026-3
author: Pierangelo Marcati
author: Kenji Nishihara
published in: Journal of Differential Equations
publication date: 2003-07-30T00:00:00Z
review text: The paper refers to \(L^p\)-\(L^q\) estimates for the solution of the Cauchy problem for the damped wave equations of the form \(V_{tt}- V_{xx}+ V_t= 0\), \(x\in\mathbb{R}^1\), \(t> 0\), \((V,V_t)|_{t=0}= (V_0, V_1)(x)\), corresponding to the solution for the parabolic equation \(\varphi_t- \varphi_{xx}= 0\), \(\varphi|_{t=0}= (V_0+ V_1)(x)\). The following types of estimates are obtained: \[ \begin{multlined}\Biggl\|(V- \varphi)(.,t)- e^{-t/2}{V_0(.+t)+ V_0(.-t)\over 2}\Biggr\|_{L^p}\leq\\ Ct^{-{1\over 2}({1\over q}-{1\over p})-1}\|V_0, V_1\|_{L^q},\quad t\geq 1,\quad 1\leq q\leq p\leq\infty.\end{multlined} \] As an application, the authors investigate the Cauchy problem for the one-dimensional compressible flow through a porous media.
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 1957270
zbMATH Keywords: diffusive phenomenon
zbMATH Keywords: \(p\)-system with damping
zbMATH Keywords: nonlinear diffusion wave
Wikidata QID: Q58619759
reviewed by: Claudia Simionescu-Badea
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1957270