Global existence for a wave equation on $R^n$

Perikles G. Papadopoulos; Nikolaos M. Stavrakakis

doi:10.3934/dcdss.2008.1.139

Article Contents

2008, Volume 1, Issue 1: 139-149. Doi: 10.3934/dcdss.2008.1.139

This issue Previous Article Higher order two-point boundary value problems with asymmetric growth Next Article 3D steady compressible Navier--Stokes equations

Global existence for a wave equation on $R^n$

Perikles G. Papadopoulos^1, and
Nikolaos M. Stavrakakis^1,

1.
Department of Mathematics, National Technical University, Zografou Campus 157 80, Athens, Hellas

Received: August 31, 2006

Revised: December 31, 2006

Published: February 2008

Abstract Related Papers Cited by

Abstract

We study the initial value problem for some degenerate non-linear dissipative wave equations of Kirchhoff type: $ u_{t t}-\phi (x)||\grad u(t)||^{2\gamma}\Delta u+\delta u_{t} = f(u),x\in R^n,t\geq 0,$ with initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N \geq 3, delta > 0, \gamma\geq 1$, $f(u)=|u|^{a}u$ with $a>0$ and $(\phi (x))^{-1} =g (x)$ is a positive function lying in $L^{N/2}(R^n)\cap L^{\infty}(R^n)$. If the initial data $\{ u_{0},u_{1}\}$ are small and $||\grad u_{0}||>0$, then the unique solution exists globally and has certain decay properties.

Keywords:

Mathematics Subject Classification: 35A07, 35B30, 35B40, 35B45,35L15, 35L70,35L80.

Citation:

Article Metrics

HTML views() PDF downloads(69) Cited by(0)

Global existence for a wave equation on $R^n$

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

Global existence for a wave equation on $R^n$

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content