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March  2008, 1(1): 139-149. doi: 10.3934/dcdss.2008.1.139

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, Greece, Greece

Received  September 2006 Revised  January 2007 Published  December 2007

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: Quasilinear Hyperbolic Equations, Generalised Sobolev Spaces, Blow-Up, Dissipation, Global Solution, Concavity Method, Kirchhoff Strings, Unbounded Domains, Weighted $L^p$ Spaces., Potential Well.
Mathematics Subject Classification: 35A07, 35B30, 35B40, 35B45,35L15, 35L70,35L8.
Citation: Perikles G. Papadopoulos, Nikolaos M. Stavrakakis. Global existence for a wave equation on $R^n$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 139-149. doi: 10.3934/dcdss.2008.1.139
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