Global existence for a wave equation on $R^n$
Perikles G. Papadopoulos Nikolaos M. Stavrakakis
Discrete & Continuous Dynamical Systems - S 2008, 1(1): 139-149 doi: 10.3934/dcdss.2008.1.139
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

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