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