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Stability of the heat and of the wave equations with boundary timevarying delays
On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms
1.  Department of Mathematics and Statistics, Federal University of Campina Grande, 58109970, Campina Grande, PB, Brazil 
2.  Department of Mathematics  State University of Maringá, 87020900 Maringá, PR, Brazil 
3.  Department of Mathematics, State University of Maringá, Maringá, PR, 87020900, Brazil 
4.  Department of Mathematics, University of NebraskaLincoln, Lincoln, NE, 685880130, United States 
5.  Department of Mathematics, University of NebraskaLincoln, Lincoln, NE 68588 
$u_{t t}$ $ \Delta u + u_t^{m1}u_t = F_u(u,v) \text{ in }\Omega\times ( 0,\infty )$,
$v_{t t}$$  \Delta v + v_t^{r1}v_t = F_v(u,v) \text{ in }\Omega\times( 0,\infty )$,
where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n=1,2,3$ with a smooth boundary $\partial\Omega=\Gamma$ and $F$ is a $C^1$ function given by
$ F(u,v)=\alphau+v^{p+1}+ 2\beta uv^{\frac{p+1}{2}}. $
Under some conditions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain several results on the global existence, uniform decay rates, and blow up of solutions in finite time when the initial energy is nonnegative.
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