### Asymptotic behavior of solutions of some nonlinearly damped wave equations on $\mathbb R^N$

#### Abstract

We discuss the asymptotic behavior of solutions of the nonlinearly

damped wave equation

$$

u_{tt} +\delta \vert u_t\vert ^{m-1}u_t -\phi (x)\Delta u = \lambda

u\vert u\vert ^{\beta -1}, \quad x \in \mathbb R^n, \ t \geq 0,

$$

with the

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$ and $(\phi

(x))^{-1} =g (x)$ is a positive function lying in

$L^{p}(\mathbb R^n)\cap L^{\infty}(\mathbb R^n)$, for some $p$. We prove

blow-up of solutions when the source term dominates over the

damping, and the initial energy is assumed to be positive. We also

discuss global existence energy decay of solutions.

damped wave equation

$$

u_{tt} +\delta \vert u_t\vert ^{m-1}u_t -\phi (x)\Delta u = \lambda

u\vert u\vert ^{\beta -1}, \quad x \in \mathbb R^n, \ t \geq 0,

$$

with the

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$ and $(\phi

(x))^{-1} =g (x)$ is a positive function lying in

$L^{p}(\mathbb R^n)\cap L^{\infty}(\mathbb R^n)$, for some $p$. We prove

blow-up of solutions when the source term dominates over the

damping, and the initial energy is assumed to be positive. We also

discuss global existence energy decay of solutions.

#### Keywords

Semilinear hyperbolic equations; blow-up; nonlinear dissipation; potential well; concavity method; unbounded domains; generalized Sobolev spaces

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