### Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$

DOI: http://dx.doi.org/10.12775/TMNA.2001.006

#### Abstract

We discuss the asymptotic behaviour of solutions for the nonlocal

quasilinear hyperbolic problem of Kirchhoff Type

$$ u_{tt}-\phi (x)\Vert\nabla u(t)\Vert^{2}\Delta u+\delta u_{t} =

|u|^{a}u,\quad x\in {\mathbb 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 $N \geq 3$, $\delta \geq 0$ and

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

$L^{N/2}(\mathbb R^N)\cap L^{\infty}(\mathbb R^N )$. When the

initial energy $ E(u_{0},u_{1})$, which corresponds to the

problem, is non-negative and small, there exists a unique global

solution in time. When the initial energy $E(u_{0},u_{1})$ is

negative, the solution blows-up in finite time. A combination of

the modified potential well method and the concavity method is

widely used.

quasilinear hyperbolic problem of Kirchhoff Type

$$ u_{tt}-\phi (x)\Vert\nabla u(t)\Vert^{2}\Delta u+\delta u_{t} =

|u|^{a}u,\quad x\in {\mathbb 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 $N \geq 3$, $\delta \geq 0$ and

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

$L^{N/2}(\mathbb R^N)\cap L^{\infty}(\mathbb R^N )$. When the

initial energy $ E(u_{0},u_{1})$, which corresponds to the

problem, is non-negative and small, there exists a unique global

solution in time. When the initial energy $E(u_{0},u_{1})$ is

negative, the solution blows-up in finite time. A combination of

the modified potential well method and the concavity method is

widely used.

#### Keywords

Quasilinear hyperbolic equations; global solution; blow-up; dissipation; potential well; concavity method; unbounded domains; Kirchhoff strings; generalised Sobolev spaces; weighted $L^p$ spaces

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