## Topological Methods in Nonlinear Analysis

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

#### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 17, Number 1 (2001), 91-109.

Dates
First available in Project Euclid: 22 August 2016

https://projecteuclid.org/euclid.tmna/1471875799

Mathematical Reviews number (MathSciNet)
MR1846980

Zentralblatt MATH identifier
0989.35091

#### Citation

Papadopoulos, Perikles G.; Stavrakakis, Nikos M. Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$. Topol. Methods Nonlinear Anal. 17 (2001), no. 1, 91--109. https://projecteuclid.org/euclid.tmna/1471875799

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