Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$
Keywords
Quasilinear hyperbolic equations, global solution, blow-up, dissipation, potential well, concavity method, unbounded domains, Kirchhoff strings, generalised Sobolev spaces, weighted $L^p$ spacesAbstract
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.Downloads
Published
2001-03-01
How to Cite
1.
PAPADOPOULOS, Perikles G. and STAVRAKAKIS, Nikos M. Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$. Topological Methods in Nonlinear Analysis. Online. 1 March 2001. Vol. 17, no. 1, pp. 91 - 110. [Accessed 18 February 2025].
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