Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$ - Papadopoulos, Perikles G.; Stavrakakis, Nikos M. (Topological Methods in Nonlinear Analysis, 2001)
We discuss the asymptotic behaviour of solutions for the nonlocalquasilinear hyperbolic problem of Kirchhoff Type $$ u_{tt}-φ (x)\Vert abla u(t)\Vert^{2}Δ u+δ u_{t} = |u|^{a}u,\quad x\in {\mathbb R}^N,\ t≥ 0,$$with initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) =u_1 (x)$, in the case where $N ≥ 3$, $δ ≥ 0$ and $(φ (x))^{-1} =g (x)$ is a positive function lying in $L^{N/2}(\mathbb R^N)\cap L^{∞}(\mathbb R^N )$. When theinitial energy $ E(u_{0},u_{1})$, which corresponds to theproblem, is non-negative and small, there exists a unique globalsolution in time. When the initial energy $E(u_{0},u_{1})$ isnegative, the solution blows-up in finite time. A combination ofthe modified potential well method and the concavity method iswidely used.