Global existence and blow-up results for an equation of Kirchhoff type on R^N - Papadopoulos, Perikles G.; Stavrakakis, Nikos M. (Topological Methods in Nonlinear Analysis, 2001)
We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type u_tt-φ (x)abla u(t)^2Δ u+δ u_t = |u|^au,xR^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(R^N)L^∞(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.