Global existence and blow-up of solutionsfor a higher-order kirchhoff-type equation with nonlinear dissipation

Abstract

This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipation u t t + ( ∫ Ω ׀ D m u ׀ 2 d x ) q ( − Δ ) m u + u t ׀ u t ׀ r = ׀ u ׀ p u , x ∈ Ω , t > 0 , in a bounded domain, where m < 1 is a positive integer, q, p, r < 0 arepositive constants. We obtain that the solution exists globally if p ≤ r, while if p > max{r, 2q}, then for any initial data with negative initial energy, the solution blowsup at finite time in L p +2 norm.