Global existence, asymptotic behaviour, and global non-existence of solutions for damped non-linear wave equations of Kirchhoff type in the whole space

Abstract

We study the global existence, asymptotic behaviour, and global non-existence (blow-up) of solutions for the damped non-linear wave equation of Kirchhoff type in the whole space: utt+ut=(a+b∥∇u∥2γ)Δu+∣u∣αu in ℝN×ℝ+ for a, b⩾0, a+b>0, γ⩾1, and α>0, with initial data u(x, 0)=u0(x) and ut(x, 0)=u1(x). Copyright © 2000 John Wiley & Sons, Ltd.