About the solvability behaviour for special classes of nonlinear hyperbolic equations (Q1863444)

Global existence for small data solutions to a special type of semilinear wave equations with time-dependent coefficient in the principle part of the form \[ u_{tt}-a^2(t)\Delta u=u_{t}^2-a^2(t)\left(\nabla u\right)^2 \] with \(u(0,x),u_t(0,x)\in C_0^\infty(\mathbb R^n)\) is investigated. Three cases of \(a(t)\) are considered. For \(a(t)=t^l\), \(l>0\), small data solutions exist globally. For \(a(t)=t^{-2}\exp(t^{-1})\), there exist arbitrary small \(C_0^\infty\)-data leading to blowing up solutions. This means that an increasing coefficient may have an improving influence on global existence for small data solutions. For the slowly oscillating coefficient \(a(t)=\lambda(t)b(t)\) with positive and smooth \(\lambda(t)\) (for example \(\lambda(t)=\exp(t^\alpha)\) or \(\lambda(t)=(1+t)^l\), \(\alpha,l>0\)) and with positive, smooth and periodic \(b(t)\) small data solutions exist globally.