Global existence with large data for a nonlinear weakly hyperbolic equation (Q2770423)

The authors of this interesting paper investigate a semilinear degenerate wave equation \(u_{tt}-a(t)\triangle u=-f(u)\) on \([0,T]\times \mathbb R_x^n \) for cases \(n=1,2\) under initial condition \(u(0,x)=u(x)\), \(u_t(0,x)=u_1(x)\) (large data) in the case of a subcritical defocusing power nonlinearity. Here \(|f(s)|=C(1+|s|)^p\), \(|f'(s)|=C(1+|s|)^{p-1}\), \(\int_0^sf(\sigma)d\sigma =F(s)\geq C|s|^{p+1}\) (\(C>0\), \(p\geq 1\)). The coefficient \(a(t)\geq 0\) is piecewise \(C^2\) and increasing for \(t>t_0\) and vanishes of finite order for \(t<t_0\). The main result is that the problem under consideration has a unique solution \(u\in C^2([0,T],C^{\infty }(\mathbb R))\) \((p<(k+3)/(k-1))\) and any \(p\) if \(k\leq 1\). The second result is that there exists a unique \(C^2([0,T],C^{\infty }(\mathbb R))\) solution, under similar conditions, for any initial data \(u_0,u_1\in C^{\infty} (\mathbb R)\), provided \(p<1+4/(\sqrt{2k^2+8k+4}-2)\).