On a class of semilinear weakly hyperbolic equations (Q2465921)

The author is interested in a non-autonomous semilinear wave equation with changing and eventually vanishing speed of propagation, and consider the Cauchy problem \[ u_{tt}(x, t)- a(t)\Delta_x u(x,t)= -b(t)|u(x, t)|^{p-1} u(x,t),\quad x\in\mathbb{R}^n,\tag{1} \] \[ u(x,0)= u_0(x),\quad u_t(x, 0)= u_1(x),\tag{2} \] where \(a(t),b(t): \mathbb{R}\to \mathbb{R}\) are given two continuous functions. In this paper the author discusses the global solvability of (1)--(2) assuming large initial data.