Global and unique weak solutions of nonlinear wave equations (Q1901650)

The existence and uniqueness of global weak solutions of the problem \({\mathcal L} u+ f(u)= 0\), \(u(0)= u_0\), \(u_t(0)= u_1\) is studied, where \(\mathcal L\) is a linear wave operator and the nonlinearity \(f\) has the so-called critical growth at infinity, typically, \[ f(u)= |u|^\sigma u,\quad \sigma= {4\over N- 2}, \] where \(N\) is the spatial dimension. The main result of the paper states that all weak solutions \((u, u_t)\) of the problem are continuous in time with values in the energy space \(H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)\). In particular, the energy is a continuous function of time.