Global and unique weak solutions of nonlinear wave equations. (English) Zbl 0841.35067

Kapitanski, Lev

Math. Res. Lett. 1, No. 2, 211-223 (1994).

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.

Reviewer: E.Feireisl (Praha)

MSC:

35L70	Second-order nonlinear hyperbolic equations
35L15	Initial value problems for second-order hyperbolic equations
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

nonlinear wave equations; critical growth at infinity