Long time behaviour of solutions of nonlinear wave equations. (Solutions en grand temps d’équations d’ondes non linéaires.) (French) Zbl 0830.35088

Zuily, Claude

Séminaire Bourbaki. Volume 1993/94. Exposés 775-789. Paris: Société Mathématique de France, Astérisque. 227, 107-144 (Exp. No. 779) (1995).

The author presents a survey on the Cauchy problem for nonlinear wave equations, focussing both on small data results and on semilinear dispersive wave equations. In the first part the classical existence results are recalled. Here, and similarly for the results presented afterwards, a sketch of the proof is given. Each result is complemented by extensive comments. Then the global existence results for small data as well as the estimates on the life span in the general case are formulated. In the second part dispersive semilinear wave equations of the type \(u_{tt} - \Delta u=f(u)\), with nonnegative \(\int^u_0 f(t)dt\), are studied, typically \(f(u) = u |u |^{p - 1}\). Global existence results for arbitrary (smooth) data are presented, in particular for the critical exponent \(p^* = {n + 2 \over n - 2}\), \(n\) being the space dimension. The recent result of Shatah and Struwe for \(3 \leq n \leq 7\), \(p = p^*\), and its proof are discussed extensively. A long list of references, going beyond those referred to in the text, is provided.
For the entire collection see [Zbl 0811.00012].

Reviewer: R.Racke (Konstanz)

MSC:

35L70	Second-order nonlinear hyperbolic equations
35B40	Asymptotic behavior of solutions to PDEs
35A07	Local existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

blow-up; critical exponent; semilinear dispersive wave equations; life span