On the existence of global solutions of nonlinear wave equations. (Zur Existenz globaler Lösungen nichtlinearer Wellengleichungen.) (German) Zbl 0798.35107

In the last decade the existence theory for small global-in-time solutions of initial-value problems to fully nonlinear evolution equations and systems has been strongly developed. This clearly written survey article gives an introduction to this theory, presents many of the original results and proofs. The relevant ideas are best explained by the example of the initial-value problem for the nonlinear wave equation. It is well known for the nonlinear wave equation that in general one cannot expect to obtain a global classical solution. Therefore, a general global existence theorem can only be proved under special assumptions on the nonlinearity and on the initial data. The result is a theorem which is applicable for small initial data, assuming a certain degree of vanishing of the nonlinearity near zero. The necessary degree depends on the space dimension, being strongly connected with the asymptotic behavior of solutions to the associated linearized problem. The presentation of the theory is made using the classical method of continuation of local classical solutions with the help of a priori estimates obtained for small initial data. The proof of the a priori estimates represents the non-classical part of the approach. It requires in particular the idea of using decay of solutions of the associated linearized problem. This general idea can be used to prove the existence of small global-in-time solutions not only for the wave equation but for many nonlinear evolution equations. The article contains a lot of examples, remarks and references, which will make it pleasant to read by interested people and specialists.
Reviewer: S.Jiang (Bonn)


35L70 Second-order nonlinear hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35L45 Initial value problems for first-order hyperbolic systems