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The lifespan of classical solutions of nonlinear hyperbolic equations. (English) Zbl 0632.35045

Pseudo-differential operators, Proc. Conf., Oberwolfach/Ger. 1986, Lect. Notes Math. 1256, 214-280 (1987).
[For the entire collection see Zbl 0606.00012.]
This paper is concerned with the life span of classical solutions of nonlinear hyperbolic equations. First the author gives a self-contained and somewhat simplified exposition of F. John’s methods, which allow one to determine the time of blowup asymptotically in two dimensional case. For radial solutions of the nonlinear wave equation in 3 space dimensions John has proved an upper bound for life span. A slightly modified version of his arguments are given in this paper. Then the rest of the paper is devoted to proving existence theorems.
To prove the existence theorems the author first constructs approximate solutions and rewrites the Cauchy problem as a Cauchy problem for the difference between the exact and the approximate solution. The approximate solution is sufficiently accurate to allow existence to be proved by the methods of John-Klainerman and Klainerman in the form given by Hörmander. The limitation in the existence theorem comes from the blowing up of the approximate solution which is obained by solving essentially a scalar first order differential equation.
Reviewer: J.Wang

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L60 First-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L15 Initial value problems for second-order hyperbolic equations

Citations:

Zbl 0606.00012