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Zbl 0892.35102
Delort, Jean-Marc
On the existence time of the semilinear Klein-Gordon equation in dimension one. (Sur le temps d'existence pour l'équation de Klein-Gordon semi-linéaire en dimension 1.) (French)
[J] Bull. Soc. Math. Fr. 125, No.2, 269-311 (1997). ISSN 0037-9484

In the paper under consideration the Cauchy problem for a second order semilinear Klein-Gordon equation is studied in the case of one space variable $x$. The nonlinear term $F$ is a polynomial of the unknown function $u$ and its gradient $(\partial_tu, \partial_xu)$. The Cauchy data are $\varepsilon>0$ small and have a weak decay at infinity, i.e. they belong to some Sobolev class $H^N(\bbfR)$, $N\ge 3$. Moreover, the polynomial $F$ is a linear combination of bilinear forms verifying a Kosecki type null condition. The main result asserts that there exists a unique solution with life-span time $T_\varepsilon\ge c\varepsilon^{-4} | \log \varepsilon |^{-G}$, $c=\text {const}>0$. A global existence result is valid if $F\equiv f(u) \in C^\infty$, $f(u)= O(u^2)$, $u\to 0$.
[P.Popivanov (Sofia)]

MSC 2000:: *35L70 Second order nonlinear hyperbolic equations
35L15 Second order hyperbolic equations, initial value problems
35B40 Asymptotic behavior of solutions of PDE

Keywords: Kosecki type null condition; life-span time

Cited in: Zbl 0962.35023

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