×

zbMATH — the first resource for mathematics

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) Zbl 0892.35102
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 \(H^N(\mathbb{R})\), \(N\geq 3\). 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 \(T_\varepsilon\geq c\varepsilon^{-4} | \log \varepsilon |^{-G}\), \(c=\text{const}>0\). A global existence result is valid if \(F\equiv f(u) \in C^\infty\), \(f(u)= O(u^2)\), \(u\to 0\).

MSC:
35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI Link Numdam EuDML
References:
[1] BONY (J.-M.) . - Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles nonlinéaires , Ann. Sci. École Norm. Sup., t. 14, 1981 , p. 209-256. Numdam | MR 84h:35177 | Zbl 0495.35024 · Zbl 0495.35024
[2] BONY (J.-M.) . - Second microlocalization and propagation of singularities for semilinear hyperbolic equations , Hyperbolic equations and related topics (Katata/Kyoto, 1984). - Academic Press, Boston, 1986 , p. 11-49. MR 89e:35099 | Zbl 0669.35073 · Zbl 0669.35073
[3] BOURGAIN (J.) . - Fourier transforms restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , I, II, Geom. Funct. Anal., t. 3, 1993 , p. 107-156, 202-262. MR 95d:35160a | Zbl 0787.35097 · Zbl 0787.35097
[4] FANG (Y.-F.) et GRILLAKIS (M.G.) . - A priori estimates for the 2-d wave equation , Commun. Part. Diff. Eqs, t. 21, 1996 , p. 1643-1665. MR 97f:35018 | Zbl 0861.35053 · Zbl 0861.35053
[5] GEORGIEV (V.) et POPIVANOV (P.) . - Global solutions to the two-dimensional Klein-Gordon equations , Commun. Part. Diff. Eqs, t. 16, 1991 , p. 941-995. MR 92g:35140 | Zbl 0741.35039 · Zbl 0741.35039
[6] HÖRMANDER (L.) . - Non-linear Hyperbolic Differential Equations , Lectures Notes in Lund, preprint, 1986 - 1987 .
[7] KENIG (C.) , PONCE (G.) et VEGA (L.) . - The Cauchy problem for the Korteweg-de-Vries equation on Sobolev spaces of negative indices , Duke Math. J., t. 71, 1993 , p. 1-21. Article | MR 94g:35196 | Zbl 0787.35090 · Zbl 0787.35090
[8] KENIG (C.) , PONCE (G.) et VEGA (L.) . - A bilinear estimate with applications to the KdV equation , J. Amer. Math. Soc., t. 9, 1996 , p. 573-603. MR 96k:35159 | Zbl 0848.35114 · Zbl 0848.35114
[9] KLAINERMAN (S.) . - Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions , Comm. Pure Appl. Math., t. 38, 1985 , p. 631-641. MR 87e:35080 | Zbl 0597.35100 · Zbl 0597.35100
[10] KLAINERMAN (S.) et MACHEDON (M.) . - Smoothing estimates for null forms and applications , Duke Math. J., 1996 , p. 99-131. Article | MR 97h:35022 | Zbl 0909.35094 · Zbl 0909.35094
[11] KOSECKI (R.) . - The Unit Condition and Global Existence for a Class of Nonlinear Klein-Gordon Equations , Jour. Diff. Eq., t. 100, 1992 , p. 257-268. MR 93k:35178 | Zbl 0781.35062 · Zbl 0781.35062
[12] MORIYAMA (K.) , TONEGAWA (S.) et TSUTSUMI (Y.) . - Almost Global Existence of Solutions for the Quadratic Semilinear Klein-Gordon Equation in One Space Dimension , preprint, 1996 . · Zbl 0925.35139
[13] OZAWA (T.) , TSUTAYA (K.) et TSUTSUMI (Y.) . - Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions , Math. Z., t. 222, 1996 , p. 341-362. Article | MR 97e:35112 | Zbl 0877.35030 · Zbl 0877.35030
[14] SHATAH (J.) . - Normal forms and quadratic nonlinear Klein-Gordon equations , Comm. Pure Appl. Math., t. 38, 1985 , p. 685-696. MR 87b:35160 | Zbl 0597.35101 · Zbl 0597.35101
[15] SIMON (J.C.H.) et TAFLIN (E.) . - The Cauchy problem for nonlinear Klein-Gordon equations , Commun. Math. Phys., t. 152, 1993 , p. 433-478. Article | MR 94d:35110 | Zbl 0783.35066 · Zbl 0783.35066
[16] YORDANOV (B.) . - Blow-up for the one-dimensional Klein-Gordon Equation with a cubic nonlinearity , preprint, 1996 .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.