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Lifespan and blow-up of solutions of the quasilinear wave equations in two dimensions. II. (Temps de vie et comportement explosif des solutions d’équations d’ondes quasi-linéaires en dimension deux. II.) (French) Zbl 0844.35102

[For part I see Ann. Sci. Ec. Norm. Super., IV. Ser. 28, No. 2, 225-251 (1995).]
The author considers the quasilinear wave equation in two dimensions, and demonstrates the existence of a function \(T^a_\varepsilon\) named “asymptotic life time” of the solution, with the two properties
i) \(\forall N\in \mathbb{N}\), \(T_\varepsilon\geq T^a_\varepsilon- \varepsilon^N\), for \(0< \varepsilon\leq \varepsilon_N\).
ii) There is \(C> 0\) so that \(1/C(T_\varepsilon- t)\leq |\nabla^2 u|_{L^\infty}\leq C(1/(T^a_\varepsilon- t))\),
where \(u\) is the solution of the wave equation, \(T_\varepsilon\) is the life time and \(\varepsilon> 0\) the size parameter. The result is obtained in two steps: First, an approximate solution is constructed. Then it is proved that the real solution has almost the same life time as the approximate solution and an estimation of the difference between real and approximate solutions is given by using the energy inequality.
Reviewer: V.A.Sava (Iaşi)

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35L05 Wave equation
35B40 Asymptotic behavior of solutions to PDEs
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