Lifespan and blow-up of solutions of the quasilinear wave equations in two dimensions. II
[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.
scientific article; zbMATH DE number 572043
scientific article