Blowup of small data solutions for a quasilinear wave equation in two space dimensions. (English) Zbl 1080.35043

Alinhac, Serge

Ann. Math. (2) 149, No. 1, 97-127 (1999).

Summary: For the quasilinear wave equation \(\partial_t^2u - \Delta u = u_t u_{tt},\) we analyze the long-time behavior of classical solutions with small (not rotationally invariant) data. We give a complete asymptotic expansion of the lifespan and describe the solution close to the blow-up point. It turns out that this solution is a “blow-up solution of cusp type,” according to the terminology of the author.

MSC:

35L70	Second-order nonlinear hyperbolic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

Cauchy problem; Hörmander’s conjecture