The Lifespan of Solutions to Nonlinear Systems of High Dimensional Wave Equation

OF HIGH DIMENSIONAL WAVE EQUATION Vladimir Georgiev1 and Hiroyuki Takamura2 and Zhou Yi3 1Dipartimento di Matemati a "L.Tonelli, Università di Pisa, Via F. Buonarroti 2, 56127 Pisa, Italy. e-mail: georgiev dm.unipi.it 2Institute of Mathemati s, University of Tsukuba Tsukuba Ibaraki 305-8571, Japan. e-mail: takamura math.tsukuba.a .jp 3Institute of Mathemati s, Fudan University Shanghai 200433, China. e-mail: yizhou fudan.a . n Abstra t In this work we study the lifespan of solutions to p-q system in the higher dimensional ase n ≥ 4. A suitable lo al existen e urve in the p-q plane is found. The urve hara terizes the lo al solutions in Sobolev spa e H with s ≥ 0. Further, some lower and upper bounds of the lifespan of lassi al solutions are found too. The work is an extension of the work [7℄, where a suitable global existen e small data urve is studied. In the sub riti al ase, we give almost pre ise results for the lower bounds of the lifespan by using a suitable weighted Stri hartz estimate for the higher dimensional wave equation. 1. Introdu tion The system of the following wave equations (1.1) { ∂ t u−∆u = f(v), ∂ t v −∆v = g(u), (here ∆ is the Lapla e operator in R;n ≥ 2) an be onsidered as an evolution problem asso iated with the Hamiltonian system (1.2) { −∆u = f(v), −∆v = g(u). The study of the existen e of small data solutions for the Cau hy problem (1.3)  ∂ t u−∆u = |v|p, p > 1, ∂ t v −∆v = |u|q, q > 1,