Global and almost global existence for general quasilinear wave equations in two space dimensions

Abstract

For 2-D quasilinear wave equations of the form □u=F(u,Du,D2u), we show that the corresponding Cauchy problem admits a unique global classical solution with small initial data, if both the quadratic nonlinearity and the cubic nonlinearity satisfy the corresponding null conditions. We also prove that the classical solution is almost global, i.e., the lifespan of solution Tε≥exp⁡(cε−2), where ε>0 denotes the amplitude of the initial data and c is a positive constant, provided that the quadratic nonlinearity satisfies the null condition and ∂uβF(0,0,0)=0, β=3,4. These results extend Alinhac's seminal work Alinhac (2001) [9] to the general case in a unified way.