Global small data smooth solutions of 2-D null-form wave equations with non-compactly supported initial data

Abstract

For the 2-D nonlinear wave equations □u=F(∂u,∂2u) with initial data (u(0,x),∂tu(0,x))=(εu0(x),εu1(x)), where x=(x1,x2), x0=t, ∂=(∂0,∂1,∂2), ε>0 is small enough, u0(x),u1(x)∈C0∞(R2), and the smooth nonlinearity F(∂u,∂2u)=O(|∂u|2+|∂2u|2), when F(∂u,∂2u) satisfies the null conditions, S. Alinhac in [2] shows that the smooth solution u exists globally. The proof relies on the compactness of the support of (u0(x),u1(x)). Recently, for a class of quasilinear wave equations □u=Nαβμν∂αβ2u∂μν2u or □u=Aα∂α(|∂tu|2−|∇u|2) with small and non-compactly supported initial data, where Nαβμν and Aα are constants (0≤α,β,μ,ν≤2), when the related null condition hold, the authors in [6] prove the global existence of solution u. In this paper, we will prove the global existence for the general 2-D null-form wave equations □u=F(∂u,∂2u) with non-compactly supported initial data. The new key ingredient is to establish a class of weighted L∞-L∞ estimates of solution w to the 2-D linear wave equation □w=f(t,x) instead of the usual L∞-L2 estimates used in [2], [6] and so on. From this, we also get a better time-decay rate for the “good derivatives” of solution u of nonlinear wave equations.