Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains

Abstract: For 3-D quadratically quasilinear wave equations with or without null conditions in exterior domains, when the compatible initial data and Dirichlet boundary values are given, the global existence or the optimal existence time of small data smooth solutions have been established in early references. However, for 2-D quadratically quasilinear wave equations with null conditions in exterior domains, due to the slower space-time decay rates of solutions, it is unknown whether the global small solutions exist. In this paper, we are concerned with this unsolved problem and show the global existence for a class of 2-D quadratically quasilinear wave equations. Our main ingredients include: Firstly, through finding two good unknowns we can transform such a class of quadratically quasilinear wave equations into certain manageable cubic nonlinear forms; Secondly, some crucial pointwise estimates are derived for the initial boundary value problem of 2-D linear wave equations in exterior domains; Thirdly, we establish some precise space-time decay estimates on the good derivatives of solutions as well as the solutions themselves.