Hyperbolic trapped rays and global existence of quasilinear wave equations
The authors considered the initial boundary value problem of the system of wave equations: \[ \square u= Q(du, d^2u),\;(t,x)\in\mathbb{R}_+\times \mathbb{R}^3\setminus{\mathcal K},\;u(t,\cdot)|_{\partial{\mathcal K}}= 0,\;u(0,\cdot)= f,\;\partial_t u(0,\cdot)= g,\tag{1} \] where \(\square= (\square_{c_1},\dots, \square_{c_D})\) with \(\square_{c_I}= \partial^2_t- c^2_I\Delta\) and \(c_I> 0\), \(I= 1,\dots, D\), \(Q(du, d^2u)\) are quadratic quasilinear nonlinearities satisfy a form of the null condition, the obstacle \({\mathcal K}\) is smooth and compact. The main purpose of this paper is to prove the global solvability to (1) under the smallness of initial data in a weighted Sobolev space. In particular, they can drop the star-shaped hypothesis and handle nontrapping obstacle as well as any obstacle that has exponential local decay rate of energy for \(H^2\) data for the linear equation.
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