Global existence of solutions to nonlinear wave equations by weighted Strichartz inequalities.

摘要

We study the global existence of the solution to the nonlinear wave equation with small data under the assumption of spherical symmetry. More specifically, we consider the equation □u = Fl( u, Du) with small and radially symmetric data in the Minkowski space M = RxR n where F satisfies a certain condition for existence. First, using a conformal transformation, the Minkowski space M = RxR n is compactly embedded in a compact subset of the Einstein universe E = R x Sn. This method is called the "conformal compactification of Minkowski space." Then a weighted version of the Strichartz estimates on the Einstein universe is proved to show the global existence of a solution. This estimate is proved for the linear inhomogeneous wave equation with zero Cauchy data. Then a standard iteration argument will bring the result of global existence. This work basically simplifies and extends some of the existing results.