Global and Unique Weak Solutions of Nonlinear Wave Equations

Abstract

While the uniqueness of the energy class solutions of semi-linear wave equations with supercritical nonlinearity remains an open problem, there has been a considerable progress recently in understanding of the critical and subcritical cases (see [Brenner & von Wahl], [Pecher], [Ginibre & Velo 1-3], [Ginibre, Soffer & Velo], [Kapitanski 1, 4-6], [Struwe 1, 2], [Grillakis 1-3], [Shatah & Struwe]). In particular, the following result is proved in [Kapitanski 4, 5]. Let L be a second order hyperbolic differential operator of the form (the sum over the repeated indices is assumed)