Well-posedness in the energy space for non-linear system of wave equations with critical growth

Abstract

The authors consider the well-posedness in energy space of the critical non-linear system of wave equations with Hamiltonian structure $$ \left\{ \begin{gathered} u_{tt} - \Delta u = - F_1 (|u|^2 ,|v|^2 )u, \hfill \\ v_{tt} - \Delta v = - F_2 (|u|^2 ,|v|^2 )v, \hfill \\ \end{gathered} \right. $$ where there exists a function F(λ, µ) such that $$ \frac{{\partial F(\lambda ,\mu )}} {{\partial \lambda }} = F_1 (\lambda ,\mu ),\frac{{\partial F(\lambda ,\mu )}} {{\partial \mu }} = F_2 (\lambda ,\mu ). $$ By showing that the energy and dilation identities hold for weak solution under some assumptions on the non-linearities, we prove the global well-posedness in energy space by a similar argument to that for global regularity as shown in “Shatah and Struwe’s paper, Ann. of Math. 138, 503–518 (1993)”.