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

Authors:

Institution:

(1) Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, 100088, P. R. China;(2) Department of Mathematics, Xidian University, P.O.Box 245, Xi’an, 710071, P. R. China

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)”. Supported by NSF of China (No. 10571016) and Special Funds for Major State Basic Research Projects of China

Keywords:

dilation identity Besov space energy solution Strichartz estimates local energy identity

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