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

作者：Miao Chang Xing^*; Bin Zhu You

来源：Acta Mathematica Sinica-English Series, 2008, 24(1): 17-26.

DOI：10.1007/s10114-007-1031-8

• dilation identity
• Besov space
• energy solution
• Strichartz estimates
• local energy identity

The authors consider the well-posedness in energy space of the critical non-linear system of wave equations with Hamiltonian structure
{ u(tt) - Delta u = -F-1(vertical bar u vertical bar(2), vertical bar v vertical bar(2))u, v(tt) - Delta v = -F-2(vertical bar u vertical bar(2), vertical bar v vertical bar(2))v,
where there exists a function F(lambda, mu) such that
delta F(lambda,mu)/delta lambda = F-1(lambda,mu), delta F(lambda,mu)/delta 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)".

• 出版日期2008-1
• 单位西安电子科技大学