摘要
Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions,provided that the initial data are compactly supported and sufficiently small in Sobolev norm.In this work,Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions.A natural question then arises whether the time-growth is a true phenomenon,despite the possible conservation of basic energy.In the present paper,we establish that the top-order energy of the solutions in Alinhac theorem remains globally bounded in time.
基金
supported by the China Postdoctoral Science Foundation(2021M690702)
The author Z.L.was in part supported by NSFC(11725102)
Sino-German Center(M-0548)
the National Key R&D Program of China(2018AAA0100303)National Support Program for Young Top-Notch Talents
Shanghai Science and Technology Program[21JC1400600 and No.19JC1420101].