Global Existence for Systems of Nonlinear Wave Equations in Two Space Dimensions
1993; Kyoto University; Volume: 29; Issue: 6 Linguagem: Inglês
10.2977/prims/1195166427
ISSN1663-4926
Autores Tópico(s)Stability and Controllability of Differential Equations
ResumoWe consider the Cauchy problem for the system of nonlinear wave equations (*) (df -AX =Fl(u,u',u} in (0,°°)xR 2 , / = !,-••,# with initial data «,(0,;c) = £0, (*) , (d,ut )(0,jc) = ey/((jt), where Ft (i = 1 , ,AO are smooth functions of degree 3 near the origin (u,u',u) = Q,<t)l,il/l < E C ~ ( R 2 ) and £ is a small positive parameter. We assume that Fl(i = , , N ) are independent of u for any j * i . In the previous paper, the author showed the global existence of the small solution to the Cauchy problem (*) assuming that the cubic parts of the nonlinear terms satisfy Klainerman's null condition and that the nonlinear terms are independent of u)ukulu.m for any j,k,l,m = , , N . In this paper, we show the global existence without imposing further assumptions than the null condition on the cubic parts of the nonlinear terms.
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