The uniform existence of solutions for two-dimensional and three-dimensional semilinear wave equations with oscillatory data

Abstract

For two-dimensional and three-dimensional semilinear wave equations, I prove the uniform existence of solutions with oscillatory initial data. Hence I solve an open question in one paper of J. L. Joly, G. Metivier and J. Rauch. Nonlinear geometric optics provide an asymptotic description in the limit 8 > 0 of solutions to the oscillatory initial value problem (1) DU + f(u, VU) = 0, L := 92_ (2) atu(0,x) =-e1-jaj(x)e e ) j = 0,)1, where (3) aj E Co (B(0, Ro); C), , E C (RT; R), and (4) d~p(x) $0, when e suppaoUsuppal. In particular, one has existence on an independent time interval and this is true for all dimensions. J. L. Joly, G. Metivier and J. Rauch showed that for n > 4, one consequence of focusing effects is that this uniform domain of existence may not exist when d~o vanishes in the support of the initial data. When the dimension n = 1, classical estimates of Haar type imply uniform local solvability. The explicit formulas for the solutions show that in dimensions no larger than 3 the family of solutions of the linear initial value problem with the same initial data is uniformly bounded on spacetime for 0 0 the family uC is bounded in L' ([O,T] x Rn). Received by the editors May 7, 1997. 1991 Mathematics Subject Classification. Primary 35L70. (@)1999 American Mathematical Society 195 This content downloaded from 157.55.39.207 on Thu, 20 Oct 2016 04:08:47 UTC All use subject to http://about.jstor.org/terms