OSCILLATIONS AND CONCENTRATION EFFECTS IN SEMILINEAR DISPERSIVE WAVE EQUATIONS

摘要

Let (u(n)) be a sequence of smooth solutions to a dispersive nonlinear wave equation, partial derivative(t)(2)u(n)-Delta u(n)+f(u(n))=0 in R1 divided by 3 with uniformly compactly supported Cauchy data converging weakly to 0 in H-1(R(3))xL(2)(R(3)). Let (upsilon(n)) be the sequence of solutions to ther linear wave equation with the same Cauchy data. We show that u(n)-upsilon(n) goes strongly to 0 in the energy space C([0, T], H-1)boolean AND C-1([0,T], L(2)) if f is a subcritical nonlinearity. In the critical case f(u)=u(5), we show that this property is equivalent to upsilon(n)-->0 in L(i)nfinity([0,T], L(6)). Then we give sharp sufficient conditions on microlocal measures associated to the data. The proof relies on a microlocal version of P.-L. Lions' con centratiou-compacity. (C) 1996 Academic Press, lnc. [References: 26]