ON SQUEEZING AND FLOW OF ENERGY FOR NONLINEAR-WAVE EQUATIONS

We study the nonlinear wave equation u - delta(2) Delta u + f(u) = 0 u nder n-dimensional periodic boundary conditions (n = 1, 2, 3) in a Sob olev phase-space H-3 = H-s x H-s-1(T-n) = {(u, u)(x)}. In [K1] we inte rpreted the ''energy transition to higher frequencies'' problem for th is (and similar) equations as a squeezing: time-t flow maps of the equ ation with large t ''squeeae'' r-balls in H-s to narrow cylinders form ed by vector-functions such that the norms of their first Fourier coef ficients are less then some small gamma. We proved in [K1] a version o f Gromov's (non)squeezing theorem: the phenomenon stated above is impo ssible for gamma < r if H-s is chosen to be the symplectic phase space of the equation, H-s = H-1/2. In this paper we show that in smooth ph ase spaces Ha(s), s greater than or equal to 5, the squeezing is typic al if the dispersion delta is small. We use this result and some relat ed statements to obtain a (rather rough) picture of qualitative behavi our of solutions of the phi(4)-equation u - delta(2) Delta u + u(3) = 0.