Finite Energy Travelling Waves for Nonlinear Damped Wave Equations

E(u,ut) = f |Vu|2 + m\u\2 + |ut|2 dx — [ |u|"+1 dx, (1.3) 2 JRn a+ 1 JR„ which represents a Lyapunov function of the problem, i.e., it is decreasing along any nonstationary trajectory of (1.1). Employing the potential-well arguments of PAYNE-SATTINGER [16] one observes that any solution of (1.1) emanating from sufficiently small initial data exists globally for all t e R+ and tends to zero with growing t. On the other hand, LEVINE [13, Part IV] showed that solutions corresponding to certain large data (with negative energy) blow up at a finite time. In this paper, we shall deal with localized, finite energy states, i.e., with solutions to (1.1) that belong to neither class discussed above. A standard example of such a solution is, of course, a nonzero solution to the stationary problem: -Aw + mw=\w\a~1w, wgH1(Rn). (1.4)