An example of nonlinear wave equation whose solutions decay faster than exponentially

Abstract

There are many papers which treat the lower boundedness of solutions of nonlinear wave equations (cf. [ 1, 2, 3, 5, etc.). From these works it seems reasonable to believe that the solutions of most of nonlinear wave equations would decay at most exponentially, namely, it would hold that E(u(r)) 3 C,,e “, t 3 0, for some C,>O and A > 0 depending on the initial datum (u(O), u,(O)), where E(u(r)) denotes the energy associated with the equation under consideration. In this note we give an example of the wave equation with certain singular nonlinearity whose solutions decay much faster than exponen- tially. Our example is; u,I--u,, + IUI I’u,+ 1uJ -1z1=o in IxR+ u(.u, 0) = ug, u,(x, 0) = u, and 4 ;/ = 0, where I is a bounded interval in R and r, s( are constants such that O<r<atl. In fact, we consider a more general equation: