Long-time behavior and convergence for semilinear wave equations on ℝN

We prove that any bounded solution (u,ut) of u tt+dut-Δu+f(u) = 0, u = u(x, t), xεℝN, N≥3, converges to a fixed stationary state provided its initial energy is appropriately small. The theory of concentrated compactness is used in combination with some recent results concerning the uniqueness of the so-called ground-state solution of the corresponding stationary problem. © 1997 Plenum Publishing Corporation.