Localized Solutions of a Semilinear Parabolic Equation with a Recurrent Nonstationary Asymptotics

We examine the behavior of positive bounded, localized solutions of semilinear parabolic equations ut = ∆u + f(u) on RN . Here f ∈ C1, f(0) = 0, and a localized solution refers to a solution u(x, t) which decays to 0 as x → ∞ uniformly with respect to t > 0. In all previously known examples, bounded, localized solutions are convergent or at least quasiconvergent in the sense that all their limit profiles as t → ∞ are steady states. If N = 1, then all positive bounded, localized solutions are quasiconvergent. We show that such a general conclusion is not valid if N ≥ 3, even if the solutions in question are radially symmetric. Specifically, we give examples of positive bounded, localized solutions whose ω-limit set is infinite and contains only one equilibrium.