Large-time behavior of bounded radial solutions of parabolic equations on R^N: Part II—convergence for initial data with a linearly stable limit at infinity

We consider the Cauchy problem (Formula presented.) where N≥2, f is a C1 function satisfying minor nondegeneracy conditions, and u0 is a radially symmetric function having a finite limit ζ as |x|→∞. We have previously proved that if ζ is a stable equilibrium of the equation ξ˙=f(ξ) and the solution u is bounded, then u is quasiconvergent: its ω-limit set with respect to the topology of Lloc∞(RN) consists of steady states. In the present paper, we consider the case when ζ is linearly stable: f(ζ)=0 and f′(ζ)<0. Under this condition, we show that if the solution of the above Cauchy problem is bounded, then it converges, locally uniformly with respect to x∈RN, to a single steady state.