ON BOUNDED RADIAL SOLUTIONS OF PARABOLIC EQUATIONS ON RN: QUASICONVERGENCE FOR INITIAL DATA WITH A STABLE LIMIT AT INFINITY

We consider the Cauchy problem for the nonlinear heat equation u(t) = Delta u + f(u), x is an element of R-N, t > 0, where N >= 2 and f is a C-1 function satisfying minor nondegeneracy conditions. Our goal is to describe the large-time behavior of bounded solutions whose initial data are radially symmetric and have a finite limit zeta as vertical bar x vertical bar -> infinity. In the present paper, we examine the following two cases: f(zeta) not equal 0, or f(zeta) = 0 and zeta is a stable equilibrium of the equation (xi) over dot = f(xi). We prove that bounded solutions with such initial data are quasiconvergent: as t -> infinity, they approach a set of steady states in the topology of L-loc(infinity)(R-N).