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

We consider the Cauchy problem \( \begin{aligned}&u_t=\Delta u+f(u),&\quad&x\in \mathbb R^N,\ t>0,\\&\,u(x,0)=u_0(x),&\quad&x\in \mathbb R^N, \end{aligned}\) $$\begin{array}{cccc}& u_t=\mathrm{\Delta }u+f(u),\hfill & \hfill & x\in {\mathbb{R}}^N,t>0,\hfill \\ \hfill & u(x,0)=u_0(x),\hfill & \hfill & x\in {\mathbb{R}}^N,\hfill \end{array}$$ where \(N\ge 2\) $N\ge 2$ , f is a \(C^1\) $C^1$ function satisfying minor nondegeneracy conditions, and \(u_0\) $u_0$ is a radially symmetric function having a finite limit \(\zeta \) $\zeta$ as \(|x|\rightarrow \infty \) $|x|\to \infty$ . We have previously proved that if \(\zeta \) $\zeta$ is a stable equilibrium of the equation \(\dot{\xi }=f(\xi )\) $\dot{\xi }=f(\xi )$ and the solution u is bounded, then u is quasiconvergent: its \(\omega \) $\omega$ -limit set with respect to the topology of \(L_{loc}^\infty ({\mathbb R}^N)\) $L_{\mathit{loc}}^{\infty }({\mathbb{R}}^N)$ consists of steady states. In the present paper, we consider the case when \(\zeta \) $\zeta$ is linearly stable: \(f(\zeta )=0\) $f(\zeta )=0$ and \(f'(\zeta )<0\) $f^{\prime }(\zeta )<0$ . Under this condition, we show that if the solution of the above Cauchy problem is bounded, then it converges, locally uniformly with respect to \(x\in {\mathbb R}^N\) $x\in {\mathbb{R}}^N$ , to a single steady state.