We consider the Cauchy problem  \begin{align*}   &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{align*} where $N\ge 2$, $f$ is a $C^1$ function satisfying minor nondegeneracy conditions, and $u_0$ is a radially symmetric function having a finite limit $\zeta$ as $|x|\to\infty$. We have previously proved that if $\zeta$ is a stable equilibrium of the equation $\dot\xi=f(\xi)$ and the solution $u$ is bounded, then $u$ is quasiconvergent: its $\omega$-limit set with respect to the topology of $L_{loc}^\infty(\mathbb R^N)$ consists of steady states. In the present paper, we consider the case when $\zeta$ is linearly stable: $f(\zeta)=0$ and $f'(\zeta)<0$. Under this condition, we show that if the solution of the above Cauchy problem is bounded, then it converges, locally uniformly with respect to $x\in\mathbb R^N$, to a single steady state.