Large-time behavior of solutions of parabolic equations on the real line with convergent initial data

We consider the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ We prove that if a solution $u$ of this equation is bounded and its initial value $u(x,0)$ has distinct limits at $x=\pm\infty,$ then the solution is quasiconvergent, that is, all its limit profiles as $t\to\infty$ are steady states.