This paper studied the initial and boundary value problems to a kind of non-
linear parabolic equations: u_t-f(u)_xx=0, x∈R₊, f'(u)>0, u(0,t)=u_-, t≥0; u (+∞,0)=u₊. The general case u_-≠u₊ is considered. It is proved that under some
smallness conditions the solutions of this problem tend to the self-similar solution
as the time t goes to infinity. Furthermore, the optimal decay rate is also obtained which
reads (1+t)^-(1/4).