The eternal solutions of parabolic equations with boundary condition

In this paper, we study the parabolic equations of the form $$\{\begin{array}{rcll}Lu(y,t)& =& f,& (y,t)\in Q,\\ u(y,t)& =& 0,& (y,t)\in \mathrm{\partial }Q,\\ u(y,t)& & isuniformlyboundedfrombelow,& (y,t)\in Q,\end{array}$$ where $Q=\mathrm{\Omega }\times \mathbb{R}\subset {\mathbb{R}}^{n+1}$ and $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded Lipschitz domain with $0\in \mathrm{\Omega }$. Here $L$ is a general second order uniformly parabolic differential operator in non-divergence form or divergence form. For $f=0$, we establish the structure of the solution space, which is one dimensional and the solutions in this space grow exponentially at one end and decay exponentially at the other. For $f\mathrm{\ne }0$, we show that all solutions can be presented by the solutions corresponding to the homogenous equations($f=0$) and a bounded special solution of the inhomogeneous equations. Our method is based on maximum principle in $Q$ and the Harnack type inequalities.