The eternal solutions of parabolic equations with boundary condition

Abstract

In this paper, we study the parabolic equations of the form $$ \left\{ \begin{array}{rcll} Lu(y,t) &=& f, \qquad &(y,t)\in Q,\\ u(y,t)&=& 0, \qquad &(y,t)\in \partial Q, \\ u(y,t)&& \hspace{-8mm}\mbox{is uniformly bounded from below}, \qquad &(y,t)\in Q, \end{array} \right. $$ where $Q=\Omega\times\mathbb{R}\subset\mathbb{R}^{n+1}$ and $\Omega\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain with $0\in\Omega$. 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 $f\neq0$, 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.