# Asymptotic behavior of Green function for a parabolic equation associated to a Dirichlet form on metric spaces.

### Abstract

In this paper, we are interesting by the asymptotic behavior of Green function for the parabolic problem

\begin{equation}

\left\{

\begin{array}{ll}

H u - V u + \frac{\partial u}{\partial t} =0,\quad \hbox{in } X\times \mathbb{R}_{+},\\

u(x,0)= u_{0}(x),~~x\in X .

\end{array}

\right.

\end{equation}

where $u_{0}\in L^{2}(X,m)$, $X$ is a complete locally compact metric space, $H$ is a selfadjoint operator associated with a regular Dirichlet form $\mathcal E$ and $m$ is a Radon measure on non-empty Borel subspace $U$ of $X$ such that $m(U)>0$ and $V$ belongs to a class of time dependant potentials which can be written as a nonlinear combination of derivatives functions.