Existence Results for a Heat Flow with Exponential Nonlinearity on the Finite Graph

We investigate the following heat flow \[ \begin{cases} e^{4\pi u^{2}} \frac{\partial u}{\partial t} - \Delta u = \lambda(t) u e^{4\pi u^{2}} + \nu(t), &(x,t) \in V \times (0,+\infty), \ u_{0} = u(x,0), &x \in V, \end{cases} \] where $G = (V,E)$ is a connected finite graph. We obtain that the heat flow has a unique solution. Moreover, there exists some sequence of solution $u(\,\cdot\,,t)$ that converges to $u_{\infty}$ as $t \to \infty$, which is the solution to the corresponding elliptic equation \[ -\Delta u_{\infty} = \lambda_{\infty} u_{\infty} e^{4\pi u_{\infty}^{2}} + \nu_{\infty}. \]