Existence and Nonexistence of Global Solutions to the Parabolic Equations on Locally Finite Graphs

Let u be a function on the connected locally finite graph \(G=(V, E)\) $G=(V,E)$ , \(\Delta \) $\mathrm{\Delta }$ be the usual \(\mu \) $\mu$ -Laplacian. We consider the following parabolic equation on locally finite graphs \(\begin{aligned}\left\{ \begin{array}{lll} u_t(t,\, x)=\Delta u(t,\, x)+h(x)u^{1+\alpha }(t,\, x),\quad (t,\, x)&{}{\in }&{}(0,\, +\infty )\times V,\\ u(0,\, x)=u_0(x),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x&{}{\in }&{}V,\end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{u}_t(t,x)=\mathrm{\Delta }u(t,x)+h(x)u^{1+\alpha }(t,x),(t,x)\hfill & \in \hfill & (0,+\infty )\times V,\hfill \\ u(0,x)=u_0(x),x\hfill & \in \hfill & V,\hfill \end{array}\right)\end{array}$$ where \(\alpha >0\) $\alpha >0$ , h(x) is a bounded function satisfying \(h_0=\inf _{x\in V}h(x)>0\) $h_0={inf}_{x\in V}h(x)>0$ and \(h_1=\sup _{x\in V}h(x)<+\infty \) $h_1={sup}_{x\in V}h(x)<+\infty$ , \(u_0(x)\) $u_0(x)$ is a bounded, nonnegative and nontrivial initial value. Firstly, we prove that there exists sufficiently small \(t_0>0\) $t_0>0$ such that the above-mentioned equation has a unique nonnegative solution in \([0,\ t_0]\) $[0,t_0]$ . Secondly, motivated by Lin–Wu (Calc. Var. Partial Differ. Equ., 2017), under the curvature condition and the general volume growth condition, by heat kernel estimate, we prove that the nonnegative solutions blow up in finite time if \(0<m\alpha \le 2\) $0<m\alpha \le 2$ , and that there exists a global nonnegative solution for a small enough initial value if \(m\alpha >2\) $m\alpha >2$ .