Blow-up for a semilinear heat equation with Fujita’s critical exponent on locally finite graphs

Let $$G=(V,E)$$ be a locally finite, connected and weighted graph. We prove that, for graph satisfying curvature dimension condition $$CDE'(n,0)$$ uniform polynomial volume growth of degree m, all non-negative solutions the equation $$\partial _tu=\Delta u+u^{1+\alpha }$$ blow up in finite time, provided that $$\alpha =\frac{2}{m}$$ . also consider blow-up problem under certain conditions initial value. These results complement our previous work joined with Lin.