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

中文翻译：

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局部有限图上带有 Fujita 临界指数的半线性热方程的爆破

令\(G=(V,E)\)是一个局部有限、连通且加权的图。我们证明，对于满足曲率维数条件\(CDE'(n,0)\)和均匀多项式体积增长m 次的图，方程\(\partial _tu=\Delta u+u ^{1+\alpha }\)在有限时间内爆炸，前提是\(\alpha =\frac{2}{m}\)。我们还考虑了体积增长和初始值在某些条件下的爆破问题。这些结果补充了我们之前与 Lin 合作的工作。