AI 中文总结
该研究针对无限加权图上半线性热不等式的Fujita现象，建立积分体积增长等准则，证明了超解存在性与柯西解的等价性，在特定图类上验证了准则的尖锐性。
AI 中文摘要
我们研究无限加权图上由变速度拉普拉斯算子生成的半线性热不等式的Fujita现象。假设该图带有一个合适的适配路径度量，我们建立了一个积分体积增长准则，强制开圆柱$(0,\infty)\times V$上的每个非负整体经典超解消失，且不假设初始值或迹。我们证明，此类非平凡超解存在当且仅当方程对某个非零点源数据具有正整体柯西解。互补的热核构造表明，当相同的体积积分收敛且存在匹配的锚定热核上界时，所有足够小的点源数据都存在整体柯西解。这在整数格和一族对数扰动的加权半直线上证明了尖锐性；在后一例子中，甚至迭代对数的指数都能决定存在-不存在的二选一情况。一个更精细的不存在准则将内在体积增长与内在环域的容度耦合起来。其证明结合了抛物检验、拉普拉斯预解式约化以及预解式流的路径wise分解。这些不存在结果不要求体积加倍性质、庞加莱不等式、热核界或随机完备性。
英文摘要
We study the Fujita phenomenon for semilinear heat inequalities generated by variable-speed Laplacians on infinite weighted graphs. Assuming that the graph carries a proper adapted path metric, we establish an integral volume-growth criterion forcing every nonnegative global classical supersolution on the open cylinder $(0,\infty)\times V$ to vanish, without assuming an initial value or trace. We prove that a nontrivial supersolution of this kind exists if and only if the equation has a positive global Cauchy solution for some nonzero point-source datum. A complementary heat-kernel construction gives global Cauchy solutions for all sufficiently small point-source data when the same volume integral converges and a matching anchored heat-kernel upper bound is available. This proves sharpness on integer lattices and on a family of logarithmically perturbed weighted half-lines; in the latter examples, even the exponent of an iterated logarithm can determine the existence--nonexistence alternative. A finer nonexistence criterion couples intrinsic volume growth with the capacity of intrinsic annuli. Its proof combines parabolic testing, a Laplace--resolvent reduction, and a pathwise decomposition of resolvent currents. The nonexistence results require no volume-doubling property, Poincaré inequality, heat-kernel bound, or stochastic completeness.
