发表机构
Politecnico di Milano（米兰理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对度量测度空间上的半线性热方程，建立了不依赖随机完备性的爆破准则，并利用热核加权函数空间构造压缩映射，统一证明了多项式与指数热核衰减下的整体存在性，推广至黎曼流形、度量图及加权图。
AI 中文摘要
我们研究了度量测度空间上具有时变反应系数的半线性热方程的有限时间爆破与整体存在性。对于凸非线性，我们建立了一个爆破准则，该准则通过其线性演化保留了对初始数据的依赖，并且不需要随机完备性。在额外的守恒性和热核假设下，我们还证明了与多项式体积增长相关的临界Fujita指数处的爆破。我们的主要整体存在性结果基于一个热核加权的函数空间，在该空间中温和解映射成为压缩映射。这产生了一个统一的构造，涵盖了多项式和指数热核衰减、一般的局部Lipschitz非线性以及时变反应系数，而不需要凸性或单调性假设。特别地，整体存在性准则仅使用非线性在解所探索的范围上的行为。应用包括黎曼流形、度量图，以及在单独的离散框架内的加权图。这些论证，尤其是支撑整体存在的函数构造，是新的，并且即使对于上述经典底层空间也产生了新的结果。
英文摘要
We study finite-time blow-up and global existence for semilinear heat equations with time-dependent reaction coefficients on metric measure spaces. For convex nonlinearities, we establish a blow-up criterion that retains the dependence on the initial datum through its linear evolution and does not require stochastic completeness. Under additional conservativity and heat kernel assumptions, we also prove blow-up at the critical Fujita exponent associated with polynomial volume growth. Our main global existence result is based on a heat-kernel-weighted functional space in which the mild solution map becomes a contraction. This yields a unified construction covering both polynomial and exponential heat kernel decay, general locally Lipschitz nonlinearities, and time-dependent reaction coefficients, without convexity or monotonicity assumptions. In particular, the global existence criterion only uses the behaviour of the nonlinearity on the range explored by the solution. Applications include Riemannian manifolds, metric graphs, and, within a separate discrete framework, weighted graphs. The arguments, especially the functional construction underlying global existence, are new and lead to new results even for the classical underlying spaces considered above.
