We study monotone finite difference approximations for a broad class of reaction-diffusion problems, incorporating general symmetric L\'evy operators. By employing an adaptive time-stepping discretization, we derive the discrete Fujita critical exponent for these problems. Additionally, under general consistency assumptions, we establish the convergence of discrete blow-up times to their continuous counterparts. As complementary results, we also present the asymptotic-in-time behavior of discrete heat-type equations as well as an extensive analysis of discrete eigenvalue problems.

翻译：本研究针对包含一般对称Lévy算子的广泛反应-扩散问题，探讨其单调有限差分逼近方法。通过采用自适应时间步长离散化策略，我们推导出此类问题的离散Fujita临界指数。此外，在一般相容性假设下，我们建立了离散爆破时间向其连续对应量收敛的理论。作为补充性成果，本文还给出了离散热型方程的渐近时间行为分析，并对离散特征值问题进行了系统性研究。