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：本文研究了一类包含一般对称Lévy算子的反应-扩散问题的单调有限差分逼近。通过采用自适应时间步长离散化方法，我们推导了这类问题的离散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解的收敛性。作为补充结果，我们还给出了离散热型方程的渐近时间行为，并对离散特征值问题进行了广泛分析。