AI 中文总结
该研究针对无限加权图上带梯度吸收项的粘性哈密顿-雅可比方程，结合Hardy与Hölder不等式，建立总质量衰减阈值，结果适用于含整数格等的多类图。
AI 中文摘要
我们研究无限加权图上带梯度吸收项的非线性扩散方程非负解的长时间行为。该方程结合了离散p-拉普拉斯扩散项与哈密顿-雅可比型非线性吸收项，用于建模离散结构上同时发生扩散和非线性阻尼的过程。在图满足多项式体积增长条件且参数满足特定序关系的假设下，我们建立了解的总质量的精确衰减估计。我们证明，当吸收指数低于临界阈值时，总质量随时间趋于无穷而衰减至零。该临界阈值明确依赖于图的体积增长率和扩散指数，且与连续欧氏环境中类似方程的Fujita型阈值一致。我们的证明结合了Hardy不等式和Hölder不等式，以及适配图几何的精心选取的随时间变化的尺度参数。所得结果即使在线性扩散情形下也是新的，且自然推广至一大类图，包括整数格和有限生成幂零群的Cayley图。
英文摘要
We study the long-time behavior of nonnegative solutions to a nonlinear diffusion equation with gradient absorption on infinite weighted graphs. The equation combines a discrete p-Laplacian diffusion term with a nonlinear absorption term of Hamilton-Jacobi type, modeling processes where both diffusion and nonlinear damping occur on discrete structures. Under the assumptions that the graph satisfies a polynomial volume growth condition and that the parameters satisfy certain ordering relations, we establish sharp decay estimates for the total mass of the solution. We prove that the total mass decays to zero as time tends to infinity whenever the absorption exponent lies below a critical threshold. This critical exponent depends explicitly on the volume growth rate of the graph and the diffusion exponent, and coincides with the Fujita-type threshold known for analogous equations in the continuous Euclidean setting. Our proof combines Hardy and Hölder inequalities with a carefully chosen time-dependent scaling parameter adapted to the graph geometry. The results are new even in the linear diffusion case and extend naturally to a broad class of graphs, including integer lattices and Cayley graphs of finitely generated nilpotent groups.
