Large time behavior of the solution to the Cauchy problem for viscous Hamilton-Jacobi equation on infinite graphs
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.