Large time behavior of the solution to the Cauchy problem for the discrete p-Laplacian with density on infinite graphs

We consider the Cauchy problem for the nonstationary discrete p-Laplacian with inhomogeneous density \r{ho}(x) on an infinite graph which supports the Sobolev inequality. For nonnegative solutions when p > 2, we prove the precise rate of stabilization in time, provided \r{ho}(x) is a non-power function. When p > 2 and \r{ho}(x) goes to zero fast enough, we prove the universal bound. Our technique relies on suitable energy inequalities and a new embedding result.