Moser–Nash kernel estimates for degenerate parabolic equations

In this paper we deal with the Cauchy problem associated to a class of nonlinear degenerate parabolic equations, whose prototype is the parabolic p-Laplacian (2>p>∞). In his seminal paper, after stating the Harnack estimates, Moser proved almost optimal estimates for the parabolic kernel by using the so called ‘Harnack chain’ method. In the linear case sharp estimates come by using Nash's approach. Fabes and Stroock proved that Gaussian estimates are equivalent to a parabolic Harnack inequality. In this paper, by using the DiBenedetto–DeGiorgi approach we prove optimal kernel estimates for degenerate quasilinear parabolic equations. To obtain this result we need to prove the finite speed of propagation of the support and to establish optimal estimates. Lastly we use these results to prove existence and sharp pointwise estimates from above and from below for the fundamental solutions.