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The Cauchy problem for the fast $p-$Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation $u_t=Δ_p u$ on the whole Euclidean space, in the so-called &#34;good fast diffusion range&#34; $\tfrac{2N}{N+1}<p<2$. It is well-known that non-negative solutions behave for large times as $\mathcal{B}$, the Barenblatt (or fundamental) solution, which has an explicit expression. We prove the so-called Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates of nonnegative solutions in terms of $\mathcal{B}$. This can be considered the nonlinear counterpart of the celebrated Gaussian estimates for the linear heat equation. We characterize the maximal (hence optimal) class of initial data such that the GHP holds, by means of an integral tail condition, easy to check. The GHP is then used as a tool to analyze the fine asymptotic behavior for large times. For initial data that satisfy the same integral condition, we prove that the corresponding solutions behave like the Barenblatt with the same mass, uniformly in relative error. When the integral tail condition is not satisfied we show that both the GHP and the uniform convergence in relative error, do not hold anymore, and we provide also explicit counterexamples. We then prove a &#34;generalized GHP&#34;, that is, pointwise upper and lower bounds in terms of explicit profiles with a tail different from $\mathcal{B}$. Finally, we derive sharp global quantitative upper bounds of the modulus of the gradient of the solution, and, when data are radially decreasing, we show uniform convergence in relative error for the gradients. To the best of our knowledge, analogous issues for the linear heat equation $p=2$, do not possess such clear answers, only partial results are known.