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

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We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation $u_t=\Delta_p u$ 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.