Local smoothing effects, positivity, and Harnack inequalities for the fast \(p\)-Laplacian equation (Q981626)

The paper deals with qualitative and quantitative properties of local weak solutions of the fast \(p\)-Laplacian equation \(\partial_t u=\Delta_pu\), with \(1<p<2\). The main results are positivity and boundedness estimates for locally defined solutions in domains of \(\mathbb R^n \times [0,T]\). These estimates imply an intrinsic Harnack inequality when \(1<p \leq \frac{2n}{n+1}\), i.e., for a very fast diffusion range. The boundedness results can be extended to the case \(p=1\). Moreover, the existence for the so-called large solutions for any \(1<p<2\) are given and the properties of the solutions are studied. Finally, a new local energy inequality for a suitable norm of the gradient of the solution is established.