Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation

By Matteo Bonforte, Razvan Gabriel Iagar, Juan Luis V'azquez. 2009

54 citations recorded by the source corpus.

We study qualitative and quantitative properties of local weak solutions of the fast p-Laplacian equation, ∂tu=Δpu, with 1<p<2. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of Rn×[0,T]. We combine these lower and upper bounds in different forms of intrinsic Harnack inequalities, which are new in the very fast diffusion range, that is when 1<p⩽2n/(n+1). The boundedness results may be also extended to the limit case p=1, while the positivity estimates cannot. We prove the existence as well as sharp asymptotic estimates for the so-called large solutions for any 1<p<2, and point out their main properties. We also prove a new local energy inequality for suitable norms of the gradients of the solutions. As a consequence, we prove that bounded local weak solutions are indeed local strong solutions, more precisely ∂tu∈Lloc2.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/48fa3b8c592700a5e2dae43523e1c61751cb1f3a