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

AbstractWe 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