Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case

Abstract

AbstractWe study a generalized class of supersolutions, so-called p-supercaloric functions, to the parabolic p-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for $$p\ge 2$$ p ≥ 2 , but little is known in the fast diffusion case $$1<p<2$$ 1 < p < 2 . Every bounded p-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic p-Laplace equation for the entire range $$1<p<\infty $$ 1 < p < ∞ . Our main result shows that unbounded p-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case $$\frac{2n}{n+1}<p<2$$ 2 n n + 1 < p < 2 . The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case $$1<p\le \frac{2n}{n+1}$$ 1 < p ≤ 2 n n + 1 and the theory is not yet well understood.