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

Abstract

We 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.

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

“…Consequently, u 1 (t, x) ≤ u(t 2 , x) for all t > 0 and x ∈ R N . We acknowledge that, despite the comparison principle, it has not been stated in the form (43) in [37], it has been used in this form in [37,Proposition III.7.1]. Hence, we do not claim any originality for the above result.…”

“…Those are precisely the class of solution that we consider in Theorems 6 and 5 (notice that assumption (i) guarantees it). For the existence result for another notion of solutions, namely p-caloric ones, we refer to a recent paper [43].…”

Elliptic Harnack's inequality for a singular nonlinear parabolic equation in non‐divergence form

“…In this section, we construct an explicit viscosity supersolution $$v$$ to (1.1) in $$B_{R}(0)\times (0,\infty )$$ that takes infinite lateral boundary values and vanishes at the bottom of the cylinder. Recently infinite point source solutions have been constructed for the supercritical $$p$$‐parabolic equation in [13]. While it is straightforward to check that our function is a supersolution, it may not be immediately clear how one obtains its expression and therefore we present the derivation.…”

Elliptic Harnack's inequality for a singular nonlinear parabolic equation in non-divergence form

“…In this section we construct an explicit viscosity supersolution v to (1.1) in B R (0)×(0, ∞) that takes infinite lateral boundary values and vanishes at the bottom of the cylinder. Recently infinite point source solutions have been constructed for supercritical p-parabolic equation in [GKM21]. While it is straightforward to the check that our function is a supersolution, it may not be immediately clear how one obtains its expression and therefore we present the derivation.…”