Interior Harnack Estimates: The State-of-the-Art for Quasilinear Singular Parabolic Equations

Abstract

In this paper we give some historical information about elliptic and parabolic Harnack inequalities. Then we state the main results known for Harnack inequalities of solutions to quasilinear degenerate parabolic equations. Lastly we focus our attention on Harnack inequalities of solutions to quasilinear singular parabolic equations where the theory did important steps forward in the last few years but still there are some points to be fully understood.Mathematics Subject Classification. Primary 35K67; Secondary 35K92, 35K20.

Regularity results for a class of doubly nonlinear very singular parabolic equations

“…Consider, for instance, the singular equation (1.3). It is known that any bounded solution is regular (see, for instance, the monograph [18]), but Harnack inequality, the existence of potential and the boundedness of any weak solution happen only in the supercritical range p > 2N N +2 , with N standing for the dimension of the space domain (see, for instance, the monograph [24]) and these phenomena look like to be highly correlated (see, for instance, the review papers [27] and [28]). When p is in the subcritical range (1 < p ≤ 2N N +2 ) only some suitable L r − L ∞ estimates can be proved (see [22]) and the Harnack estimates are degenerating in a very weak form (see [2,22,23] and [35]).…”

Harnack’s inequality for doubly nonlinear equations of slow diffusion type

“…Loosely speaking, solutions of non-homogeneous equations behave like solutions of the heat equation in an intrinsic time scale. A counterexample [11] shows that a Harnack estimate with t w independent of u is false. Since the proof in [7] relies on comparison with explicit solutions, it cannot be adapted for general quasilinear equations.…”

“…The same method was used by Kuusi [23] to obtain weak Harnack estimates for super-solutions of nonlinear degenerate parabolic equations. For an extensive overview regarding the parabolic p-Laplace equation and the porous medium equation with the definition of weak solution involving u m+1 2 , we refer to the monograph [10] by DiBenedetto, Gianazza and Vespri and the survey [11] by Düzgün, Fornaro and Vespri. Harnack's inequality for the prototype doubly nonlinear equation…”