Self-improving property of the fast diffusion equation

Abstract

We show that the gradient of the m-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse Hölder inequality in suitable intrinsic cylinders. Relying on an intrinsic Calderón-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for m ∈ (n−2) + n+2 , 1 . Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for m ≥ 1 (see [GS16] in the list of references) to the singular case. In particular, an intrinsic metric that depends on the solution itself is introduced for the singular regime.

Higher integrability for the singular porous medium system

“…We point out that independently of us, Gianazza & Schwarzacher [12] proved the higher integrability result in the scalar case for nonnegative solutions in the fast diffusion range (1.4). In contrast to [12], we prove the higher integrability regardless of whether the solution is non-negative or signed in the scalar case, or vector-valued in the case of systems. In another point, our results are also different.…”

“…Instead of an inhomogeneity given by a bounded function f , we consider a right-hand side in divergence form div F with F ∈ L σ for some σ > 2. In [12] the boundedness assumption on f is imposed to ensure that weak solutions are bounded. Here, we are able to deal with unbounded solutions.…”

Higher integrability for obstacle problem related to the singular porous medium equation

“…The method of intrinsic scaling is also applied to the study of regularity problems of porous medium type equations of the form ∂ t uu m = 0, m > 0, where u = u(x, t) and (x, t) ∈ R n+1 . There are two different approaches to the study of higher integrability of weak solutions to porous medium equations, one is the approach developed by Gianazza and Schwarzacher [9,10], the other is the approach developed by Bögelein et al [1,2]. In fact, the second method can also be used to study the vector-valued weak solutions of porous medium system.…”

“…In [7] the authors introduced a certain intrinsic cylinders of the form Q r,θr 2 (z 0 ) = B r (x 0 ) × t 0θ r 2 , t 0 + θ r 2 , θ ≈ u 1-m , (1.1) and used this kind of cylinders to study the boundedness, the Hölder continuity and the Harnack inequality of nonnegative weak solutions to porous medium equations. The intrinsic scaling method in [10] is very close to this kind of idea and the intrinsic cylinder takes the form Q r,θr 2 (z 0 ) = B r (x 0 ) × t 0θ r 2 , t 0 + θ r 2 , θ ≈ -- In fact, if we use a dimensional analysis, then the relation in (1.2) actually implies [θ ] = [u] 1-m , which is similar to (1.1). This enables us to use some ideas from [7] in this approach.…”

“…This enables us to use some ideas from [7] in this approach. Motivated by the proof of [7,Proposition B.4.1], Gianazza and Schwarzacher obtained a mean value result for nonnegative weak solutions [10,Proposition 5.2], which is a key ingredient in the proof of the reverse Hölder inequality in the degenerate regime. Therefore, the approach in [10] relies heavily on the regularity of the solution and this is the weak point in the study of the higher integrability.…”