Hölder estimates for singular non-local parabolic equations

Abstract

In this paper, we establish local Hölder estimate for non-negative solutions of the singular equation (M.P) below, for m in the range of exponents ( n−2σ n+2σ , 1). Since we have trouble in finding the local energy inequality of v directly. we use the fact that the operator (−△) σ can be thought as the normal derivative of some extension v * of v to the upper half space, [CS], i.e., v is regarded as boundary value of v * the solution of some local extension problem. Therefore, the local Hölder estimate of v can be obtained by the same regularity of v * . In addition, it enables us to describe the behaviour of solution of non-local fast diffusion equation near their extinction time.1

Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

“…It is very peculiar that the solution becomes identically zero in finite time. This is the so-called finite-time extinction phenomenon which is typical of some ranges of fast diffusion, see [39] for standard diffusion and [19,26] for fractional diffusion.…”

Regularity theory for singular nonlocal diffusion equations

“…The main technical novelty with respect to that paper is that, instead of the quadratic energies that were used there, which coincide with the ones which are adequate to treat linear problems, here we need to use a "nonlinear" energy adapted to β, since in our case β ′ (0) = 0. When L = (−∆) σ/2 and β is a power, this energy coincides with the one used in [12]. Let us notice however that our treatment of the energy differs from the one therein, which is what allows us to consider sign changing solutions.…”

Local continuity and asymptotic behaviour of degenerate parabolic systems

“…By an argument similar to the Lemma 4.2 of [KL2], we have the following lemma Lemma 3.6. If (3.16) is violated, then there exists a time level…”