$1$-Dimensional Harnack Estimates

Let $u$ be a non-negative super-solution to a $1$-dimensional singular parabolic equation of $p$-Laplacian type ($1<p<2$). If $u$ is bounded below on a time-segment $\{y\}\times(0,T]$ by a positive number $M$, then it has a power-like decay of order $\frac p{2-p}$ with respect to the space variable $x$ in $\mathbb R\times[T/2,T]$. This fact, stated quantitatively in Proposition 1.1, is a"sidewise spreading of positivity"of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an effect is based on geometrical ideas.


Introduction
Let E = (α, β) and define E −τo,T = E × (−τ o , T ], for τ o , T > 0. Consider the non-linear diffusion equation is a local, weak super-solution to (1.1), if for every compact set K ⊂ E and every sub-interval [t 1 , for all non-negative test functions This guarantees that all the integrals in (1.3) are convergent. These equations are termed singular since, for 1 < p < 2, the modulus of ellipticity |u x | p−2 → ∞ as |u x | → 0.
Remark 1.1 Since we are working with local solutions, we consider the domain E −τo,T = E ×(−τ o , T ], instead of dealing with the more natural E T = E ×(0, T ], in order to avoid problems with the initial conditions. The only role played by τ o > 0 is precisely to get rid of any difficulty at t = 0, and its precise value plays no role in the argument to follow. for some y ∈ E, and for some M > 0. Letρ There existsσ ∈ (0, 1), that can be determined a priori, quantitatively only in terms of the data, and independent of M and T , such that (1.5)

Novelty and Significance
The measure theoretical information on the "positivity set" in {y} × (0, T 2 ] implies that such a positivity set actually "expands" sidewise in R × [ T 4 , T 2 ], with a power-like decay of order p 2−p with respect to the space variable x. Although considered a sort of natural fact, to our knowledge this result has never been proven before; it is the analogue of the power-like decay of order 1 p−2 with respect to the time variable t, known in the degenerate setting p > 2 (see [2], [3, Chapter 4, Section 4], [7]). As the t − 1 p−2 -decay is at the heart of the Harnack estimate for p > 2, so Proposition 1.1 could be used to give a more streamlined proof of the Harnack inequality in the singular, super-critical range 2N N +1 < p < 2. This will be the object of future work, where we plan to address the general N -dimensional case.
The proof is based on geometrical ideas, originally introduced in two different contexts: the energy estimates of § 2 and the decay of § 3 rely on a method introduced in [8] in order to prove the Hölder continuity of solutions to an anisotropic elliptic equation, and further developed in [5,6]; the change of variable used in the actual proof of Proposition 1.1 was used in [4].

Further Generalization
Consider partial differential equations of the form where the function A : E −τo,T × R × R → R is only assumed to be measurable and subject to the structure condition where 1 < p < 2, C o and C 1 are given positive constants. It is not hard to show that Proposition 1.1 holds also for weak super-solutions to (1.6)-(1.7), since our proof is entirely based on the structural properties of (1.1), and the explicit dependence on u x plays no role. However, to keep the exposition simple, we have limited ourselves to the prototype case.

Energy Estimates
Let u be a non-negative bounded, weak super-solution in E −τo,T , assume and let ω be a positive parameter. Without loss of generality we may assume that 0 ∈ (α, β). For ρ sufficiently small, so that (−ρ, ρ) ⊂ (α, β), let a ∈ (0, 1), H ∈ (0, 1] parameters that will be fixed in the following, There exists a positive constant γ = γ(p), such that for every cylinder Q(y) = B ρ (y)×(0, T ] ⊂ E −τo,T , and every piecewise smooth, cutoff function ζ vanishing on ∂B ρ (y), such that 0 ≤ ζ ≤ 1, and ζ t ≤ 0, Proof -Without loss of generality, we may assume y = 0. In the weak formulation of (1.1) take ϕ = G(u)ζ p as test function, with for an absolute constant γ 1 independent of ρ, and ζ 2 is monotone decreasing, and satisfies for an absolute constant γ 2 independent of T . It is easy to see that we have Modulo a Steklov averaging process, we have The second term on the right-hand side vanishes, as ζ(x, T ) = 0. Therefore, an application of Young's inequality yields where we have taken into account that ζ t ≤ 0. Therefore, we conclude where the first term on the left-hand side is non-negative, since for 0 < s < 1 − a < 1 the function f (s) = (s+a) 2−p t is monotone decreasing, and f (1 − a) = 1 1−a > 1.

Remark 2.2
Even though in the next Section H basically plays no role, we chose to state the previous Proposition with an explicit dependence also on H for future applications.

A Decay Lemma
Without loss of generality, we may assume µ − = 0. Let M = ω, L ≤ M 2 , and suppose that u(0, t) > M ∀t ∈ (0, Now, let s o be an integer to be chosen, and define Then, for any ν ∈ (0, 1), there exists a positive integer s o such that |{t ∈ (0, Proof -Take t ∈ A so : by definition, there existsx ∈ B ρ 2 such that u(x, t) < L/2 so . On the other hand, by assumption u(0, t) > 2L, and therefore, u(0, t) + (L/2 so ) > L. Hence and we obtain If we integrate with respect to time over the set A so , we have Apply estimates (2.1) with a = 1 2 so , Hω = HM = L. The requirement H ≤ 1 is satisfied, since L ≤ M 2 . They yield With these choices, we have

|Q|.
If we require L ≤ T ρ p 1 2−p , and we substitute it back in the previous estimate, Therefore, if we want that |A so | ≤ ν|(0, The previous result can also be rewritten as Now letρ be such that and assume that Bρ ⊂ (α, β). Then Lemmas 3.1-3.2 can be rephrased provided (3.1) holds, and B(x) cρ ⊂ (α, β).

A DeGiorgi-Type Lemma
Assume that some information is available on the "initial data" relative to the cylinder B 2ρ (y) × (s, s + θρ p ], say for example u(x, s) ≥ M for a.e. x ∈ B 2ρ (y) for a constant δ ∈ (0, 1) depending only upon p, and independent of M and ρ.
Remark 4.1 Lemma 4.1 is based on the energy estimates and Proposition 3.1 of [1], Chapter I which continue to hold in a stable manner for p → 1. These results are therefore valid for all p ≥ 1, including a seamless transition from the singular range p < 2 to the degenerate range p > 2.

Proof of Proposition 1.1
Fix y ∈ E, defineρ as in (3.2), and choose a positive parameter C ≥ 4, such that the cylindrical domain This is an assumption both on the size of the reference ball B (y) and on T : nevertheless, we can always assume it without loss of generality. Indeed, if (5.1) were not satisfied, we would decompose the interval (0, T 2 ] in smaller subintervals, each of width τ , such that (5.1) is satisfied working withρ replaced by The only role of C is in determining a sufficiently large reference domain which contains the smaller ball we will actually work with, and will play no other role; in particular the structural constants will not depend on C. Now, introduce the change of variables and the new unknown function

Returning to the Original Coordinates
In terms of the original coordinates and the original function u(x, t), this implies where the time t 1 corresponding to τ 1 is computed from (5.2) and (5.7), and dist(x, y) = 2cρ. Now, apply Lemma 4.1 with M replaced by M o over the cylinder B cρ 2 (x) × t 1 , t 1 + θ(cρ) p . By choosing the assumption (4.2) is satisfied, and Lemma 4.1 yields

A Remark about the Limit as p → 2
The change of variables (5.2) and the subsequent arguments, yield constants that deteriorate as p → 2. This is no surprise, as the decay of solutions to linear parabolic equations is not power-like, but rather exponential-like, as in the fundamental solution of the heat equation. Nevertheless, our estimates can be stabilised, in order to recover the correct exponential decay in the p = 2 limit. However, this would require a careful tracing of all the functional dependencies in our estimates, and we postpone it to a future work.