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

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for non-negative solutions u : T → R ≥0 with parameters m ∈ (0, ∞) and p ∈ (1, ∞). If m = 1, (1.1) reduces to the parabolic p-Laplace equation, whereas for p = 2 we retrieve the porous medium equation. Doubly nonlinear equations of type (1.1) are classified as doubly degenerate if m > 1 and p > 2, singular-degenerate if m > 1 and p ∈ (1, 2), degeneratesingular if m ∈ (0, 1) and p > 2 and doubly singular if m ∈ (0, 1) and p ∈ (1, 2). Furthermore, depending on the behavior of solutions, we distinguish between slow diffusion equations with m( p −1) > 1 and fast diffusion equations with m( p −1) < 1. The qualitative difference between both cases stems from the fact that in the former one solutions might have a compact support, while this is not possible in the latter one. In the present paper, we treat the complete slow diffusion range p(m − 1) > 1, which includes the doubly degenerate case and the singular-degenerate and degenerate-singular slow diffusion case. In the literature, (1.1) often appears in equivalent forms; cf. [17][18][19][20]28,33]. More precisely, we note that formally (1.1) is a transformation of is weakly differentiable with respect to the space variable, whereas the second one claims this for u m (in the case m < 1 only the latter one makes sense). For the prototype porous medium equation the equivalence of both notions of solutions has been shown in [6]. It is still an open problem if the same is true for doubly nonlinear equations and porous medium type equations with a general structure.
Harnack estimates play a crucial role in the regularity theory of partial differential equations. In the elliptic setting, essential contributions are due to Moser [25] for linear elliptic equations and Serrin [29] and Trudinger [31] for quasilinear elliptic equations. In the parabolic setting, the first results have been obtained by Hadamard [16] and Pini [27] for non-negative solutions of the heat equation. For the heat equation Harnack's inequality takes the form with waiting time 2 . Moser [26] showed that this result is true for linear parabolic equations as well and demonstrated the necessity of the waiting time. Later, Trudinger [32] proved Harnack inequalities for quasilinear parabolic equations and the homogeneous doubly nonlinear equation ∂ t u p−1 − div |Du| p−2 Du = 0 with p > 1. Using an approach based on mean value inequalities for suitable De Giorgi classes, Gianazza and Vespri [14] gave a proof that extends to more general operators A(x, t, u, Du) instead of |Du| p−2 Du. Finally, simplifying an approach originally introduced by Moser, Kinnunen & Kuusi [22] obtained Harnack's inequality for the homogeneous doubly nonlinear equation, where the Lebesgue measure is replaced by a more general Borel measure. In the case of non-homogeneous nonlinear equations, the situation is more involved. DiBenedetto [7] proved that non-negative weak solutions of the parabolic p-Laplace equation and the porous medium equation satisfy an intrinsic Harnack inequality of the form with t w = cu(x o , t o ) 2− p p for the parabolic p-Laplace equation and t w = cu(x o , t o ) 1−m 2 for the porous medium equation. These Harnack inequalities are called intrinsic, because the waiting times depend on the solution itself. 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. Nearly 20 years later, this problem was overcome by DiBenedetto, Gianazza & Vespri [9], whose proof only uses measure theoretical tools. The main novelty is the socalled Expansion of Positivity. 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 has first been proved by Vespri [33] for the full range of parameters p > 1 and m + p > max{2, 3 − p n }. The proof uses explicit constructions involving the Barenblatt solution and therefore cannot be applied to more general structures. For the doubly degenerate case Fornaro and Sosio [12] generalized the result to weak solutions of where the operators A and B fulfill the conditions with p ≥ 2, positive constants c 0 , c 1 , c 2 , c and a function satisfying an (m − 1)-growth condition with m ≥ 1. They used a definition of weak solution involving u m+1 2 . A weak Harnack inequality for super-solutions can be found in [24]. For the case of fast diffusion equations, we refer to the articles by Fornaro, Sosio and Vespri [13] and Vespri and Vestberg [34].
In this paper we prove Harnack's inequality for the entire slow diffusion range and thereby close the gap for the by now missing singular-degenerate and degenerate-singular slow diffusion cases. Furthermore, we work with a definition of weak solution involving u m , which is new even for the doubly degenerate case and the slow diffusion porous medium equation.

Setting
We consider non-negative weak solutions to the doubly nonlinear equation with m > 0. For the vector field A : T × R × R n → R n we assume that A is measurable with respect to (x, t) ∈ T for all (u, ξ) ∈ R × R n and continuous with respect to (u, ξ) for a.e. (x, t) ∈ T . Moreover, we assume that A satisfies the following growth and ellipticity conditions for p > 1 and structure constants 0 < ν ≤ L < ∞. We demand that for some σ > n+ p p−1 and that the parameters m and p satisfy m( p − 1) > 1 which means that we are in the slow diffusion range. In the following we abbreviate We now give the precise definition of weak solution to (1.3) that we use throughout the paper.

Definition 1.1 Assume that the vector field
is a non-negative weak sub(super)-solution to the doubly nonlinear equation (1.3) if and only if the identity holds true for any testing function ϕ ∈ C ∞ 0 ( T , R ≥0 ). If u is a weak sub-and super-solution it is called a weak solution.
We are now in the position to formulate the main result of our paper: Theorem 1.2 Let m > 0, p > 1 with m( p − 1) > 1 and u be a continuous, non-negative, weak solution to (1.3) in the sense of Definition 1.1, where the vector field A satisfies (1.4) and F satisfies (1.5) Then, there exist constants c o , γ > 1 depending only on n, m, p, L, ν and σ such that for all cylinders Note that the continuity assumption in Theorem 1.2 is not restrictive. The Harnack inequality continues to hold for a.e. point (x o , t o ) ∈ T if we state it for an arbitrary non-negative, weak solution to (1.3). However, for the sake of a neater exposition of the result, we prefer to state it for continuous solutions.

Plan of the paper
In Sect. 2 we collect some auxiliary tools. Using u m − a m for some level a as test function in (a mollified version of) the definition of weak sub-and super-solutions, we derive certain Caccioppoli inequalities in Sect. 3. For convenience of the reader we state all intermediate results for weak sub-respectively super-solutions instead of weak solutions, so that it becomes clear what the minimal assumptions are. Next, in Sect. 4 we show that weak sub-solutions to (1.3) are locally bounded and give a quantitative estimate. In Sect. 5 we prove so-called De Giorgi type lemmas. Loosely speaking, the first lemma shows that if a super-solution u to (1.3) is smaller than some level M only on a small enough proportion of a suitable cylinder, then u is larger than M 2 a.e. on a smaller cylinder contained in the first one. The second lemma gives an analogous statement for sub-solutions in the case that u is larger than a fixed level only on a small enough proportion of the bigger cylinder and consequently smaller than a fraction of the level on the smaller cylinder. The proofs of the statements rely in particular on the Caccioppoli estimates. In Sect. 6 we prove Expansion of Positivity of non-negative weak super-solutions. The conclusion of the section is that if Here, the constants b, κ ∈ (0, 1) depend only on the data and α. In the proof, the Caccioppoli estimates and the first De Giorgi type lemma are used. Finally, in Sect. 7 we deduce the intrinsic Harnack inequality stated in Theorem 1.2. To show the forward inequality, i.e. the second inequality in (1.8), after a transformation we use the second De Giorgi type lemma and iteratively apply Expansion of positivity. Subsequently, we prove that the forward inequality implies the backward Harnack inequality, i.e. the first inequality in (1.8). Actually, a more general version of the backward Harnack inequality is shown in Sect. 7.

Notation
First, we introduce some notation used throughout the paper. For functions defined on T , we denote the time slice at time t

Mollification in time
Since weak solutions do not possess a time derivative in general we have to use mollification.
To this end, for v ∈ L 1 ( T , R N ) and h > 0 we define the following mollification in time which formally satisfies the ordinary differential equation Using the same technique as in [30,Lemma 3.6], we conclude that any sub(super)-solution to (1.3) in the sense of Definition 1.1 satisfies the mollified version of (1.6), for any ϕ ∈ C ∞ 0 ( T , R ≥0 ).

Transformation
The following Lemma is an easy consequence of a change of variables.
The next Lemma shows that the product of a non-negative weak super-solution u with a non-decreasing C 1 -function γ is a super-solution to a modified equation. A similar argument has already been used in [9]. (1.3) in × I associated to A, F and γ ∈ C 1 (I )∩C 0 (I ) is non-decreasing and satisfies 1 C ≤ γ ≤ C on I for a constant C ≥ 1. Then, the functioñ u := γ u is a non-negative weak super-solution to (1.3) in × I associated to the vector-field

Lemma 2.3 Let ⊂ R n be bounded and open and I ⊂ R an open interval. Assume that u is a non-negative weak super-solution to
Proof In the following we abbreviate I : By assumption γ , u and therefore also [[u]] h are non-negative. By an approximation argument we may use γ ϕ as testing function in the mollified weak formulation (2.4) on the interval I instead of (0, T ). This leads to Passing to the limit h ↓ 0 with the help of Lemma 2.1 and taking into account that spt(γ ϕ) is compact in the last term on the right-hand side, this leads to which is in view of the definition ofũ and F equivalent to Recalling the definition of A, this yields the claim.
Combining the last two lemmata leads to the following statement, which is used in the proof of the expansion of positivity.

Corollary 2.4 Let T > 0 and u a non-negative weak super-solution to (1.3) in B × (0, T ) associated to A and F. Further, assume that I ⊂ R is an open interval, that
is an increasing C ∞ -diffeomorphism and that γ ∈ C 1 (I ) ∩ C 0 (I ) is non-decreasing and satisfies 1 C ≤ γ ≤ C on I for some constant C ≥ 1. Then, the function v(x, τ ) := γ (τ ) · u(x, (τ )) is a non-negative weak super-solution to (1.3)

Auxiliary lemmata
For a function v ∈ W 1,1 and k < the next lemma gives a local estimate for the product of the measures of superlevel sets {v > } and sublevel sets {v < k} in terms of the L 1 -norm of Dv on the intersection of their complements, cf.

Lemma 2.6
For any α > 0, there exists a constant c = c(α) such that, for all a, b ≥ 0, the following inequality holds true: The next lemma summarizes all properties we need concerning the boundary term b defined in (2.1).

Lemma 2.7
Let m > 0. There exists a constant c = c(m) such that for every u, a ≥ 0 we have Proof The proof of (i) can be found in [4, Lemma 2.3] for m ≥ 1 and in [3, Lemma 3.4] for 0 < m < 1. The inequalities in (ii) are a consequence of (i) and Lemma 2.6.
The following iteration lemma is a well known result and can be found for instance in [8, Chap. I.4, Lemma 4.1].

Caccioppoli inequalities
In this section we derive energy estimates that are crucial in the course of the paper. We start with the energy estimates for weak super-solutions.
Since the claimed estimates are local in nature, we may assume without loss of generality that ). An approximation argument shows that the mollified weak formulation (2.4) extends to non-negative testing functions ϕ ∈ L p (0, T ; W and u(0) ∈ L m+1 ( ) by the assumptions on u, growth condition (1.4) and Lemma 2.1. We therefore find that holds true for any ϕ ∈ L p (0, T ; W We choose as testing function in the mollified version (3.3) of the differential equation. For the first term on the left hand side we have where the meaning of I ε and II ε is clear in this context. We let h ↓ 0 also in the diffusion term. For the resulting integral we use assumptions (1.4) and Young's inequality to obtain Inserting the preceding estimates into (3.3), we conclude that Now, we pass to the limit ε ↓ 0 in the preceding inequality. Since u ∈ C 0 ([0, T ]; L m+1 ( )), Altogether, we deduce the estimate for any t 1 ∈ s (t o ). Finally, taking the supremum over t 1 ∈ s (t o ) in the first term and passing to the limit t 1 ↑ t o in the second term yields inequality (3.1).
In order to prove (3.2) we choose ϕ(x, t) = η p (x)ψ ε (t)(u m (x, t) − a m ) − as testing function in (3.3), where η is defined as before and For the diffusion term and the right side the same arguments as in the proof of (3.1) are applicable. Therefore by passing to the limits h ↓ 0 and ε ↓ 0 we obtain Omitting the second term on the left side, choosing t 1 = t o − s and taking the supremum over t 2 ∈ s (t o ) leads to (3.2).
Similarly, we obtain energy estimates for sub-solutions. However, in the course of the paper we only need the analogue of (3.1).

Proof
The proof is analogous to the one of the energy estimate (3.1). Here, we choose the testing function with the positive part of u m − a m instead of the negative one. Similar arguments as in the proof of (3.1) then lead us to inequality (3.5).

Local boundedness of non-negative weak sub-solutions
In this section we establish that non-negative weak sub-solutions to (1.3) are locally bounded. We argue by a parabolic version of De Giorgi classes.  Throughout the proof, we use the short-hand notation Furthermore, for a quantity k ≥ 1 to be chosen later on, we consider levels Since u m ∈ L p ( T ) by definition, Y i is finite for any i ∈ N 0 . The idea of proof is to show a recursive estimate for Y i . To this aim we first use Hölder's inequality to obtain where the definition of I is clear in this context. First, by the Gagliardo-Nirenberg inequality from Lemma 2.9 we infer for a constant c = c(n, m, p). We now consider the integrand in the first integral on the right-hand side. For u ≥ k i+1 we have with the abbreviatioñ Therefore, in view of Lemma 2.7 (ii) we obtain Using this inequality above and applying the Caccioppoli inequality (3.5) from Lemma 3.2, yields with c = c( p), so that m, p, ν, L). Further, we have that which together with k ≥ 1 implies that Finally, the preceding computations together with 0 < ≤ 1 and with a constant c = c(n, m, p, ν, L). Inserting this inequality into (4.1) and using (4.2), we conclude that where we used the abbreviations Since σ > n+ p p−1 , we have that γ > 0. Choosing k ≥ 1 large enough, such that with a suitable constant c = c(n, m, p, ν, L, σ ), we find that Thus, the assumptions of Lemma 2.8 are satisfied. Consequently we find that Y i → 0 as i → ∞, which implies u ≤ k a.e. in 1 2 Q 0 . The claim of the theorem now follows by an application of Young's inequality.

De Giorgi type lemmas
In this section we will prove certain De Giorgi type lemmata for weak sub-and supersolutions. We start with the one for super-solutions. Lemma 5.1 Let m > 0, p > 1 with m( p − 1) > 1 and u be a bounded non-negative weak super-solution to (1.3) in the sense of Definition 1.1, where the vector-field A satisfies (1.4) and F ∈ L σ ( T ) for some σ > n+ p p−1 . Moreover, consider z o ∈ T and , θ, M > 0 such that Then, there exists ν 1 ∈ (0, 1) depending only on n, m, p, ν, L, σ and θ , such that: If Proof For i ∈ N 0 define radii i and times τ i by as well as levels To shorten notation, we introduce At this stage, we use the Caccioppoli inequality (3.1). Since 0 ≤ u < k i on A i and M 2 ≤ k i ≤ M and by Lemma 2.7 (ii), we estimate the term involving b on the left-hand side by , while for the one on the right-hand side we obtain by Lemma 2.7 Thus, we conclude that where in the second last line we used assumption (5.1). Note that c = c(m, p, ν, L, θ). Next, we use Hölder's inequality with exponents n+2 n and n+2 2 , the Gagliardo-Nirenberg inequality from Lemma 2.9 with r = 2 and p and the preceding estimate. This leads to with a constant c = c(n, m, p, ν, L). Moreover, due to Lemma 2.6 we have Combing the preceding estimates yields with a constant c = c(n, m, p, ν, L, θ). Dividing the above inequality by |Q i+1 |, using the fact that |Q i | |Q i+1 | = c(n, p) shows that  ∈ (0, 1), such that Then, there exists ν 2 ∈ (0, 1) depending only on n, m, p, ν, L, σ, θ, a and ζ such that: If and In the following we will apply the Caccioppoli inequality (3.5) from Lemma 3.2. Using the definition of A i and Lemma 2.7 (i), (ii) and the fact that 1 Thus, by the Caccioppoli inequality (3.5) and assumption (5.4), we obtain for a constant c = c(m, p, ν, L, θ, ζ ). Similarly as before, we use Hölder's inequality, the Gagliardo-Nirenberg inequality from Lemma 2.9 with r = 2 and p and the last estimate to conclude for a constant c = c(n, m, p, ν, L, θ, ζ ). Notice that Combining the preceding two estimates leads to with a constant c = c(n, m, p, ν, L, θ, a, ζ ). By completely the same reasoning as in the proof of Lemma 5.1 we infer that Y i → 0 as i → ∞, provided we choose ν 2 ∈ (0, 1) small enough in dependence on the data.

Expansion of positivity
In this section, we prove the so called Expansion of Positivity of a non-negative weak supersolution u. The Expansion of Positivity is crucial in the proof of Harnack's inequality. In a first step we show the following lemma, which ensures a certain propagation of positivity in measure. and are satisfied, then Proof In the following we abbreviate Q 0 := Q + ,δ M −d p (z o ) with δ ∈ (0, 1) to be chosen later. The idea of the proof is to show that if (6.1) and (6.2) are valid, then , which is equivalent to (6.3). Therefore in a first step we let s ∈ (0, 1) and compute To estimate the first term on the right hand side we use the Caccioppoli inequality (3.2) from Lemma 3.1. Taking r = (1 − s) and a = M leads to c( p, ν, L). Recalling the definition of the boundary term b from (2.1) we estimate the left hand side by for ε ∈ (0, m m+1 ) to be chosen later. For the first term on the right-hand side, we use again the definition of b and assumption (6.1) to obtain Further, we have that and in view of assumption that (6.2) we obtain

Remark 6.2
From the proof of Lemma 6.1 we observe that ε and δ are monotonically increasing with respect to α.
The preceding lemma at hand, we are now able to prove the Expansion of Positivity for non-negative weak super-solutions to the doubly degenerate equation (1.3).

Application of lemma 6.1
For j ∈ N to be chosen later in dependence on n, m, p, ν, L, σ and α we define Note that Now we fix ∈ (0, 0 ] and assume that (6.6) is satisfied and that

From
Step 6.1 we deduce that for any τ ∈ [0, τ 0 ] there holds and In particular, letting is a non-negative weak super-solution to and Using the growth assumptions (1.4) of A together with the definition of the functions and γ we compute that A satisfies the growth and ellipticity conditions Defining we observe that k 0 = εγ (τ )s(τ )M for any τ ∈ [0, τ 0 ] and therefore inequality (6.9) can be rewritten as Moreover, the parameter b in Proposition 6.3 is monotonically increasing with respect to α. This can be seen from the definition b = δε d 2 −( j +1)d , where j is decreasing and ε and δ are increasing with respect to α; see Remark 6.2.

Harnack's inequality
We are now ready to prove our main result, Theorem 1.2. In the following section, the second (forward in time) inequality of (1.8) is shown. In a subsequent step, we ensure the validity of the first (backward in time) inequality of (1.8).

Forward inequality
Let c o ≥ 1 to be fixed later, consider ( Moreover, assume that > 0 is small enough so that T will only be needed in the proof of the backward Harnack inequality. Finally, we define the rescaled function where (x,t) : A straightforward computation shows that v is a bounded, continuous, non-negative weak super-solution of The main step towards Theorem 1.2 is the following lemma. After returning to the original variables this proves the intrinsic forward Harnack inequality, i.e. the second inequality of (1.8). Indeed, if Lemma 7.1 is valid, we obtain that which shows the second inequality of (1.8) for γ = max 1 γ 0 |B 1 (0)| 1/σ , 1 γ 1 .

Backward inequality
With the intrinsic forward Harnack inequality on hand, we are able to show the intrinsic backward Harnack inequality, i.e. the first inequality of (1.8). Actually, in the following we will prove the more general version with positive constants c 1 , c 2 such that c 1 γ c 2 −2 > 1, which implies the first inequality of (1.8) by choosing c 1 = c 2 = 2. We have already fixed (x o , t o ) ∈ T with u(x o , t o ) > 0. Now we assume that B 9 (x o ) × (t o − 4θ p , t o + 4θ p ) T . Moreover, let c 1 , c 2 > 0 be positive constants such that c 1 γ c 2 −2 > 1 and suppose that alternative (1.7) is not valid, i.e.
In order to prove the backward Harnack inequality, we consider two alternatives. First, we assume that with γ as in the right-hand side of (1.8). Our aim is to prove that (7.18) implies Since d > 0, γ > 1 and c 1 γ c 2 −1 > 1, this implies (t 1 − 2θ p , t 1 + 2θ p ) ⊂ (t o − 4θ p , t o + 4θ p ) T . Thus, we are able to apply the forward Harnack inequality with (x o , t o ) replaced by (x , t 1 ). This leads to which contradicts (7.17), or In view of (7.18) and the facts that x o ∈ B (x ) and t 1 +θ p < t o , this yields the contradiction . Therefore (7.18) implies (7.19). It remains to treat the case where (7.18) is violated. This means that there exists t ∈ (t o − 2θ p , t o ) such that u(x o , t) = c 1 γ c 2 −1 u(x o , t o ). We define τ as the largest value with this property (note that u is continuous) and let Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.