Parabolic Harnack estimates for anisotropic slow diffusion
aa r X i v : . [ m a t h . A P ] D ec PARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOWDIFFUSION
SIMONE CIANI, SUNRA MOSCONI, AND VINCENZO VESPRI
Abstract.
We prove a Harnack inequality for positive solutions of a parabolic equation withslow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problemto a Fokker-Planck equation and construct a self-similar Barenblatt solution. We exploittranslation invariance to obtain positivity near the origin via a self-iteration method and deducea sharp anisotropic expansion of positivity. This eventually yields a scale invariant Harnackinequality in an anisotropic geometry dictated by the speed of the diffusion coefficients. As acorollary, we infer H¨older continuity, an elliptic Harnack inequality and a Liouville theorem.
MSC 2020 : 35K59, 35K65, 60J60, 74N25
Key Words : Anisotropic diffusion, Fundamental solution, Harnack inequality, Intrinsic ge-ometry, Fokker-Planck equation
Contents
1. Introduction 12. Preliminaries 62.1. Functional inequalities 62.2. Scaling properties 72.3. Energy inequality and consequences 82.4. Comparison principles 102.5. L p -solutions 113. Barenblatt fundamental solutions 123.1. Construction of a Barenblatt solution 133.2. Properties of the Barenblatt solutions 144. Proof of Theorem 1.1 16References 21 Introduction
We are concerned with solutions of the model parabolic anisotropic equation(1.1) ∂ t u = N X i =1 ∂ i (cid:0) | ∂ i u | p i − ∂ i u (cid:1) , satisfied in a suitably weak sense in Ω × (0 , T ), Ω ⊆ R N for suitable choice of p i >
2. These kindof equations raised increasing interest in the last decades as they present an interesting feature,namely an anisotropic diffusion with orthotropic structure. Besides its inherent mathematicalinterest, (1.1) appears in modelling materials such as earth’s crust or wood, where the velocityof propagation of diffusion varies according to the different orthogonal directions. From themathematical point of view, the principal part in (1.1) arises as the Euler-Lagrange equationof a functional with non-standard growth , i.e. of the type Z F ( ∇ u ) dx, where 1 C ( | z | p − F ( z ) C ( | z | q + 1) for some p < q , as opposed to the standard growth condition p = q . Starting from thepioneering examples in [17,25], it soon became apparent that the regularity theory for solutionsof the corresponding Euler-Lagrange elliptic equation is much more delicate and rich than thestandard one. Since then, the elliptic regularity theory grew in considerable size. Even ifthis has not always been the case, the general principle underlying to the theory is that mostregularity results can be recovered when the power gap q − p in the non-standard growthcondition is small. Since it would be impossible to collect here all the contributions, we referto the surveys [28, Section 6] and [26] for a general overview of the subject and comprehensivebibliographic references.As the non-standard elliptic theory matured, its parabolic counterpart became a researchthˆeme as well. The delay in development was considerable, mainly because already the isotropicproblem with p i ≡ p = 2 presented great difficulties, solved in full generality only a decadeago through the work of Di Benedetto and collaborators (see [14] and the literature therein).Nevertheless, parabolic equations with non-standard growth were considered well before that(see e.g. [22]), giving rise to a large amount of results on existence, well-posedness, L ∞ -estimatesand diffusion analysis. For an extensive bibliography on this research, we refer to [4] and forthe theory of variational solutions to [26, Section 12] and the references therein.Despite some partial results, however, the regularity theory for parabolic anisotropic equa-tions is still largely unexplored. Up to our knowledge, local H¨older continuity of solution of(1.1) was unknown, as well as the validity of a suitable (necessarily intrinsic) parabolic Har-nack inequality. The latter is precisely the aim of this paper, where we are going to prove thefollowing result. Theorem 1.1.
Let u > be a local weak solution to (1.1) in Ω × [ − T, T ] and suppose that (1.2) ∀ i = 1 , . . . , N < p i < ¯ p (cid:18) N (cid:19) ¯ p := (cid:18) N N X i =1 p i (cid:19) − < N. Then, there exist two constants
B, C depending only on N and the p i ’s such that (1.3) 1 B sup K ρ ( M ) u ( · , − B M − ¯ p ρ ¯ p ) u (0 , B inf K ρ ( M ) u ( · , B M − ¯ p ρ ¯ p ) , M = u (0 , /C, whenever B M − ¯ p ρ ¯ p < T and K B ρ ( M ) ⊆ Ω , being (1.4) K r ( M ) := N Y i =1 (cid:8) | x i | < M ( p i − ¯ p ) /p i r ¯ p/p i / (cid:9) . Let us make some comments on the statement, significance and proof of the previous theorem.-
The intrinsic geometry . A parabolic Harnack inequality for a non-homogeneous equa-tion such as (1.1) cannot hold in classical form. This was first understood for the parabolic p -Laplacian equation(1.5) ∂ t u = ∆ p u through an analysis of the so-called Barenblatt fundamental solutions, a family of explicitsolution encompassing most of the features which distinguish the classical heat equation (andits quasilinear non-degenerate counterpart) from (1.5). The correct formulation of the Harnackinequality for (1.5) was found in the seminal paper [10] when p >
2, and it has an intrinsic form. To explain briefly this term let us focus on the p > ARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION 3 at that point. The parabolic nature of the equation allows for such a control to hold only aftera positive time delay (in the case of lower bounds) has passed. For the heat equation this waiting time only depends on the size of the region where we seek for the lower bound and noton the solution, while for the parabolic p -Laplacian equation (1.5), its length also depends onthe size of the solution at the chosen point: the term intrinsic refers (not only, but mainly) tothis phenomenon.In the case of (1.5), the value of the solution at the chosen point only affects the waitingtime while, for the anisotropic equation (1.1), it turns out to determine the full shape, or geometry , of the region where the control is available. This is apparent from the definition (1.4)of the intrinsic cubes : indeed, in K r ( M ), r plays the rˆole of an anisotropic radius , while M prescribes the anisotropic geometry . To justify the first statement, notice that the Lebesguemeasure of K r ( M ) is always r N , regardless of M . Regarding the second, one can followthe well-known principle that higher exponents give slower diffusion , so that lower values of M ≃ u (0 ,
0) squeeze K r ( M ) in directions of slower-than-average diffusion ( p i − ¯ p >
0) andstretch it in directions of faster-than-average diffusion ( p i − ¯ p < Barenblatt solutions . One of the main byproducts of our proof is the construction of afamily of self-similar Barenblatt solutions for (1.1) and the analysis on their basic properties.Self-similar solutions are by now a classical thˆeme and have been extensively studied in variousparabolic nonlinear frameworks, see e.g. [35, Ch. 16] and the therein cited literature. Theirrˆole turned out to be pivotal in understanding the general behaviour of solutions and hasoften been an important stepping-stone for treating more general equations and formulatingsensible statements on the general expected results: compare the classical works of Pini [32]and Hadamard [18], later generalised in the linear measurable setting by Moser [30] or, inthe singular/degenerate case, the first works [10], [15] employing the Barenablatt solutions,generalised in [12, 13].For equation (1.1), the explicit form of the Barenblatt solutions is however unknown atpresent, and their existence is obtained through an abstract approach. Naturally, we cannotassume any a-priori regularity and the method heavily relies on the identification of the naturalscalings of (1.1) mentioned above, allowing to formulate the right notion of self-similarity. Moredetails on the difficulties that this approach involves will be made in the outline of the proofbelow.-
Assumptions . The main condition required in the Harnack inequality is (1.2). On onehand, p i > i means that we are settling ourselves in the slow diffusion regime . The mainfeature of this framework is that, for example, solutions of (1.5) for p > p i < ¯ p (1 + 1 /N ) requires that the powers p i are not too sparse, following theabove mentioned principle in problems with non-standard growth. Local boundedness holdsin the larger range p i < ¯ p (1 + 2 /N ), but we are not aware of counterexamples if this conditionis violated. It would be interesting to know wether the Harnack inequality holds true also for p max ∈ [¯ p (1 + 1 /N ) , ¯ p (1 + 2 /N )) but, if so, its proof likely requires different techniques thanthe ones employed here.A few comments on the constants C, B in the statement. As mentioned above, the Barenblattsolution we use is constructed in an abstract way and we don’t know if a uniqueness theorem(up to translation and scaling) holds. The constants depend on a lower bound on the Barenblatt
S. CIANI, S. MOSCONI, AND V. VESPRI solution, hence, ultimately on the choice of the latter. Thus, they are rather undeterminedfrom the quantitative point of view.Finally, the number u (0 ,
0) is not a-priori well-defined for a weak solution. However, thanksto [16, Corollary 4.3], any solution of (1.1) under assumptions (1.2) possesses an essentiallyu.s.c. representative, allowing to give a meaning to u (0 , u (0 , >
0, for otherwise the claimed bounds trivially holds (assuminginf ∅ = + ∞ , sup ∅ = −∞ ).- Outline of the proof . As already mentioned, our first task is to build a family ofBarenblatt solutions. We find all the natural scalings of (1.1) and construct a bijection betweensolutions of (1.1) and solutions of an anisotropic Fokker-Plank equation (see e.g. [7] for a similarapproach). We then seek for a stationary solution of the latter, which is found through a fixedpoint argument and comparison principles. Here, the slow diffusion regime plays a pivotal rˆolein recovering sufficient compactness to apply Shauder fixed point theorem. Let us note thatwe rely on a weak continuity result (Lemma 3.1, point 3) of independent interest, which wewere not able to find in the literature.At this stage, the stationary solution of the Fokker-Plank equation is a rather irregular objectof little use. However, exploiting its correspondence to a Barenblatt self-similar solution of (1.1)and using a self-iteration method based on comparison principles and translation invariance, weare able to prove a positive lower bound in a small neighbourhood of the origin. Transferringthe bound to the Barenblatt solution, we find a quantitative expansion of positivity rate for it.We then proceed in a manner reminiscent of the proof in [10] of the Harnack inequality for(1.5), namely finding a positivity set and then expand it forward in time through comparisonwith Barenblatt solutions. For the first step, we actually employ a simplification describedin [12], which makes use of the so-called
Clustering Lemma of [11]. We have to face two maindifficulties: the intrinsic geometry of the problem, contrary to what happens in most instancesof the theory, involves not only the time variables but also, and mainly, the spatial ones (inan anisotropic way). Secondly, even disregarding the geometry, the natural intrinsic cubes asper (1.4) come from a pseudo-metric rather than from a metric. To face the first difficultywe heavily rely on the natural transformations leaving (1.1) invariant; for the second one, weprove an abstract pseudo-metric version of the so-called Krylov-Safonov trick, of independentinterest (Lemma 4.1).-
Consequences of Theorem 1.1 . A first corollary of the Harnack inequality is the H¨oldercontinuity of solutions of (1.1), whose detailed proof is described in [9]. However, much moreregularity is to be expected, as suggested by the ellitpic case briefly discussed below.An intrinsic Harnack inequality immediately follows from Theorem 1.1 for solutions of(1.6) N X i =1 ∂ i ( | ∂ i u | p i − ∂ i u ) = 0 . Even in the elliptic case, homogeneity is still missing, suggesting that any scale invariantHarnack inequality must be of intrinsic form, as in the parabolic case. We state our Harnackinequality for (1.1) in the following corollary.
Corollary 1.2.
Let K r ( M ) be as in (1.4) and p i as in (1.2) . There exist C, B > , dependingon N and the p i ’s such that if u > weakly solves (1.6) in K Bρ ( M ) , where M = u (0) /C > ,then (1.7) B − sup K ρ ( M ) u u (0) B inf K ρ ( M ) u, M = u (0) /C ARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION 5
Notice that condition (1.2) on the powers p i is in fact of parabolic nature, tied to theproof of [16]. The proof of the elliptic Harnack inequality under the more natural condition p max < N ¯ p/ ( N − ¯ p ) is the object of future work. The scale invariance of the Harnack inequality,i.e., the fact that the constants in (1.7) do not depend on the radius or the solution, is crucialwhen dealing with Liouville-type theorems like the one below, proved in a standard way in thelast section. Corollary 1.3.
Under assumption (1.2) , any weak solution of (1.6) in the whole R N boundedfrom below is constant. - Comparison with previous results . Local boundedness of the solutions of (1.1) hasbeen first proved in [29] under the condition p max < ¯ p (1 + 2 /N ). Some early regularity resultsin the plane are considered in [24], and regularity for parabolic problems with non-standardgrowth of p ( x ) type are contained in [1, 3, 36]. The p ( x ) growth condition does not cover thesimple equation (1.1) and we are not aware of proofs of the H¨older continuity of solutions ofthe latter in general dimensions (see [6, Remark 1.4] for a discussion of previous attempts), letalone of the Harnack inequality.In the elliptic setting much more is known regarding the regularity of solutions of (1.6), or formore general non-standard equations, see [26, Sections 5 and 6] for the relevant literature. Themost up-to-date result for (1.6) is in [6], where the Lipschitz regularity of its bounded solutionsis proved for any choice of p i >
1. The Harnack inequality for non-standard elliptic problemshas been the object of various works: [2, 5, 19, 20, 23, 31, 34] focus on isotropic equations withnon-standard growth of p ( x )-type, while [21, 27] deal with energies with Uhlenbeck structureand non-standard growth. However, none of the frameworks considered therein cover theanisotropic equation (1.6): indeed, its Euler-Lagrange equation is degenerate/singular on theunion of the coordinate axes, while non-standard functionals of p ( x )- or Uhlenbeck-type exhibitthis problem only at the origin. Moreover, as already remarked, the relevant feature of (1.7)lies in its scale invariance, while (quite naturally) this is not to be expected for the problemsconsidered in the cited works, where either the constant depends on u and r or there is anadditional term of non-homogeneous type.- Structure of the paper . Section 2 collects preliminary results, most of which are modifi-cations of well-known theorems. The most relevant part is subsection 2.2, where we set up thegeometry related to the natural scaling of the equation. In Section 3 we build the Barenblattsolution and study its positivity set. Section 4 contains the proof of the main theorem, splittedin several lemmas.-
Notations :- For ξ ∈ R and p > ξ ) p − = | ξ | p − ξ .- If E is a measurable subset of R N , we denote by | E | its Lebesgue measure.- For r > K r (¯ x ) stands for the cube {| x i − ¯ x i | r/ i = 1 , . . . , N } and we write K r = K r (0); the standard cylinder is denoted by Q − r = K r × ( − r , | K | = | Q − | = 1.- Given T ∈ (0 , + ∞ ] and Ω ⊂ R N a rectangular domain, we let Ω T = Ω × (0 , T ) while S T denotes the stripe S T = R N × (0 , T ); more generally, for s < t we will set Ω s,t = Ω × ( s, t )and S s,t = R N × ( s, t ).- For a measurable u , by inf u and sup u we understand the essential infimum and supre-mum, respectively; when u is defined on all of R N , we let k u k p = k u k L p ( R N ) for1 p ∞ ; when u : E → R and a ∈ R , we omit the domain when considering S. CIANI, S. MOSCONI, AND V. VESPRI sub/super level sets, letting (cid:2) u R a (cid:3) = (cid:8) x ∈ E : u ( x ) R a (cid:9) ; if u is defined on some Ω T ,we let u t ( x ) = u ( x, t ) while ∂ i u = ∂∂x i u , ∂ t u = ∂∂t u denote the distributional derivatives.2. Preliminaries
In this section we will discuss the functional analytic setting we pose ourselves in, the scalingproperties of the solutions of (1.1), some basic energy estimates and the resulting anisotropic DeGiorgi type lemma, comparison principles for (1.1) and the associated Fokker-Planck equationand solvability of the Cauchy problem for (1.1) for suitable initial data. Most of the materialis standard, except maybe the discussion in section 2.2.2.1.
Functional inequalities.
Given p = ( p , . . . , p N ), with p i > i = 1 , . . . , N and Ω arectangular domain, we define W , p o (Ω) := { v ∈ W , o (Ω) | ∂ i v ∈ L p i (Ω) } ,W , p loc (Ω) := { v ∈ L loc (Ω) | ∂ i v ∈ L p i loc (Ω) } , and for s < t L p ( s, t ; W , p (Ω)) := { v ∈ L ( s, t ; W , (Ω)) | ∂ i v ∈ L p i (Ω s,t ) } ,L p loc ( s, t ; W , p loc (Ω)) := { v ∈ L loc ( s, t ; W , loc (Ω)) | ∂ i v ∈ L p i loc (Ω s,t ) } ,L p loc ( s, t ; W , p o (Ω)) := { v ∈ L loc ( s, t ; W , o (Ω)) | ∂ i v ∈ L p i loc (Ω s,t ) } . A function u ∈ L ∞ loc ( s, t ; L loc (Ω)) ∩ L p loc ( s, t ; W , p loc (Ω))is a local weak solution of (1.1) in ( s, t ) × Ω, if for all s < t < t < t and any ϕ ∈ C ∞ loc ( s, t ; C ∞ o (Ω)) it holds Z Ω u t ϕ t dx − Z Ω u t ϕ t dx + Z t t Z Ω ( − u ∂ t ϕ + N X i =1 ( ∂ i u ) p i − ∂ i ϕ ) dx dt = 0 . By an approximation argument the latter actually holds for any test function ϕ ∈ W , loc ( s, t ; L loc ( R )) ∩ L p loc (0 , T ; W , p o ( R ))for any rectangular domain R ⊂⊂ Ω. By a local weak solution of (1.1) in S ∞ , we mean afunction u ∈ C ( R + ; L loc ( R N )) ∩ L p loc ( R + ; W , p loc ( R N )) such that for each T > ⊂⊂ R N , u is a weak local solution in Ω T . Finally, by an L p solution of (1.1) in S T , we mean a localweak solution u in S T such that u ∈ ∩ Ni =1 L p i ( S T ).Next we recall the following anisotropic embedding, obtained, e.g., from [16, Theorem 2.4] with σ = 2, α i ≡ θ = ¯ p/ ¯ p ∗ and the generalised Young inequality. Lemma 2.1 (Parabolic anisotropic Sobolev embedding) . Let Ω ⊆ R N be a bounded rectangulardomain. There exists a constant C = C ( N, p ) < + ∞ such that for any u ∈ L (0 , T ; W , o (Ω)) it holds (2.1) Z Ω T | u | l dx dt C (cid:18) sup t ∈ [0 ,T ] Z Ω | u | ( x, t ) dx + Z Ω T N X i =1 | ∂ i u | p i dx dt (cid:19) ( N +¯ p ) /N whenever (2.2) 2 NN + 2 ¯ p := (cid:0) N N X i =1 p i (cid:1) − < N, l := ¯ p (cid:0) N (cid:1) . ARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION 7
By applying the so-called
Local Clustering lemma in [11] to min { u, } + δ and letting δ ↓
0, weget the following alternative form, which will be used in the following.
Lemma 2.2 (Local clustering) . Let u ∈ W , ( K ρ ) satisfy for some constants ¯ C > , ¯ α ∈ (0 , Z K ρ | D ( u − − | dx ¯ C ρ N − & | [ u > ∩ K ρ | > ¯ α | K ρ | . Then for every λ, ν ∈ (0 , there exists y ∈ K ρ and a number ε = ε ( λ, ν, ¯ C, ¯ α, N ) ∈ (0 , suchthat y + K ερ ⊆ K ρ and | [ u > λ ] ∩ ( y + K ερ ) | > (1 − ν ) | K ερ | . Moreover, ε can be chosen arbitrarily small. Scaling properties.
We omit the proof of the following proposition, which is just a directcomputation.
Proposition 2.3.
Let u weakly solve (1.1) in Ω T . For M, ρ > define T ρ,θ ( y, s ) = (cid:0) θ ( p i − ¯ p ) /p i ρ ¯ p/p i y i , θ − ¯ p ρ ¯ p s (cid:1) . Then the transformed function (2.3) T ρ,θ u ( y, s ) = 1 θ u (cid:0) T ρ,θ ( y, s ) (cid:1) weakly solves (1.1) in T ρ,θ (Ω) . Due to the latter proposition, it is convenient to set(2.4) σ := N (¯ p −
2) + ¯ p α := Nσ , and α i := N (¯ p − p i ) + ¯ pσ p i , (notice that, under assumption (1.2), α i > i = 1 , . . . , N ), so that T ρ,θρ − N u ( y, s ) = ρ σ α θ u (cid:0) θ ( p i − ¯ p ) /p i ρ σ α i y i , θ − ¯ p ρ σ s (cid:1) . The previous scaling suggests the natural geometry where to settle problem (1.1). More pre-cisely, we define the intrinsic anisotropic cube as K ρ ( θ ) := T ρ,θ ( K ) , K ρ := K ρ (1)(here and in what follows we will use the same symbol T ρ,θ to denote the action of T ρ,θ on thespace variables only) and the intrinsic anisotropic cylinders as Q − ρ ( θ ) := T ρ,θ ( Q − ) , Q − ρ := Q − ρ (1) . Notice that in the anisotropic cubes the parameter ρ prescribes the size, while θ determinesits anisotropic geometry: indeed, the volume of K ρ ( θ ) does not depend on θ , since for each θ, ρ > |K ρ ( θ ) | = ρ N . The following property can be readily checked:(2.5) T ρ,θ ( K R ) = K R ρ ( R θ ) , T ρ,θ ( Q − R ) = Q − R ρ ( R θ ) , and in particular it holds K R ( R ) = K R and Q − R ( R ) = Q − R . An important special case of thetransformation (2.3) is obtained when v does not depend on the time variable and θ = ρ − N :using the notations in (2.4) we define(2.6) T ρ v ( y ) := T ρ,ρ − N v ( y ) = ρ σα v (cid:0) ρ σα i y i (cid:1) . S. CIANI, S. MOSCONI, AND V. VESPRI
By a change of variables one can check that T ρ : L ( R N ) → L ( R N ) is an isometry, andmoreover(2.7) ( T ρ,ρ − N u ) s = T ρ u ρ σ s As we will see, reasonable solutions of (1.1) preserve the L -norm in time, and therefore wewill say that u is a self-similar solution of (1.1) in S ∞ if T ρ,ρ − N u = u for all ρ > u ( y, s ) := e αs u ( e α i s y i , e s ) , and the inverse Ψ w ( x, t ) = t − α w ( t − α i x i , log t ) . Clearly if u is defined on R N × [ t , + ∞ ), t > u is defined on R N × [log t , + ∞ ) andvice-versa if w is defined on R N × [ s , + ∞ ), Ψ w is defined on R N × [ e s , + ∞ ). Due to (2.7), itholds(2.9) (Φ u ) s = T e s/σ u e s , (Ψ w ) t = T t /σ w log t By [8], Φ brings solutions of (1.1) in S t , ∞ , to solutions of the anisotropic Fokker-Planckequation(2.10) ∂ s w = N X i =1 ∂ i (cid:2) ( ∂ i w ) p i − − α i y i w (cid:3) in S log t , ∞ and Ψ does the opposite. Using (2.7), (2.9) together with T ρ T ρ = T ρ ρ , we seethat for a solution u of (1.1) in S ∞ (Φ T ρ,ρ − N u ) s = T e s/σ (cid:0) T ρ,ρ − N u (cid:1) e s = T e s/σ T ρ u ρ σ e s = T ρe s/σ u ρ σ e s = (Φ u ) σ log ρ + s for every ρ >
0, from which we readily infer the following proposition.
Proposition 2.4.
A function u is a self-similar solution of (1.1) in S ∞ if and only if Φ u isa stationary solution of (2.10) in R N +1 . In the following we will call a self-similar solution to (1.1) in S ∞ a Barenblatt fundamentalsolution , and we will denote it with B , in analogy with the literature about the p -Laplacian.Moreover, we will henceforth use coordinates ( x, t ) for the prototype equation (1.1) and ( y, s )for the Fokker-Planck equation (2.10).2.3. Energy inequality and consequences.
The following energy estimate for solutions of(1.1), is well known, see e.g. [16, Lemma] for a proof.
Lemma 2.5 (Energy inequality) . Let u be a local weak solution of (1.1) in K R × [ s , s ] . Then,for each test function of the form η ( x, t ) = N Y i =1 η p i i ( x i , t ) , η i ∈ C ∞ ( s , s ; C ∞ c ( − R/ , R/ we have, for some C = C ( N, p ) > , (2.11) Z K R ( u t − k ) ± η t dx (cid:12)(cid:12)(cid:12)(cid:12) t = s t = s + 1 C N X i =1 Z s s Z K R | ∂ i ( η ( u − k ) ± ) | p i dx dτ Z s s Z K R ( u − k ) ± | ∂ t η | dx dτ + C N X i =1 Z s s Z K R | ( u − k ) ± | p i | ∂ i η | p i dx dτ. In a standard way we can prove a de Giorgi-type Lemma.
ARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION 9
Lemma 2.6 (De Giorgi Lemma) . Let u > be a local weak solution to (1.1) in Q − and p obey (2.2) . Then for every a ∈ (0 , there exist µ a > depending on a, p and N such that | [ u a ] ∩ Q − | µ a | Q − | ⇒ inf Q − / u > a. If, in addition, it holds u in Q − , | [ u > a ] ∩ Q − | µ a | Q − | ⇒ sup Q − / u a. Proof.
We give a brief proof of the second statement, the first one being analogous. Let, for n ∈ N , ρ n = (cid:18)
12 + 12 n +1 (cid:19) , k n = a (cid:18) − n +1 (cid:19) , K n = K ρ n Q − n = Q − ρ n . We apply (2.11) to ( u − k n ) + with η n of the stipulated form with η n = 1 in Q n +1 , η n = 0outside Q n and 0 η n , | ∂ t η n | + |∇ η n | C n . Since η n ( · , − ≡
0, the energy inequality (2.11) together with the bound | u | Z K n ( u t − k n ) ( η n ) t dx + 1 C N X i =1 Z Q − n | ∂ i ( η n ( u − k n ) + ) | p i dx dτ C c n (cid:26) Z Q − n ( u − k n ) dxdt + N X i =1 Z Q − n | ( u − k n ) + | p i dx dτ (cid:27) C c n | [ u > k n ] ∩ Q − n | (2.12)for all t ∈ [ − ρ n , C = C ( N, p , a ) and c = c ( p ). Let A n = [ u > k n ] ∩ Q − n . ByChebyshev’s inequality and the assumptions on η n it holds, for l given in (2.2) (cid:0) a n +1 (cid:1) l | A n +1 | = ( k n − k n +1 ) l | A n +1 | Z Q − n +1 | ( u − k n ) + | l dx dτ Z Q − n | ( u − k n ) + η n | l dx dτ and chaining Sobolev’s embedding (2.1) on the right and (2.12) (notice that η n η n ), we get a l n l | A n +1 | C (cid:18) sup t ∈ ( − ρ n , Z K n ( u t − k n ) ( η n ) t dx + N X i =1 Z Q − n | ∂ i ( η n ( u − k n ) + ) | p i dx dτ (cid:19) N +¯ pN C c n | A n | p/N , for some bigger C , c . By the fast convergence Lemma [14, Lemma 5.1 chap. 2], if | A | issufficiently small (depending on N, p and a ), | A n | → n → ∞ , implying the claim. (cid:3) Remark 2.7.
Applying the transformation T ρ,θ and recalling (2.5), we get a similar statementfor any solution in Q ρ ( θ ) − . For example, if u > Q − ρ ( θ ), for every a ∈ (0 , ν a = ν ( a, N, p ) > | [ u a θ ] ∩ Q − ρ ( θ ) | ν a |Q − ρ ( θ ) | ⇒ inf Q − ρ/ ( θ/ u > a θ. Comparison principles.
We consider in this section the Cauchy problem for (1.1),namely(2.13) ( ∂ t u = P Ni =1 ∂ i (( ∂ i u ) p i − ) weakly in Ω T ,u t → u strongly in L (Ω)and a similar one for the Fokker-Planck equation (2.10). Given two solutions u, v of thisproblem, we say that u > v on the parabolic boundary of Ω T if ( u − v ) − ∈ L p (0 , T ; W , p (Ω))and u > v . From the monotonicity of the principal part in (1.1) we have the followingclassical comparison principle. Proposition 2.8. (Local comparison principle) Let Ω be a bounded rectangular domain, u, v be weak solutions of (2.13) in Ω T . If u > v on the parabolic boundary of Ω T , then u > v in Ω T . Next we provide a comparison principle for the class of L p -solutions, that will prove to beuseful for next purposes. We sketch the proof inasmuch the rˆole of greater integrability can befully exploited. Proposition 2.9.
Let u, v be two L p solutions of (2.13) in S T , satisfying u > v for u , v ∈ L ( R N ) . Then u > v in S T .Proof. First notice that if u is an L p solution of (2.13) in S T with u ∈ L ( R N ), then u ∈ L p (0 , T ; W , p (Ω)). Indeed, by the energy estimate (2.11) with a standard cut-off, we deducethat N X i =1 k ∂ i u k L pi ( S T ) C (cid:0) k u k + N X i =1 k u k L pi ( S T ) (cid:1) , and similarly for v . Secondly, we test the equations for u and v with ( v − u ) + ζ , where ζ acut-off function between the balls B R and B R , independent of time and such that | ∂ i ζ | C/R ,0 ζ
1. Subtracting the resulting integral equalities and using u > v we have, for every t > Z B R ∩ [ v > u ] ζ ( v − u ) ( x, t ) dx + N X i =1 Z t Z B R ∩ [ v > u ] ζ (cid:0) ( ∂ i v ) p i − − ( ∂ i u ) p i − (cid:1) ∂ i ( v − u ) dx dτ = − N X i =1 Z t Z B R ∩ [ v > u ] ( v − u ) (cid:0) ( ∂ i v ) p i − − ( ∂ i u ) p i − (cid:1) ∂ i ζ dx dτ N X i =1 CR Z t Z B R ∩ [ v > u ] ( | ∂ i v | p i − | u | + | ∂ i v | p i − | v | + | ∂ i u | p i − | v | + | ∂ i u | p i − | u | ) dx dτ CR N X i =1 k ∂ i v k p i L pi ( S T ) + k v k p i L pi ( S T ) + k ∂ i u k p i L pi ( S T ) + k u k p i L pi ( S T ) by Young’s inequality. By the initial argument and the assumptions, the sum on the rightis finite, while by the monotonicity of the operator both terms on the left are non-negative.Hence for any t < T the left hand side vanishes for R → + ∞ , giving the claim. (cid:3) As a corollary, we have the following comparison principle for solutions to the Fokker-Planckequation.
ARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION 11
Corollary 2.10.
Let v, w be L p -solutions of the Cauchy problem for the Fokker-Planck equation (2.10) satisfying w > v and w , v ∈ L ( R N ) . Then w > v in S T .Proof. It suffices to recall that the law (2.8) establishes an isomorphism between L p ( S ,e T )-solutions of the prototype equation (1.1) and L p ( S T ) solutions to the Fokker-Planck equation(2.10) and the initial data coincide. (cid:3) Remark 2.11.
It is worth noting that, while an elliptic comparison principle holds true as wellfor the stationary counterpart of (1.1), this is no longer the case for the stationary counterpartof the Fokker-Planck equation (2.10). This can be seen considering (already in the isotropiccase), the Barenblatt solutions of the p -Laplacian equation, which solve the stationary Fokker-Planck equation and contradict the elliptic comparison principle for (2.10).2.5. L p -solutions. We next consider the Cauchy problem for (1.1), with bounded and a com-pactly supported initial datum, attained in L . This problem can be read as(2.14) ( ∂ t u = P Ni =1 ∂ i (cid:0) ( ∂ i u ) p i − (cid:1) in S T ,u = g ∈ L ( R N ) supp g ⊂ ¯ B R , g ∈ L ∞ ( B R ) . We show in this section that this problem has a unique L p -solution, by a standard approxima-tion technique relying on the monotonicity of the operator. Proposition 2.12.
Problem (2.14) has a unique L p -solution which takes the initial datum g in L .Proof. We let, for n > diam(supp w ), B n = {| x | < n } and consider the boundary valueproblems(2.15) v n ∈ C ([0 , T ); L ( B n )) ∩ L p (0 , T ; W , p ( B n )) ∂ t v n − P Ni =1 ∂ i (( ∂ i v n ) p i − ) = 0 , in B n × (0 , T ) ,v n ( · ,
0) = g. We regard them as defined in the whole S T by extending them to be zero on | x | > n . Theproblems (2.15) can be uniquely solved by a monotonicity method (see for instance [22, Example1.7.1]), and give solutions v n satisfying(2.16) sup t ∈ [0 ,T ] Z R N | v n ( x, t ) | dx + 2 N X i =1 Z S T | ∂ i v n | p i dx dt = k g k , ∀ n ∈ N , and thus v n ∈ L ∞ ((0 , T ) , L ( R N )) uniformly. Notice that by the local comparison principle inΩ T , k v n k ∞ k g k ∞ hence by dominated convergence v n ( · , t ) → g in L ( R N ) implies v n ( · , t ) → g in L p i ( R N ) as t →
0, for i = 1 , . . . , N . In the weak formulation of (2.15) we take (modulo aStekelov averaging process) the testing function ( v n ) p j − , j = 1 , . . . , N , obtaining ∀ t ∈ (0 , T )(2.17) Z R N | v n | p j p j ( x, t ) dx + ( p j − N X i =1 Z S t | ∂ i v n | p i | v n | p j − dx dτ = Z R N | g | p j p j dx, implying v n ∈ ∩ Ni =1 L ∞ ((0 , T ); L p i ( R N )) , uniformly in n. This estimate, together with (2.16), provides an uniform bound for v n in N \ i =1 L p i (0 , T ; W ,p i ( R N )) ∩ L ∞ (0 , T ; L ( R N )) ∩ L ( S T ) . A subsequence can be selected weakly ∗ converging in these spaces, an such that( ∂ i v n ) p i − → η i , weakly in L p ′ i (0 , T ; W − ,p ′ i loc ( R N )) . and we can pass to the limit in the weak formulation of the equation, identifying η i = ( ∂ i v ) p i − through Minty’s trick. Semicontinuity and (2.17) imply that v is an L p solution.In order to prove uniqueness, let u , u be two L p -solutions of (2.14). By the first step ofthe proof of Proposition 2.9, both belong to L p (0 , T ; W , p ( R N )), thus v := u − u satisfies v ∈ C ([0 , T ); L ( R N )) ∩ L p (0 , T ; W , p ( R N )) ,∂ t v − P Ni =1 ∂ i (cid:0) ( ∂ i u ) p i − − ( ∂ i u ) p i − (cid:1) = 0 , in S T ,v = 0 . Test the latter with v ζ , where ζ ∈ C ∞ c ( B R ), ζ > ζ = 1 in B R and | Dζ | C/R . For all0 < t T we have12 Z B R | v t | dx + Z t Z B R N X i =1 (cid:0) ( ∂ i u ) p i − − ( ∂ i u ) p i − (cid:1) ( ∂ i u − ∂ i u (cid:1) ζ dxdτ = − Z t Z B R N X i =1 v (cid:0) ( ∂ i u ) p i − − ( ∂ i u ) p i − (cid:1) ∂ i ζ dxdτ Using the monotonicity of the principal part on the left-hand side and H¨older’s inequality onthe right, for every t ∈ (0 , T ) Z B R | v t | dx CR N X i =1 k v k L pi ( S T ) ( k ∂ i u k L pi ( S T ) + k ∂ i u k L pi ( S T ) ) → , as R → ∞ . (cid:3) Barenblatt fundamental solutions
In this section we will build a self similar solution to (1.1), i.e., by the discussion in section2.2, a stationary solution to the Fokker-Planck equation (2.10). We will then study the posi-tivity properties of such fundamental solution, which, together with the comparison principle,will be the main tool to expand the positivity set of non-negative solutions of (1.1).By the results of section 2.5 we can define, at least for bounded compactly supported initialdata g , the operator S t g := u t , t > , where u is the unique L p solution of(3.1) ( ∂ t u = P Ni =1 ∂ i (cid:0) ( ∂ i u ) p i − (cid:1) in S , ∞ ,u = g. In terms of the Fokker-Planck equation, this also defines through (2.8) the operator(3.2) ˜ S s g := (Φ u ) s s > , giving the solution at the time s ∈ R + , of the problem(3.3) ( ∂ s w = P Ni =1 ∂ i [( ∂ i w ) p i − − α i y i w ] in S ∞ ,w = g. ARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION 13
The relation (3.2) implies that(3.4) ˜ S s g = T e s/σ S e s g, where T is given in (2.6), allowing to prove properties for ˜ S s by proving them for S t .3.1. Construction of a Barenblatt solution.
In order to state some basic properties of theoperator ˜ S s we will need the following space:(3.5) X R,M = { g ∈ L ∞ ( R N ) : 0 g M, supp g ⊆ K R } , X = [ R,M> X R . Lemma 3.1. If (1.2) holds true, the operator ˜ S s , s > defined in (3.2) has the followingproperties.(1) If g ∈ L ( R N ) and supp g ⊆ K R then for some c = c ( N, p ) it holds (3.6) supp ˜ S s g ⊆ N Y i =1 [ − R i ( s ) , R i ( s )] , R i ( s ) = 2 e − s α i R + c k g k ¯ p ( p i − / ( p i σ )1 . (2) If g ∈ X , then k ˜ S s g k = k g k and ˜ S s g k g k ∞ . In particular ˜ S s : X → X for all s > .(3) For any R, M > and s > , ˜ S s : X R,M → X is continuous when X R,M and X areequipped with the weak- L topology.Proof. Consider the corresponding problem (3.1) and the therein defined operator S t . By [16,Theorem 1.1] (notice that the branch obtained there is an L p solution and therefore coincideswith S t g by uniqueness) we know that if supp g ⊆ K R , then(3.7) supp S t g ⊆ N Y i =1 [ − R i ( t ) , R i ( t )] , R i ( t ) = 2 R + c ( t − α i k g k ¯ p ( p i − / ( p i σ )1 . Letting t = e s and using (3.4) we get the first assertion, sincesupp ˜ S s g ⊆ N Y i =1 [ − ˜ R i ( s ) , ˜ R i ( s )] , ˜ R i ( s ) = e − s α i R i ( e s ) e − sα i R + c k g k ¯ p ( p i − / ( p i σ )1 . The second statement follows from its counterpart on the corresponding solution u of (3.1):to prove conservation of mass we take advantage of the compactness of the supports of u dictated by (3.7) and test (3.1) with ϕ ∈ C ∞ c ( R N ) such that ϕ ≡ ∪ t
1. The boundedness of k g n k and standard energy estimates then give a uniform bound in L p (1 , T ; W , p o ( K ¯ R )) ∩ L ∞ (1 , T ; L ( R N )), and therefore we can suppose, up to a not relabelledsubsequence, that u n converges weakly ∗ to some u in these spaces and ( ∂ i u n ) p i − ⇀ η i in L p ′ i (1 , T ; W − ,p ′ i ( R N )). Thanks to these convergences we can pass to the limit in the weakform of the equation to get Z R N u τ ϕ τ dx − Z R N g ϕ dx − Z S ,τ u ∂ τ ϕ dx dt + Z S ,τ N X i =1 η i ∂ i ϕ dx dt = 0for all 1 < τ < T , so that it only remains to show that η i = ( ∂ i u ) p i − . We cannot directlyemploy Minty’s trick, since we are missing the strong convergence of the initial data. However,the previous estimates imply that ∂ t u n is uniformly bounded in ∩ Ni =1 L p ′ i (1 , T ; W − ,p ′ i ( K ¯ R )) andAubin-Lions theorem [33, Proposition 1.3], applied to the triple N \ i =1 W ,p i o ( K ¯ R ) ֒ → L ( K ¯ R ) → N \ i =1 W − ,p ′ i ( K ¯ R ) , i = 1 , . . . , N, provides with a subsequence (not relabelled) such that u n → u strongly in L (1 , T ; L ( K ¯ R )).Consequently, up to not relabelled subsequences, for almost every time τ ∈ [1 , T ] it holds( u n ) τ → u τ in L ( K ¯ R ). For any such τ , we look at { u n } as a sequence of solution to (1.1)on [ τ, T ] with strongly convergent initial data, hence we can use the Minty’s trick to deduce η i = ( ∂ i u ) p − on S τ,T . Since τ can be chosen arbitrarily close to 1 we obtain that u is a L p solution to (3.1) with initial datum g and from uniqueness we infer that u t = S t g for any t > (cid:3) Theorem 3.2.
Under assumption (1.2) , there exists a nontrivial stationary solution w ∈ X , to (3.3) , and therefore a Barenblatt Fundamental solution.Proof. For R , M > C ε := (cid:8) g ∈ L ( R N ) : supp g ⊂ B , g , || g || L ( R N ) = ε (cid:9) ⊆ X , . If c is given in (3.6), for ¯ s sufficiently large and ¯ ε sufficiently small it holds2 e − ¯ sα i + c ¯ ε ¯ p ( p i − / ( p i σ ) ∀ i = 1 , . . . , N, so that (3.6) implies that supp ˜ S ¯ s g ⊆ B for all g ∈ C ¯ ε . Using also point (2) of the previouslemma we have that ˜ S ¯ s C ¯ ε ⊆ C ¯ ε . Moreover, C ¯ ε with the weak L topology is compact, andby point (3) of the previous lemma, ˜ S ¯ s : C ¯ ε → C ¯ ε is continuous, so that Schauder’s theoremensures the existence of a fixed point ¯ g ∈ C ¯ ε for ˜ S ¯ s . Therefore the function ¯ w s = ˜ S s ¯ g is a times-periodic, bounded and compactly supported solution of (3.3), which can therefore be extendedto R N +1 as an aeternal solution. Consider the bounded, compactly supported function g ( y ) = sup s ∈ R ¯ w ( s, y ) , g ∈ X , , for which k g k > ¯ ε . Then ˜ S g = g > ¯ w τ for every τ ∈ R , so that by the comparison principle2.10 it holds ˜ S s g > ¯ w τ + s for any s >
0. Taking the supremum in τ ∈ R gives ˜ S s g > g , butsince k ˜ S s g k = k g k , this implies ˜ S s g = g for every s >
0, i.e., g is a stationary solution of(3.3). (cid:3) Properties of the Barenblatt solutions.
Our next aim is to prove that Barenblattsolution are positive in a quantitative way, i.e., their positivity set spreads in time in a waycontrolled by scaling. This amounts in proving that stationary non-negative solutions of theFokker Planck equation are bounded from below near the origin, which is the content of thenext theorem.
ARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION 15
Theorem 3.3.
Suppose (1.2) holds, let w ∈ X , (see (3.5) ) be a nontrivial stationary solutionof the Fokker-Planck equation (2.10) and B the corresponding Barenblatt solution of (1.1) .Then there exists ¯ η > , depending on w and the data, such that B ( x, t ) > ¯ η/t α if | x i | < ¯ η t α i for i = 1 , . . . , N . Proof.
Suppose that B is given by(3.8) B ( x, t ) = t − α w ( x i t − α i ) , t > . By [16, Corollary 4.3] we can fix a lower-semicontinuous representative of B and thus of w .Since w > x (0) and numbers δ , η > K δ ( x (0) ) w ( y ) > η . By (3.8), the latter implies for any t > B ( x, t ) > η t − α , when (cid:8) | x i − x (0) i t α i | < δ t α i (cid:9) . Consider now B λ ( x, t ) = λ t − α w (cid:0) λ (2 − p i ) /p i t − α i ( x (0) i − x i ) (cid:1) , which solves (1.1) by translation invariance and Proposition 2.3. Notice that, since w ∈ X , kB λ ( · , t ) k ∞ = λ t − α supp B λ ( · , t ) ⊆ (cid:8) | x (0) i − x i | t α i λ ( p i − /p i (cid:9) . We seek for λ > B λ and B withstarting time t = 1. We need ( kB λ ( · , k ∞ η , supp B λ ( · , ⊆ K δ ( x (0) ) , ⇐⇒ ( λ η ,λ ( p i − /p i δ / , which, being p i > i , can be solved for some λ = λ ∈ (0 , B ( x, t ) > B λ ( x, t ) > λ t − α η , for (cid:12)(cid:12) x (0) i − λ (2 − p i ) /p i t − α i ( x (0) i − x i ) (cid:12)(cid:12) < δ . We let t α = λ (2 − p ) /p > η = λ t − α η , x (1) i := x (0) i (cid:0) − t α i λ ( p i − /p i (cid:1) , δ := δ (cid:8) t α i λ ( p i − /p i : 1 i N (cid:9) (notice that, by the choice of t , it holds x (1)1 = 0), to getinf K δ ( x (1) ) B ( · , t ) > η Proceeding by induction, we will find sequences t n , η n , δ n , x ( n ) with the propertiesinf K δn ( x ( n )) B ( · , t n ) > η n , x ( n ) i = 0 for i = 1 , . . . , n so that after N steps we find inf K δN B ( · , t N ) > η N . By (3.8), this implies w ( x ) > η N t αN when | x i | < ( δ N / t α i N for i = 1 , . . . , N . We set ¯ η =min { η N , ( δ N / } and scale back to B through (3.8) again, to get the claim. (cid:3) We will from now suppose that w is a fixed stationary solution in X , of (3.3). For futurepurposes we summarise some properties derived from a scaling argument for a large family ofcorresponding Barenblatt solutions. Corollary 3.4.
Let B ( x, t ) = t − α w ( x i t − α i ) be a fixed Barenblatt Fundamental solution to (1.1) with w ∈ X , . There exists ¯ η > such that the family of Barenblatt solutions B λ ( x, t ) = T ,λ − σ/ ¯ p B ( x, t ) = λ t − α w ( λ (2 − p i ) /p i x i t − α i ) , λ > has the following properties:(1) k B λ ( · , t ) k ∞ = λ t − α ;(2) supp B λ ( · , t ) ⊆ N Y i =1 (cid:8) | x i | λ ( p i − /p i t α i (cid:9) ;(3) { B λ ( · , t ) > λ ¯ η t − α } ⊇ N Y i =1 (cid:8) | x i | ¯ η λ ( p i − /p i t α i (cid:9) . Proof of Theorem 1.1
We first consider a generalisation of what is called in literature the Krylov-Safonov argument.Unfortunately the standard proof does not appear to be easily generalized in the generalanisotropic setting we are in. However we make the following observations: the cylinders Q − ρ define a pseudo-metric by(4.1) d(( x, t ) , ( y, s )) = max {| x i − y i | ¯ p/p i , | t − s | ¯ p } , namely, it holds d( z , z ) γ (d( z , z ) + d( z , z )) , ∀ z , z , z ∈ R N +1 , for a constant γ = γ ( N, p ) > pseudo-metric constant . Notice that the cylinder¯ z + Q − ρ the bottom half part of the ball B ρ (¯ z ) with respect to this distance. Lemma 4.1.
Let ( X, d) be a pseudo-metric space with pseudo-metric constant γ and x ∈ X .For any β > there exists a constant ω = ω ( γ, β ) > such that for any bounded u : B ( x ) → R with u ( x ) > there exist x ∈ B ( x ) and r > such that (4.2) B r ( x ) ⊆ B ( x ) , r β sup B r ( x ) u ω, r β u ( x ) > /ω. Proof.
Extend u as 0 outside B ( x ) and suppose that the claim is false. For ω a parameter tobe determined depending only on β and γ , we will construct a sequence of points contradictingthe boundedness of u . Set r = 1 / (2 γ ) and choose ω > (2 γ ) β . Since r β u ( x ) > /ω , it musthold r β sup B r ( x ) u > ω. Choose x ∈ B r ( x ) such that r β u ( x ) > ω and set r = r ω − /β , so that r β u ( x ) > /ω. If B r ( x ) ⊆ B ( x ), we can similarly construct x ∈ B r ( x ) such that r β u ( x ) > /ω, r = r ω − /β . Proceed by induction to get a sequence of points and radii such that, if B r n ( x n ) ⊆ B ( x ), r βn u ( x n ) > /ω, r n = r n − ω − /β . ARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION 17 As ω >
1, the first condition contradicts the boundedness of u if all the balls B r n ( x n ) arecontained in B ( x ). This can be achieved if for any n > x , x n ) γ n − X i =0 γ i d( x i , x i +1 ) γ r ∞ X i =0 γ i ω − i/β < , which holds for γ ω − β < / (cid:3) Lemma 4.2.
Let u > solve (1.1) in Q − , and suppose that for some ¯ ν ∈ (0 , a > it holds (4.3) | [ u > a ] ∩ Q − | > (1 − ¯ ν ) | Q − | . Then for every choice of λ, ν ∈ (0 , there exist ¯ y ∈ K , ¯ t ∈ ( − , − ¯ ν/ and ε ∈ (0 , determined only by means of N, p , ν, ¯ ν, a and λ , such that ¯ y + K ε ⊂ K and | [ u ¯ t > λ a ] ∩ (¯ y + K ε ) | > (1 − ν ) | K ε | . Proof.
Choose r = r (¯ ν ) > / | [ u > a ] ∩ Q r | > | Q r | (1 − ¯ ν ) /
2. Wewrite down the energy estimates (2.11) for ( u − a ) − with η of the form prescribed therein, η > η = 1 on Q − r , η = 0 outside Q − and | ∂ t η | + | ∂ i η i | C (¯ ν ), to get, thanks to sup Q − ( u − a ) − a , N X i =1 Z Q − r | ∂ i ( u − a ) − | p i dx dt C (¯ ν, a ) . By the same argument of [12, Lemma 9.1], there exists a time level ¯ t ∈ ( − , − ¯ ν/
4] such that Z K r (cid:12)(cid:12)(cid:12)(cid:12) ∂ i (cid:18) u ¯ t a − (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) p i dx C (¯ ν, a ) r N − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) u ¯ t a > (cid:21) ∩ K r (cid:12)(cid:12)(cid:12)(cid:12) > (1 − ¯ ν ) | K r | / . By H¨older’s inequality, u ¯ t /a fullfills the assumptions of Lemma 2.2 in K r , giving the claim. (cid:3) It is worth underlining that the parameter ε in the previous statement can be made arbitrarilysmall by eventually changing the point. In the next Lemma, we suppose that an essentiallyupper semicontinuous representative for the solution has been chosen, through [16, Corollary4.3]. Lemma 4.3.
Let u > be a bounded solution of (1.1) in Q − . There exist C, δ, ε > dependingon N and p such that if u (0 , > C , (4.4) inf ¯ x + K ρ ( ερ − N ) u ¯ t > ε ρ − N for some (¯ x, ¯ t ) ∈ Q − and ρ > with ¯ x + K ρ ( ε ρ − N ) ⊆ K . Proof.
Let C = 1 /ω , where ω = ω ( N, p ) is given in Lemma 4.1 with β = N , (using the quasi-metric in (4.1)). We apply the lemma to u/C and extend u as 0 in the upper half-space. Then,(4.2) implies the existence of a point z ∈ Q − and r ∈ (0 ,
1) such that z + Q − r ⊆ Q − , r N sup z + Q − r u , r N u ( z ) > C . The solution v = T r,r − N u ( · + z ) in Q − , (with T given in (2.3)) obeys(4.5) sup Q − v , v (0) > C . We prove that (4.3) holds for ¯ ν = µ a given in Lemma 2.6 when a = C / ν depends onlyon N and p ). Indeed, if, by contradiction, we have | [ v > a ] ∩ Q − | µ a | Q − | , then since 0 v Q , Lemma 2.6 gives v (0) sup Q − / v a C , contradicting the last condition in (4.5). Therefore the thesis of Lemma 4.2 holds true for any ν, λ to be chosen and for the corresponding point z = (¯ x, ¯ t ) ∈ Q − , the following measureestimate holds at the time ¯ t | [ v ¯ t λa ] ∩ (¯ x + K ε ) | ν | K ε | , ¯ t ∈ ( − , − µ a / . Recall that for any ν , ε can be chosen arbitrarily small, so we can also suppose ν < µ a , ε − µ a > , ε − a > . We choose λ = 1 / w = T ε/ ,ε/ v ( · + z ). Since v solves (1.1) in Q − and by (2.5) it holds K ε = K ε ( ε ) = T ε/ ,ε/ ( K ), w solves (1.1) in K × (0 , ε − µ a ] and itsatisfies(4.6) | [ w ( · , ε − a ] ∩ K | ν. We propagate forward in time the information in (4.6) as follows. Write down the energyinequality for ( w − − in the subcylinder K × (0 , ν ] with 1 > η > η = 1 in K , η = 0 outside of K and | ∂ i η | C to get for all t ∈ (0 , ν ] Z K ( w t − − dx Z K ( w − − dx + C N X i =1 Z t Z K ( w τ − p i − dx dτ. By choice of the subcylinder the second term on the right is bounded
C ν , while the first termis smaller than ν due to ε − a > | [ w t ∩ K | ,hence | [ w t ∩ K | C ν ∀ t ∈ (0 , ν ] , which implies by integration | [ w ∩ Q | C ν | Q | , Q := K × (0 , ν ] . Suppose ν = 2 − n ¯ p for some n ∈ N . We partition K in 2 Nn dyadic cubes x i + K − n and considerthe corresponding intrinsically scaled cylinders Q i = ( x i + K − n ) × (0 , ν ]. On at least one ofthese cylinders it holds | [ w ∩ Q i | C ν |Q i | , and since Q i = ( x i ,
0) + T − n , Q − , we can apply Lemma 2.6 (see Remark 2.7) choosing ν suchthat Cν µ , (determining n and ε in the process, depending only on N and p ). This implies w > / x i ,
0) + T − n , Q − / and in particular w > / z + K δ for some z and δ depending on N , p alone. Rescaling back to u we get for some z ∈ Q − u > ε r − N / z + K δ εr/ ( ε r − N / , concluding the proof by suitably redefining the constants. (cid:3) We can now prove the Harnack inequality (1.3).
Proof of Theorem 1.1 . We can safely suppose that u (0 , > u (0 , C inf K ρ ( M ) u ( · , D M − ¯ p ρ ¯ p ) , M = u (0 , /C. ARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION 19
We will prove a weaker version of the latter, namely, that there exists constants,
A, B, C, D > M = u (0 , /C , then(4.8) inf K r ( M ) u ( · , D M − ¯ p r ¯ p ) > u (0 , /B if K Ar ( M ) × [ − M − ¯ p ( A r ) ¯ p , M − ¯ p ( A r ) ¯ p ] ⊆ Ω T . The parameter C is the one given in Lemma 4.3, while A, B, D will be chosen through conditionsof the form A > ¯ A ( N, p ) , B > ¯ B ( N, p ), D > ¯ D ( N, p ), so that taking the largest one willgive (4.7). More precisely, we will first choose D universally large, then choose A and B beaccordingly large.By assumption the function v = T r,M u ( T given in (2.3)) solves the equation in Q A := K A × [ − A ¯ p , A ¯ p ] and v (0 ,
0) = C . Then (4.4) holds, namely there exists (¯ x, ¯ t ) ∈ Q − such thatinf ¯ x + K ρ ( ε ρ − N ) v ¯ t > ε ρ − N for (¯ x, ¯ t ) + K ρ ( ε ρ − N ) ⊆ K − . We choose λ > s < − x, s ) defined as b λ,s ( x, t ) = B λ (¯ x + x, t − s )is a lower barrier for v in K A × [¯ t, A ¯ p ]. In order to have b λ,s ( · , ¯ t ) v ( · , ¯ t ) on K A it must hold ( supp b λ,s ( · , ¯ t ) ⊆ ¯ x + K ρ ( ε ρ − N ) , k b λ,s ( · , ¯ t ) k ∞ ε ρ − N , By Corollary 3.4, this amounts to ( λ ( p i − /p i (¯ t − s ) α i ( ε ρ − N ) ( p i − ¯ p ) /p i ρ ¯ p/p i = ǫ ( p i − ¯ p ) /p i ρ σα i ,λ (¯ t − s ) − α ε ρ − N , which holds true for s = ¯ s obeying | ¯ t − ¯ s | = ρ σ <
1, and ¯ λ = λ ( ε, N, ¯ p ) sufficiently small. Since − < ¯ s < −
1, by Corollary 3.4 it holds b ¯ λ, ¯ s ( x, t ) > ¯ λ ¯ η ( t − ¯ s ) − α > ¯ θ ( t + 1) − α for some ¯ θ = ¯ θ ( N, p ) and all t > , x ∈ N Y i =1 {| ¯ x i − x i | < ¯ η ¯ λ ( p i − /p i ( t − ¯ s ) α i } ⊇ P t (¯ x ) := N Y i =1 {| ¯ x i − x i | < ¯ η ¯ λ ( p i − /p i t α i } . We then choose ¯ τ sufficiently large so that P ¯ τ (¯ x ) ⊇ K and, consequently, A such that A ¯ p > ¯ τ and supp b ¯ λ, ¯ s ( · , t ) ⊆ K A for all t ∈ (¯ t, ¯ τ ). Since ¯ x ∈ K and the support of b λ,s ( · , t ) is decreasingin s and increasing in t , both choices can be made depending only on the parameters, henceultimately on N , p alone. The comparison principle then yields v ( · , ¯ τ ) > b ¯ λ, ¯ s ( · , ¯ τ ) > ¯ θ (¯ τ + 1) − α in K . Defining D = ¯ τ , B = C (¯ τ + 1) α / ¯ θ and rescaling back gives (4.8). It is readily verified that theparameters A, B, D such that (4.8) holds are all chosen through conditions prescribing thatthey are all sufficiently large, depending only on N and p .We next deal with the other inequality in (1.3), sketching its proof as some arguments areidentical to the previous ones (see also [9] for a different approach). We consider the inequality(4.9) sup K r ( M ) u ( · , − D M − ¯ p r ¯ p ) B u (0 ,
0) if K Ar ( M ) × [ − M − ¯ p ( A r ) ¯ p , M − ¯ p ( A r ) ¯ p ] ⊆ Ω T for M = u (0 , /C ( C given in the previous step) and suitable choices of A, B, D > D , then determine A accordingly, and B so that B D − γ > K r ( M ) u ( · , − D M − ¯ p r ¯ p ) > B u (0 , . Choose γ > N/ ¯ p and rewrite the latter in terms of v = T r,MD γ u : the resulting information is(4.10) sup K ( D − γ ) v ( · , − D γ (¯ p − ) > B v (0 , , v (0 ,
0) =
C D − γ . We fix a point x ∈ K ( D − γ ) such that v ( x , − D γ (¯ p − ) > B v (0 ,
0) =
B C D − γ and suppose that A is so large that v is a solution in ( x , − D γ (¯ p − ) + Q − . As long as B D − γ > x ∈ x + K , ¯ t ∈ [ − D γ (¯ p − − , − D γ (¯ p − ] , ¯ s ∈ [ − D γ (¯ p − − , − D γ (¯ p − − λ ( N, p ) > b ¯ λ, ¯ s centered at (¯ x, ¯ s ) is below v at the time¯ t . As before, there exists ¯ θ = ¯ θ ( N, p ) such that for any t > − D γ (¯ p − , it holds b ¯ λ, ¯ s ( · , t ) > ¯ θ ( t + D γ (¯ p − + 1) − α in P t + D γ (¯ p − (¯ x ) . We suppose that D is so large that 0 ∈ P D γ (¯ p − (¯ x ): recalling that x ∈ K ( D − γ ) and¯ x ∈ x + K , this can be done if1 + D − γ ( p i − ¯ p ) /p i D (1+ γ (¯ p − α i , ∀ i = 1 , . . . , N, i.e., for sufficiently large D , as long as the exponent of D on the left is less than the one on theright (which is positive). Recalling the definition of α i = ( N (¯ p − p i ) + ¯ p ) / ( N (¯ p −
2) + ¯ p ) p i ,through elementary algebraic manipulations, this amounts to γ ¯ p ( p i − > N (¯ p − p i ) + ¯ p which certainly holds for sufficiently large γ = γ ( N, p ), thanks to p i > i = 1 , . . . , N .If needed, we further increase A so that the comparison principle between v and b ¯ λ, ¯ s can beapplied in the time interval [¯ t, v (0 , > b ¯ λ, ¯ s (0 , > ¯ θ ( D γ (¯ p − + 1) − α and this contradicts the second condition in (4.10), as long as C D − γ < ¯ θ ( D γ (¯ p − + 1) − α . This holds for sufficiently large D if the exponent of D on the right is greater than the one onthe left, which, recalling that α = N/ ( N (¯ p −
2) + ¯ p ), amounts to γ − α (1 + γ (¯ p − γ ¯ p − NN (¯ p −
2) + ¯ p > . Since γ > N/ ¯ p the latter is true, giving the seeked contradiction. This concludes the proof of(4.9) and of the theorem, by choosing the largest between the present and previous parameters A, B, D . (cid:3) ARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION 21
Finally, we prove the Liouville theorem stated in the Introduction.
Proof of Corollary 1.3 . We suppose that sup R N u > inf R N u and let ǫ ∈ (0 , sup R N u − inf R N u ).Consider the non-negative solution v ε = u − inf R N u + ε/ x ε such that v ε ( x ε ) = ε . Up to translations, the Harnack inequality (1.7) implies that v ε B ε in x ε + K ρ ( ε/C ), for all ρ >
0. Letting ρ → + ∞ , we get v ε B ε in the whole R N ,i.e. u inf R N u + ( B − / ε in R N and letting ε → (cid:3) Acknowledgement.
All authors are members of GNAMPA of INdAM. S. Mosconi is supported by:grant PRIN n. 2017AYM8XW:
Non-linear Differential Problems via Variational, Topological and Set-valued Methods of the Ministero dell’Istruzione, Universit`a e Ricerca; grant PdR 2020-2022 - linea 2:
MOSAIC of the University of Catania; grant PdR 2020-2022 - linea 3:
PERITO of the University ofCatania.
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Acta Math. Sin. (Engl. Ser.) (2006), 793–806. 5(S. Ciani) Dipartimento di Matematica e Informatica “Ulisse Dini”, Universit`a degli Studi diFirenze, Viale Morgagni 67/A 50134, Firenze, Italy
Email address : [email protected] (S. Mosconi) Dipartimento di Matematica e Informatica, Universit`a degli Studi di Catania, VialeA. Doria 6, 95125 Catania, Italy
Email address : [email protected] (V. Vespri) Dipartimento di Matematica e Informatica “Ulisse Dini”, Universit`a degli Studi diFirenze, Viale Morgagni 67/A 50134, Firenze, Italy
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