The impact of intrinsic scaling on the rate of extinction for anisotropic non-Newtonian fast diffusion

We study the decay towards the extinction that pertains to local weak solutions to fully anisotropic equations whose prototype is \[ \partial_t u= \sum_{i=1}^N \partial_i (|\partial_i u|^{p_i-2} \partial_i u), \qquad 1<p_i<2. \] Their rates of extinction are evaluated by means of several integral Harnack-type inequalities which constitute the core of our analysis and that are obtained for anisotropic operators having full quasilinear structure. Different decays are obtained when considering different space geometries. The approach is motivated by the research of new methods for strongly nonlinear operators, hence dispensing with comparison principles, while exploiting an intrinsic geometry that affects all the variables of the solution.


Introduction
For an open bounded set Ω ⊂ R N and a positive time T , we consider anisotropic differential equations whose prototype is the following Differential operators as (∂ t − ∆ p ) above appear already in the seminal work [27], in the guise of the prototype example of operators obtained as the sum of monotone ones.They enjoy many interesting properties (see for instance the book [2]) whose interpretation has led to a rich mathematical theory (see for instance [6], [8], [30], [31]).Nonetheless, even after more than half a century, the basic regularity properties of local weak solutions to equations (1.1) remain an open problem (see for instance [1], [7], [11]).Besides the theoretical intrinsic interest and challenge, this kind of equations appear in various physical contexts (see Chap.IV of [4]), unveiling the mathematical description of diffusion processes for which the propagation has a different non-Newtonian behavior along each coordinate axis; as well as modeling electro-rheological fluids (see for instance the seminal paper [28] or the book [29]), in particular when the stress tensor is a function of an electromagnetic field that varies on each coordinate direction.This work is developed for the so-called fast diffusion regime, 1 < p i < 2 for all i ∈ {1, . . ., N }, which seems to unfold very strong properties of solutions.The precise attribute we are interested in is the property of extinction in finite time of local weak solutions to (1.1), meaning that there exists a finite time T * < T , called time of extinction, such that the solution u vanishes out from T * : This property is enjoyed by the solutions to the parabolic p-Laplacean equation and it affects preponderantly the nature and behavior of solutions (see [15] or, more in general, [5] and [14]).For instance, in [19] the authors show that a point-wise Harnack inequality cannot be found for the solutions to (1.2) in the sub-critical range 1 < p < 2N/(N + 1); while in the super-critical range 2N/(N + 1) < p < 2 the phenomenon of expansion of positivity is closely related to the singular character of the operator, that privileges the elliptic behavior to the diffusive one, as soon as the modulus of ellipticity |∇u| p−2 ∇u blows up.
To the very interesting properties of singular equations, the operator (1.1) adds the fascinating ones of anisotropy.In [22], the asymptotic behavior is studied through the analysis of self-similarity, showing that new mathematical methods need to be developed in order to overcome the strong non-uniqueness phenomena and to construct suitable barriers.In [3], the authors show that these anisotropic equations are, in a certain sense, richer than their p-Laplacean counterpart; indeed, for solutions to equations as (1.1) within the more relaxed condition 1 < p < 2 (here p is an average of p i s, see Section 3) the dichotomy finite speed of propagation/extinction in finite time is no longer valid and it is replaced by conditions on the growth exponents p i s taking into account the competition between diffusions.
Solutions to singular p-Laplacean equations as (1.2), have a decay toward extinction (see [19]) that follows the law being B ρ the ball of radius ρ and γ a positive constant depending only on the data {N, p}.In the present work we show that the decay profile of extinction of solutions to equations of the kind of (1.1) is the same as the one to the p-Laplacean if one considers a particular space-geometry, being γ a positive constant depending only on the data, and, for any fixed τ > 0 This particular space geometry, which we refer to as intrinsic geometry (see Section 2), has interesting features: although the cylinder K ρ (T * − t) degenerates in these directions x i for which p i > p when t approaches T * , it preserves its volume regardless of the time level undertaken; and more, when p i ≡ p for all i = 1, . . ., N , the set K ρ (τ ) is the classical cube.
We also show that the decay rate of a solution u to equations of the type (1.1) can be estimated within a geometry that is non-degenerative, but at the price of a more complex rate , being λ i = N (p i − 2) + p, λ = N (p − 2) + p (as usual) and γ a positive constant depending on the data.Here the geometry will be referred to as the standard geometry, being based on cubes as (1.4) Unlike the intrinsic geometry considered before, this one does not take into account the time variable.Again, when p i ≡ p for all i = 1, . . ., N , the set K ρ is the classic cube of hedge 2ρ.It is clear that the extinction rate in this case will depend on the smallness of T * − t and the maximum of the exponents in the sum.
It is the precise aim of our study to carry out an analysis of these two rates of extinction within these two different underlying geometries.The method of derivation of these decay rates has its own mathematical interest: confirming the well-known principle that the run itself can be more instructive than the final destination, we obtain the above behaviour of solutions from various Harnack-type estimates.These inequalities are found in three different topologic settings: L 1 loc (Ω), L 1 loc (Ω)-L ∞ loc (Ω) and L r loc (Ω)-L r loc (Ω) backward in time, and all of them are new for solutions to operators as (1.1) (we refer to Section 2 for the precise statements).
Here below we give an example of what we mean by Harnack-type estimates in the L 1 loc (Ω)-topology, or, in short, L 1 -L 1 Harnack-type inequality.
L 1 -L 1 Harnack-type inequality Let u be a non-negative local weak solution to (1.1) in R N × R + 0 and let ρ, t be positive fixed numbers.Then, the following two estimates hold true in their respective space configurations.
1 Let K ρ (t) be defined as in (1.3).Then there exists a constant γ(N, p i ) > 1 such that sup 0≤τ ≤t ˆKρ(t) 2 Let K ρ be defined as in (1.4).Then there exists a constant γ(N, p i ) > 1 such that

Novelty and Significance.
Origins.To the best of our knowledge, the idea of a Harnack-type estimate in the topology of L 1 loc (Ω) had its first appearance in [16] for the prototype p-Laplacean equation, and it was used in [19] with the aim of giving a bound from below to its solutions in a small cylinder, so to prove a point-wise Harnack inequality.There these integral Harnack-type estimates are first used to evaluate the time of extinction of solutions.The method has been reported in ( [15], Chap.VII) for solutions to the prototype singular equation (1 < p < 2).A proof for p-Laplacean type equations with full quasilinear structure can be found first in the paper [18] and then in the monograph [17], again with the aim of obtaining a bound from below toward the determination of a point-wise Harnack-type inequality.All these estimates are unknown for anisotropic equations such as (1.1).In contrast with the few results available in literature (see for instance [11], [22]) that use crucially the invariance and comparison properties of the prototype equation, we derive here the aforementioned Harnack-type inequalities for the full-quasilinear structure operator (see definition (3.1)-(2.2))adopting a technique that dispenses with comparison principles and treats equations that have bounded and measurable coefficients.For this whole spectrum of equations we derive the decay rate of extinction.As anticipated, in the cours d'oevre for the evaluation of the extinction rate, we derive backward L r loc (Ω)-L ∞ loc (Ω) estimates that have their own mathematical interest (see Theorems 2.4, 2.5).For their derivation, we assume that the solutions are locally bounded: this is a crucial point for the regularity theory of anisotropic p-Laplacean equations, as a condition on the spareness of the exponents p i s is necessary already for the elliptic case (see for instance [23], [24]).From the (anisotropic) parabolic point of view, the theory of local boundedness is reasonably complete, see for instance [13], [20], [26].Finally, these L r loc (Ω)-L ∞ loc (Ω) estimates are reminiscent of the isotropic case (see for instance [19]) and are obtained through the successive application of standard L r loc (Ω)-L ∞ loc (Ω) estimates (Theorems 5.4, 5.1) with backwards L r loc (Ω) ones (see Theorems 5.2, 5.5).We refer to [17] and the references therein for the isotropic counterpart.The lack of (known) regularity of solutions encumbers the research for applications on models directly intertwined with (1.1) (see [4] Chap.IV).Nonetheless, these operators reveal a very interesting picture of the underlying nonlinear analysis and competitive behaviour between different diffusions.
The role of intrinsic geometry.A satisfying study of anisotropic operators as (1.1) cannot be brought on regardless of the self-similar geometry embodied in the operator itself.This is already understood in the case of the evolutionary p-Laplacean equation, where has been shown that a Harnack inequality holds true only in a particular geometry, called intrinsic geometry.We refer to [15] and [32] for insights on this topic.Roughly speaking, in the regularity theory of diffusive p-Laplacean equations, time is linked to space by a relation that takes into account the solution itself, as t = ρ p u 2−p o , supposing u o > 0 is the value of the solution at a point.In the case of anisotropic operators behaving like (1.1), the full power of self-similar geometry is needed, and the scaling factor depending on u o enters also the in space variables.As a concrete example, in the degenerate case and for solutions u of (1.1) in S ∞ = R N × R + , a point-wise Harnack inequality takes the following form (we refer to [11]): positive constants depending only on {N, p i } .In the available literature, L 1 -L 1 Harnack-type estimates are derived for the diffusive p-Laplacean operators (see [17]) without the use of a particular intrinsic geometry.Here we overcome the difficulty of the nonhomogeneity of the operator by setting an intrinsic geometry that depends also on time, as K ρ (t) in (1.3), which considers self-similar space-cubes as In this case, the particular self-similar factor M depends on the radius and on the a priori chosen time level t, and has the interesting feature of reestablishing the homogeneity in the estimates.With a little abuse of notation, along the text we still call this geometry intrinsic geometry, because the quantity M here above is always related to some norm of u in applications (see for instance the use of (5.3) and (4.1)).A last word in honor of the standard geometry K ρ is due.Local integral L 1 -L ∞ Harnack-type inequalities hold true also in this case (see Theorems 2.8-2.2),which is when one considers M = 1; but the anisotropy is inevitably carried over into a sum of the quantities t/ρ p on the right-hand side of the estimates, with different powers depending on p i s.A novel method is also used in this case, which we believe to be useful also for other nonlinear operators.
Applications and Future Perspectives.The range of application of the Harnack-type inequalities we are about to describe is very wide.As for the main purpose of the present work, they can be used to estimate the decay of the solution at the extinction time; and, assuming an integrable initial datum u 0 L 1 (R N ) they imply a certain conservation of the mass of the solution in time.In addition, not only these Harnack-type estimates are very important for the convergence of approximating solutions when dealing with the problem of the existence (see for instance [16]), but also they proved to be useful to control the measure of level sets and to give a short proof of solutions' Hölder continuity (see for instance [12] for the isotropic case).
Method.The Harnack-type estimates that are obtained throughout the paper, for each one of the mentioned geometries, have as common starting point some general energy estimates, that are collected in the Appendix.Although these energy estimates are non-trivial, they are similar to the isotropic ones (see Section 7); hence we decided to postpone their presentation so as to leave space to what is really new in the anisotropic context.Our first step is to derive L 1 -L 1 Harnack-type estimates by means of testing the equation with negative powers of the solution and a combined nonlinear iteration.In a second step, we study the L r -L ∞ inequalities by suitably adapting the classic De Giorgi-Moser scheme; here we use the L r -norm of the solution chained with the energy estimates provided by the equation in a certain geometry.Finally, we nest these inequalities with a backward L r estimate to derive L r -L ∞ inequalities in terms of the initial datum u 0 ; combining these with the first obtained L 1 -L 1 estimates we derive the L 1 -L ∞ Harnack-type estimates given by Theorems 2.7, 2.8.
Structure of the paper.In Section 2, we define the anisotropic operators with full quasilinear structure and state the main Theorems.Then, in Section 3, we give the definition of local weak solution and the proper functional spaces for it; along with the main notation used throughout the paper.In Section 4, we present the proofs of the first two Theorems, both concerning L 1 -L 1 Harnack-type estimates, but specializing the geometry in each case.In a similar fashion, in Section 5, we provide the proofs of the backward L r -L ∞ estimates, again distinguishing the two geometries.Finally, short Section 6 concludes with the main Theorems, while the last Section, Appendix 7, presents the main energy estimates used along our analysis and some standard iteration Lemmata.

Main Results and Applications
We consider singular parabolic nonlinear partial differential equations of the form (2.1) where the functions A = (A 1 , . . ., A N ) : Ω T ×R N +1 → R N and B : Ω T ×R N +1 → R are Caratheodory functions that satisfy the structure conditions, for 1 < p i < 2, for all i = 1, . . ., N, for almost every (x, t) ∈ Ω T and for all (s, ξ) ∈ R × R N , where C o , C 1 are positive constants and C is a non-negative constant that distinguishes between the cases when the equation to be homogeneous (when C = 0) from when it is not.We will say that a positive generic constant γ depends only on the data if it depends on the parameters {N, p i , C o , C 1 }; for the summation notation we refer to Section 3.
Our main results concern the integral inequalities which, for the sake of simplicity, we state in a forward cylinder centered at the origin.First, we state the Harnack-type inequalities for the L 1 loc (Ω) norm of the solution evolving in time, sorting out the case of anisotropic intrinsic geometry from the anisotropic standard one.
Theorem 2.1 (Intrinsic L 1 -L 1 Harnack-type inequality).Let u be a non-negative, local weak solution to equation ⊂ Ω T , holds true.Then, there exists a positive constant γ depending only on the data such that, either there exists an index i ∈ {1, . . ., N } for which Let u be a non-negative, local weak solution to equation (2.1)-(2.2) in Ω T , 1 < p i < 2 for all i = 1, • • • , N .Let t, ρ > 0 be such that the inclusion K 2ρ × [0, t] ⊂ Ω T holds true.Then, there exists a positive constant γ depending only on the data such that, either there exists an index i ∈ {1, . . ., N } for which or, denoting λ i = N (p i − 2) + p, we have Remark 2.3.We remark that in Theorems 2.1 and 2.2 the constants λ, λ i can be of either sign.
Then, considering extra local regularity assumptions on u such as local boundedness and u ∈ L r loc (Ω T ), for some r > 1, we have the following L r -L ∞ estimates, valid for exponents p > 2N/(N + r).
Theorem 2.4 (Intrinsic Backwards L r -L ∞ estimate).Let u be a non-negative, locally bounded, local weak solution to (2.1)-(2.2) in Ω T , and suppose that for some r > 1 it satisfies both u ∈ L r loc (Ω T ) and Then, there exists a positive constant γ depending only on the data, such that for all cylinders (2.9) λ r = N (p − 2) + rp > 0.
Then, there exists a positive constant γ depending only on the data, such that for all cylinders either there exists an index i ∈ {1, . . ., N } for which (2.5) holds true, or (2.10) sup Remark 2.6.In the prototype degenerate case (p i > 2 for all i = 1, . . ., N ) estimates (2.8)-(2.10)hold true without the second term (and third) on the right-hand side of the inequality (see for instance [10] and [20]).Similarly, to what discussed in [16], the distinction between the two approaches relies in the consideration of solutions that are either local or global in time.With the integral Harnack estimates derived in this paper, it is possible to embark on the path of global existence of solutions to (1.1).To this aim we observe that the first term on the right hand side of (2.8) is formally the same as in the degenerate case, while the second term on the right-hand side controls the growth of the solution for large times.
Finally, we state the main results or our analysis: Harnack-type estimates considered in the topologies L ∞ loc (Ω) to L 1 loc (Ω), again distinguishing when the anisotropic geometry considered is intrinsic or standard.
Then, there exists a positive constant γ depending only on the data such that, for all cylinders either there exists i ∈ {1, . . ., N } for which (2.3) holds true, or (2.11) sup Let u be a non-negative, locally bounded, local weak solution to (2.1)-(2.2) and suppose p is in the supercritical range, i.e.
Then, there exists a positive constant γ depending only on the data such that, for all cylinders either there exists i ∈ {1, . . ., N } for which (2.5) holds true, or (2.12) sup Rates of Extinction.The fact that certain solutions to (2.1)-(2.2) with C = 0 are subject to extinction in finite time has been studied in [3] and also in [2] (we refer to [5], [14], [15], for the isotropic case, all p i ≡ p).In [3], the authors suppose u to be a solution to x ∈ Ω, with u 0 ∈ L 2 (Ω) and where a i : Ω × (0, T ) × R → R are Caratheodory functions satisfying a 0 ≤ a i (x, t, s) ≤ A 0 , for a 0 , A 0 > 0 structural constants.Within this framework, the authors show that if 1 < p < 2, being p = N/( i p i −1 ) the harmonic average of the exponents p i , then the energy solutions to (2.13) vanish in a finite time, i.e By using a weaker definition of solution (see Definition 3.1), here we assume u is a non-negative, local weak solution to (2.1)-(2.2) in Ω T , with C = 0, 1 < p i < 2 for all i = 1, . . ., N , and that there exists an extinction time T * < T for u.Then, similarly to [19], we use the L 1 -L 1 Harnack-type inequalities (2.4)-(2.6) to evaluate the decay of the L 1 loc (Ω) norm of u toward its extinction and the L 1 -L ∞ Harnack-type inequalities (2.11)-(2.12) to estimate the rate of extinction of the solution in a whole half cylinder approaching T * .These two properties require different assumptions on the exponents p i .We divide the cases distinguishing the underlying geometry.
• The mass decays within the law for a positive constant γ depending only on the data.Hence the mass u(•, τ of the solution locally decays (to zero) as (T * − τ ) 1/(2−p) in a space configuration depending on time but with unchanged measure |K ρ (T * − τ )| = (2ρ) N .
• If λ = N (p − 2) + p > 0, then the solution has the following vanishing rate: sup for a positive constant γ depending only on the data.Choosing T * /2 < t < T * , it is possible to specialize this decay to an ultra-contractive bound This estimate shows that the rate of local decay of the L ∞ -norm of the solution, in a space configuration depending on each time t, is again of the type (T * − t) 1/(2−p) but now for a different power of the radius ρ.
We observe that when t → T * the time intrinsic cube K ρ (T * − t) shrinks along the directions x k for which p k > p, while in the other directions it stretches to infinity; this particular phenomenon occurs keeping the measure |K ρ (T * − t)| unchanged.Therefore, the inclusion K 4ρ (T * − t) ⊆ Ω degenerates according to the choice of time.
Standard Anisotropic Geometry.For a positive number ρ, let us consider the anisotropic standard cube K ρ as in (1.4), for ρ > 0 such that K ρ ⊂ Ω.We can estimate the local decay of its L 1 and L ∞ norms as above, but this time in a space geometry that is time independent, paying the price of having more involved estimates.
• Description of the mass decay When considering times τ approaching T * , the mass of the solution u(•, τ ) L 1 (Kρ) decays to zero at the rate (T * − τ ) 1/(2−p N ) , while when considering larger times ( . • For any time 0 < τ < T * , and assuming that λ > 0, we have a description of the local decay of the essential supremum of the solution as sup for γ positive constant depending only on the data {C o , C 1 , C 2 , p i , N } and being Here we observe that a decay rate towards extinction, i.e. for times (T * − τ ) < 1, is given from this estimate only with the extra assumption λ i = N (p i − 2) + p > 0 for all i = 1, . . ., N , and the solution vanishes in the half-cylinder as fast as (T * − τ This behavior is confirmed by those solutions that are constant along N −1 space coordinates and behave like a p 1 or p N -Laplacian by means of the only free variable.

Functional Setting and Notation
Functional Setting.We define the anisotropic spaces of locally integrable functions as and the respective spaces of functions with zero boundary data It is known (see [6], [33]) that when p > N the embedding W 1,p (Ω) ֒→ C 0,α loc (Ω) for Ω regular enough.Therefore in this work we will consider p < N .
T and for all compact sets K ⊂⊂ Ω, it satisfies the inequality ˆK uϕ dx This last membership of the test functions, together with the structure conditions (2.2), ensure that all the integrals in (3.1) are finite.Moreover, as ϕ vanishes along the lateral boundary of Ω T , its integrability increases thanks to the following known embedding theorem.

Notation.
In what follows we introduce the notation we will be using along the text.
We shorten the notation on sums and products when they are intended for all indexes i, j, k ∈ {1, . . ., N }, .
Only when the sum runs over a different range of exponents will be further specified.
Exponents are ordered, and p stands for the harmonic average We denote by ∂ i u the weak directional space derivatives and by ∂ t u the weak time-derivative (see (7.1) for more details).Finally, ∇u = (∂ 1 u, . . ., ∂ N u).
Our geometrical setting will distinguish between two types of N -dimensional cubes: Anisotropic intrinsic cube Anisotropic standard cube We will use two exponents for the decay rates: Given a measurable function u : E ⊂ R N +1 → R, we denote by sup E u (inf E u) the essential supremum (essential infimum of u) in E with respect to the Lebesgue measure.
We denote by γ a generic positive constant that depends only on the structural data 2), and it may vary in the estimate from line to line.
Young's Inequality Convention.In our estimates we will repeatedly use Young's inequality in the following form: for q > 1 and a, b, ǫ > 0 fixed, we use the well-known inequality , and γ(ǫ) = q − 1 q 1/(q−1) q The constant ǫ will not be specified as long as it depends only on the data {p i , N, C o , C 1 }.

Proof of L 1 -L 1 Harnack estimates
In this Section we prove Theorems 2.1-2.2,dividing the argument whether the anisotropic space geometry considered is the standard or the intrinsic one.
Intrinsic Anisotropic Geometry: Proof of Theorem 2.1.We consider a fixed time-length 0 < t < T , and let ρ > 0 be small enough to allow the inclusion for the fixed quantity Let u be a non-negative local weak super-solution to (2.1) in Ω T and σ ∈ (0, 1) a number.Then, there exists a positive constant γ depending only on the data such that, either (2.3) holds true for some i = 1, . . ., N , or we have Proof.For each i = 1, . . ., N we apply Hölder's inequality to the quantity to be estimated, 2,i .Next, we estimate I 2,i by taking the supremum in time and then using Hölder's inequality In the last steps we have used the property |K ρ (t)| = (2ρ) N and the definition of ν, λ i , λ (see the statement of Theorem 2.1).Now we estimate I 1,i using the inequalities (7.9) within the considered geometry: we test indeed repeatedly, for i = 1, . . ., N , equation (2.1) with the function being ζ a smooth cut-off function between the sets K σρ (t) and K ρ (t), hence enjoying the properties .
The number ν ∈ R + is fixed, and by implementing (4.3) into (7.9)we obtain (4.4) Now we manipulate the terms of (4.4), with the aim of obtaining an homogeneous estimate similar to I 2,i .
The first term on the right is bounded from above by a similar estimate as the one for I 2,i .
The second term is the one most related with our anisotropic problem; it is here that we specialize our estimates toward homogeneity.We dominate it from above by using p i < 2, with the usual trick (u + ν) in order to give an homogeneous estimate with respect to j-th index, namely , Referring again to (4.4), each j-th term of I 3 on the right can be estimated by , where the first inequality uses (u + ν) −2 ≤ ν −2 and the last inequality is brought similarly to the one for I 2,i .Finally, collecting everything together we arrive, for each i = 1, . . ., N , to the estimate 3) is violated for all i = 1, . . ., N , then the term in squared brackets on the right-hand side is smaller than 3, recalling (4.1).Thence we go back to the initial estimate and evaluate and thereby .
Proof of Theorem 2.1 Concluded.Now we perform an iteration on σ ∈ (0, 1): we define the increasing radii and consider the family of concentric intrinsic anisotropic cubes For every n ∈ N ∪ {0}, consider time-independent cut-off functions ζ n as in (7.2) between K n and Kn , hence satisfying We test equation (2.1) with ζ n and we integrate over Kn × [τ 1 , τ 2 ], for arbitrary time levels 0 ≤ τ 1 < τ 2 ≤ t, to get (4.5) Assume condition (2.3) is contradicted for all i ∈ {1, . . ., N }; then the second term in parenthesis on the right of (4.5) is bounded above by C 1 + 1, while the third term is estimated by Putting all the pieces together we obtain the estimate (4.6) By continuity of u as a map [0, T ] → L 2 loc (Ω), we take τ 2 as the time level in [0, t] such that and τ 1 as the time level satisfying It is precisely for this choice of ordering between τ 1 and τ 2 that we need u to be a solution, and not only a super-solution.Now we evaluate the second term in (4.6) with the inequality (4.2) applied to the pair of cylinders Kn × [0, t] ⊂ K n+1 × [0, t] and develop the definition of ν to write By using Young's inequality on each i-th term with exponents 2(p i −1) and the conclusion follows from the classical iteration of Lemma 7.8.
Standard Anisotropic Geometry: Proof of Theorem 2.2.Let 0 < t < T and ρ > 0 such that the following inclusion is satisfied, To consider intermediate cylinders, for a fixed σ ∈ (0, 1] we define and Moreover, for such fixed t, ρ, we define the quantity Let u be a non-negative local weak super-solution to (2.1) in Ω T and σ ∈ (0, 1) a number.Then, there exists a positive constant γ, depending on the data, such that, either there exists an i ∈ {1, . . ., N } for which (2.5) is valid, or for all i ∈ {1, . . ., N } we have , with λ i = N (p i − 2) + p and being S = sup 0≤τ ≤t ˆKρ u(x, τ ) dx.
Proof.For σ ∈ (0, 1] we consider the cylinders We use the estimates (7.9) by testing the equation with where ζ is a cut-off function of the type (7.2), defined between K σρ and K ρ , therefore verifying This gives, for all i ∈ {1, . . ., N }, the inequalities (4.10) We estimate the various terms.The first integral on the right-hand side of (4.10) is manipulated as in (4) to get .
The second term can be estimated by using that (u + ν Σ ) p j −2 < ν to get for all i = 1, . . ., N the inequalities .
Finally the third term on the right-hand side of (4.10) is estimated, for any i, j ∈ {1, . . ., N }, as .
Collecting everything together we obtain (4.11) The second factor on the right of (4.11) is smaller than 4 if (2.5) is violated for all indexes j ∈ {1, . . ., N }, and once we observe This allows us to evaluate Proof of Theorem 2.2 concluded.
Proof.We fix ρ > 0, define the sequence of increasing radii and construct the family of concentric standard anisotropic cubes , and for any τ 1 , τ 2 ∈ [0, t], we consider the family of cylinders For each n ∈ N ∪ {0} chosen, consider ζ n (x) a cut-off function of the form (7.2) between K n and Kn that is time-independent and verifies Testing (2.1)-(2.2) with such a ζ n we obtain (4.12) for arbitrary time levels τ 1 , τ 2 ∈ [0, t].Again, by the continuity of u as a map [0, T ] → L 2 loc (Ω), we take τ 2 as the time level in [0, t] such that and set Since τ 1 is arbitrary, (4.12) yields The last term on the right-hand is dominated as follows: for all i = 1, . . ., N , and assuming that condition (2.5) is violated for all indexes.Therefore, by applying first Lemma 4.2 to the pair of cylinders Q n and Qn , for which 1 − σ ≥ 2 −(n+4) , and then Young's inequality one gets A standard iteration finishes the proof as in the case of (4.7)

Proof of the backward L r -L ∞ estimates
The proof of Theorems 2.4-2.5 rely on two estimates: L r -L ∞ estimates combined with a L r estimates backward in time; the presentation is done separately for the intrinsic and the standard geometries.
Intrinsic Anisotropic Geometry: Proof of Theorem 2.4.
Theorem 5.1 (L r loc -L ∞ loc estimates).Suppose u is a non-negative, locally bounded, local weak sub(super)solution to (2.1)-(2.2) in Ω T .Let r 1 and λ r = N (p − 2) + rp > 0.Then, there exists a positive constant γ, depending only on the data, such that ∀t > 0, ∀ρ > 0 : K 4ρ (t) × (0, t) ⊂ Ω T , either (2.3) holds for some i ∈ {1, . . ., N } or (5.1) sup Proof.Assume condition (2.3) does not hold for every i ∈ {1, . . ., N }.Let σ ∈ (0, 1) be fixed and consider the decreasing sequences of radii, for each i ∈ {1, . . ., N }, and of time levels from which one constructs the sequence of nested and shrinking cylinders be a cut-off function as in (7.2) therefore verifying for all i = 1, . . ., N , and In the weak formulation (3.1), for each n ∈ N, consider the test function ϕ n = (u − k n+1 ) + ξ n , over the cylinders Q n , for the truncation levels where k is a positive real number to be determined.By the classical energy estimate (7.4) we obtain the following bound on the energy where first we implemented the construction of the cut-off function ζ and then we have used that for each i ∈ {1, . . ., N } the condition (2.3) is violated.
The case max 1, 2N N +2 < p < 2. We estimate the energy E n from above in terms of the L 2 -norm of the truncations (u − k n ) + .Observe that for all s = 0, 1, . . ., N , having defined p 0 = 2, it holds Hence we have and taking into account as a further condition the right hand side of (5.2) now reads (5.4) Now we want to put in a chain the estimate of E n obtained in terms of (u with the anisotropic Sobolev embedding (3.2).
Here we take advantage of exponent p being in the super-critical range, p > max{1, 2N/(N + 2)}: indeed, in such a range, the number q = p(N + 2)/N is greater than 2 and we can use Hölder inequality on (u − k n+1 ) + 2 L 2 (Q n+1 ) to allow the aforementioned chaining procedure.In the embedding (3.2) we make the choices By setting 2,Qn , from the previous estimate we derive (5.5) the Fast Converge Lemma 7.7, ensures X n → 0 as n → ∞, meaning that sup for every 1 ≤ r ≤ 2 < q for which (and for sure) λ r = N (p − 2) + rp > 0.
Here we observe that a priori information on the boundedness of u was not necessary in order to get the first sup-estimate in this case.Finally, we perform a cross-iteration on σ ∈ (0, 1) as follows.Still referring to radii ρ i as in the construction above, we now consider the increasing sequences, for n ∈ N ∪ {0}, and define The previous estimate applied to the pair of cylinders Qn and Qn+1 gives us

and, by taking
In this case, the conditions λ r > 0 and 1 < p ≤ 2N/(N + 2) imply r > 2 and also q = p N +2 N ≤ 2 < r.Here we need to consider the L r -norm of the truncated functions and supposing u locally bounded, recalling q < 2 < r, we apply the anisotropic embedding (3.2) to get Now again we make a chain of inequalities, but this time using E n and Y n .By acting in a similar fashion as before and assuming (5.3), we get and therefore the aforementioned chain reads Again by the Fast Convergence Lemma 7.7, if k > 0 is taken so that Proceeding as before, one has by means of Young's inequality with ǫ = 1/2 for exponent µ = (N +p)(r−2) N (r−q) > 1.Then by iteration, taking σ = 1/2 and letting n → ∞ sup Theorem 5.2 (L r loc estimates backward in time).Let u be a non-negative, locally bounded, local weak solution to (2.1)-(2.2) and assume u ∈ L r loc (Ω T ), for some r > 1.Then there exists a positive constant γ, depending only on the data, such that either (2.3) is satisfied for some i ∈ {1, . . ., N } or Proof.Assume (2.3) fails to happen for all i ∈ {1, . . ., N }.Fix σ ∈ (0, 1) and construct the cylinders With these stipulations, a cut off function ζ, such as in (7.2), between K ρ (t) and K (1+σ)ρ (t) satisfies , and the estimates (7.5) with K 1 = K ρ (t) and K 2 = K (1+σ)ρ (t) are now written sup 0≤τ ≤t ˆKρ(t) .
Without loss of generality one can assume that, for all i = 1, . . ., N , In fact, if for some index i = 1, . . ., N Cρ and then (5.7) comes immediately.Hence sup 0≤τ ≤t ˆKρ(t) We estimate the second integral on the right-hand side by applying Hölder's inequality, .
The last integral on the right-hand side is dominated as follows using Hölder inequality and noticing that Putting the estimates all together we finally get (5.8) . Now we perform an iteration on σ: fix ρ > 0 and for n ∈ N ∪ {0} consider the increasing sequence of radii

By setting
.
We use Young's inequality in each i-th term of the sum for a constant b > 1 depending only on the data, and with these stipulations we arrive at and proof is completed once we choose ǫ = 1/2b < 1 and let n → ∞ as usual.Proof.We plug inequality (5.7) into (5.1) to obtain Then there exists a positive constant γ, depending only on the data such that, for all K ρ ×[0, t] ⊂ Ω T , either for some i ∈ {1, . . ., N } condition (2.5) is satisfied or (5.10) sup Proof.Assume condition (2.5) is violated for all indexes i ∈ {1, . . ., N }.Let σ ∈ (0, 1) be fixed and consider the decreasing sequences from which one constructs the sequence of nested and shrinking cylinders where, as usual in the standard anisotropic geometry, In the weak formulation (3.1) we consider test functions ϕ n = (u − k n+1 ) + ζ n , over the cylinders Q n , for the truncation levels where k is a positive real number to be determined (along the proof).By the energy estimates (7.4) we get (5.11) As in the proof of Theorem 5.1, from now on we distinguish between the case where p is in the super and the sub-critical ranges.We will only present how to proceed when p is in the super-critical range; the sub-critical range is treated analogously to what was done for the anisotropic intrinsic geometry but now taking into account take we are working under the assumptions related to the anisotropic standard setting.
Consider max{1, 2N N +2 } < p < 2. By observing that ρ p C p i ≤ 1, for all i ∈ {1, . . ., N }, and choosing k ≥ ν Σ , from the previous estimate (5.11) one gets Although the geometry is different, we derive a similar estimate to (5.5) by means of Hölder's inequality, so to obtain sup An analogous iteration procedure is applied considering the radius to be ρ rather than ρ i , completing thereby the proof for the super-critical range of p.
Proof.Assume (2.5) is not verified for all i ∈ {1, . . ., N }.Fix σ ∈ (0, 1) and construct the cylinders Observe that (5.13) is a natural assumption: if it is violated then, for some i ∈ {1, . . ., N }, then and (5.12) is found.Then, as in Theorem 5.2, we estimate the various terms as follows , for λ i,r = N (p i − 2) + pr, while the second term in the parenthesis of (5.14) is managed as follows i .
Plugging these estimates into (5.14)we obtain, and applying Young's inequality in each term of the sum, we get (5.15) From this point on, we perform a standard iteration on σ: for fixed ρ > 0 and n ∈ N ∪ {0}, we consider the increasing sequence of radii and the proof is completed once we choose ǫ = 1/2b < 1 and let n → ∞.
Proof.We use (5.12) to estimate the integral term at the right-hand side of (5.10) Proof.We start by considering inequality (5.1) and then estimate the integral on its right-hand side by (2.4) to get Standard Geometry.Proof of Theorem 2.8.
Proof.We combine Theorem 5.4 with r = 1 and Theorem 2.2 to get sup

Appendix
Energy Estimates.To the aim of computation, it would be technically convenient to pass from the formulation (3.1) of local weak solution to its Steklov averaged version, which allows us to perform computations under the integral sign with the approximating functions (7.1) defined for all 0 < t < T .This is the same definition as the one presented in [15] (see in particular Chapter II for more details), and we refrain from specifying further this procedure, leaving space to what is really new.
Separate Variables Test Functions.For a compact set K ⊂ Ω, we will usually test the equation , being π i the euclidean projection to the i-th component.Sometimes we will use the notation for ζ(x) as above and ξ(τ ) ∈ C 1 loc (0, T ) a function to be specified at each recurrence.
Proof.We test equation (2.1) with ϕ 3), vanishing on ∂K, for all times, and verifying ζ(τ 1 , x) = 0, for all x ∈ K.So we arrive, through a standard Steklov approximation, to being B, A i ,for all i = 1, . . ., N , the Caratheodory functions of (2.1)-(2.2).We evaluate the terms separately, using the structure conditions (2.2) and Young's inequality (3.3) on each i-th term with q = p i , q ′ = p i /(p i − 1) to get where in the last inequality we have collected the terms Choosing suitably ǫi and ǫ i small enough for all i = 1, . . ., N and joining all the previous estimates together implies, for all k ∈ R, Energy Estimates 2 -Testing with positive powers.
Lemma 7.2.Let u be a non-negative, locally bounded, local weak solution to ) be a cut-off function between K 1 and K 2 as in (7.2).Let t > 0 be any number such that the inclusion is preserved.Then, there exists a positive constant γ, depending only on the data, such that Proof.In the weak formulation (3.1) choose as a test function, defined over being ζ as in (7.2) and k ∈ R + to be determined.We observe that f (u) = 0 outside the set Now we define F (u) = ˆu k f (s) ds an integral function of f and we observe that The test function ϕ is an admissible one, modulo a Steklov approximation, thanks to the local boundedness of u: observe that Passing to the limit the in Steklov approximation, we obtain Combining all the estimates we obtain, for all s ∈ (0, t] Here we observe that, on the set [u > k], the following holds true so that we estimate for each i = 1, . . ., N, The other integral term does not involve the derivatives of the cut-off function Now we estimate from above I 3 , I 4 as Hence, choosing ǫ i and ǫi appropriately small, we obtain for all s ∈ (0, t]  2).We observe that ϕ i (x, 0) = 0, for all x ∈ K, and that the function ϕ i , adequately averaged in time, is admissible due to the choice of ζ and ) |∂ i ζ| ∈ L p i loc (Ω T ).
In the weak formulation we use Steklov averages (see for instance the monograph [17]) for the interpretation of ∂ τ u, to recover by approximation 0 ˆK uϕ i dx passing to the limit thanks to the condition u ∈ C loc (0, T ; L 2 (K)), while all the other terms in the Steklov approximation converge to the relative integrals, thanks to the structure conditions and the bound ν −α > (u + ν) −α , ν, α > 0.
We estimate I 3 and −I 4 from below by means of Young's inequality Remark 7.5.The constant γ deteriorates both as soon as p N ↑ 2 and as p 1 ↓ 1.
Remark 7.6.We observe that all the energy estimates (7.4), (7.5), (7.9) recover, when p i ≡ p, known estimates known for the isotropic p-Laplacean evolution equations (see for instance the Appendix of [18]).This is due to the simple fact that for all ξ = (ξ 1 , . . ., ξ N ) ∈ R N there exists an universal constant γ = γ(p i , N ) > 0 such that Algebraic Lemmas.Here we collect two Lemmata evolving sequences of numbers, that can both be found in [15] (see [13] for the anisotropic counterpart), useful along our proofs.

Lemma 7 . 7 .
[Fast geometric convergence Lemma] Let (Y n ) n be a sequence of positive numbers verifying Y n+1 ≤ Cb n Y 1+α n , being C > 0, b > 1 and α > 0 given numbers.Then the following logical implication holds trueY o ≤ C −1/α b −1/α 2 ⇒ lim n↑∞ Y n = 0.Lemma 7.8.[Iteration Lemma] If we have a sequence of equibounded numbers {Y n } such that, for constants I, b > 1 and ǫ ∈ (0, 1) (7.11) Y n ≤ ǫY n+1 + Ib n , then, by a simple iteration, there exists γ > 0 such that Y 0 ≤ γ I. Research Data Policy and Data Availability Statements.All data generated or analysed during this study are included in this article.
and ζi ζ p i i = ζ , in order to adjust the powers of ζ.Again we use Young's inequality for each i = 1, . . ., N to estimate