An extensive study of the regularity properties of solutions to doubly singular equations
aa r X i v : . [ m a t h . A P ] J a n AN EXTENSIVE STUDY OF THE REGULARITY OF SOLUTIONS TODOUBLY SINGULAR EQUATIONS
VINCENZO VESPRI AND MATIAS VESTBERG
Abstract.
In recent years, many papers have been devoted to the regularity of doublynonlinear singular evolution equations. Many of the proofs are unnecessarily complicated,rely on superfluous assumptions or follow an inappropriate approximation procedure.This makes the theory unclear and quite chaotic to a nonspecialist. The aim of thispaper is to fix all the misprints, to follow correct procedures, to exhibit, possibly, theshortest and most elegant proofs and to give a complete and self-contained overview ofthe theory. Introduction
This work is concerned with the regularity properties of weak solutions to doubly nonlinearequations whose model case is ∂ t u − ∇ · ( u m − |∇ u | p − ∇ u ) = 0 in Ω T := Ω × (0 , T ) , (1.1)where Ω ⊂ R n is an open bounded set, and the parameters m and p are restricted to therange p ∈ (1 , , m > , and 2 < m + p < . (1.2)The term doubly nonlinear refers to the fact that the diffusion part depends nonlinearly bothon the gradient and the solution itself. Such kind of equations describe several physicalphenomena and were introduced by [15] (see also the nice survey by Kalashnikov [13]).Moreover, these equations have an intrinsic mathematical interest because they represent anatural bridge between the more natural generalisations of the heat equation: the parabolic p -Laplace and the Porous Medium equations.Especially in recent years, many papers have been devoted to this topic. The approachesare sometimes not rigorous, sometimes not with sharp assumptions or with unnecessarilylong proofs. The natural definition of weak solutions is obtained from (1.1) by a formalapplication of the chain rule and requires that a certain power of u (rather than u itself)has a weak gradient. This is perhaps the most delicate point: too many papers devoted tothis topic do not take this aspect into account carefully, and use incorrect approximationsor non-admissible test-functions. For more details, we refer the reader to Section 2.Analogously, some results presented below, such as the L -Harnack inequality and theexpansion of positivity have been obtained previously under the assumption that the function u itself has weak gradient, see [8] and [9]. Since this is not necessarily true in our settingwe have included detailed proofs showing that the strategies developed in [8] and [9] areapplicable also without assuming the existence of ∇ u . But we do not limit ourselves tofix this aspect. We go through the regularity theory and we use a unified approach givingshorter and different proofs with respect to the ones known in literature. In this way, a Date : January 14, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Doubly nonlinear parabolic equations, H¨older continuity, Harnack inequality. reader can have a self-contained overview of the theory of doubly nonlinear singular parabolicequations. We obtain different results under various ranges for the parameters. The timecontinuity, mollified weak formulation, energy estimates, expansion of positivity and L -Harnack inequality are obtained in the full range (1.2). Local boundedness of weak solutionsis shown in the smaller range m + p > − pn − ( n − pp ) . (1.3)We recall that this range is sharp. In the special case m = 1, (1.1) becomes the singularparabolic p -Laplace equation. Then the condition (1.3) and the integrability required of u in Definition 2.1 below reduce to p > nn +2 and u ∈ L respectively, which are well-knownsharp conditions to guarantee local boundedness for this equation, see for example ChapterV of [4].The local H¨older continuity will be proven only in the so-called supercritical range m + p > − pn . (1.4)Note that (1.4) is a stricter condition than (1.3). We decided that it was too much dispersivefor the reader to prove H¨older continuity also in the sub-critical case because requires aslighty different approach (and assumptions). In the last section, we prove Harnack estimatesin the supercritical range. Note that, as proven in [6] for the p-Laplacian, this result is sharp. Acknowledgments.
M. Vestberg wants to express gratitude to the Academy of Finland.Moreover, we thank Juha Kinnunen for useful discussions and feedback during the writingof this article. 2.
Setting and main result
In order to motivate the natural definition of weak solutions, we reformulate (1.1). For-mally applying the chain rule, we can write the equation in the form ∂ t u − ∇ · ( β − p |∇ u β | p − ∇ u β ) = 0 , (2.1)where β := 1 + m − p − > . (2.2)For later reference we note that (1.3) can be expressed conveniently in terms of β , p and n as p ( β + 1)1 − β ( p − > n. (2.3)We will prove our result not only for solutions to (2.1), but for all equations of the form ∂ t u − ∇ · A ( x, t, u, ∇ u β ) = 0 , (2.4)where A ( x, t, u, ξ ) is a vector field satisfying | A ( x, t, u, ξ ) | ≤ C | ξ | p − (2.5) A ( x, t, u, ξ ) · ξ ≥ C | ξ | p (2.6)An example of an equation that satisfies these conditions is ∂ t u − n X i,j =1 ( a ij ( x, t ) β − p |∇ u β | p − u βx i ) x j = 0 in Ω T := Ω × (0 , T ) , (2.7) EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 3 where the coefficients a ij are bounded and measurable and where the matrix ( a ij ( x, t )) ni,j =1 is positive definite uniformly in ( x, t ). We arrive at the definition of weak solutions bymultiplying (2.4) by a smooth test function and integrating formally by parts. Definition 2.1.
A function u : Ω T → R is a weak solution to (2.4) if and only if u ≥ u β ∈ L p (0 , T ; W ,p (Ω)), u ∈ L β +1 (Ω T ) and ¨ Ω T A ( x, t, u, ∇ u β ) · ∇ ϕ − u∂ t ϕ d x d t = 0 , (2.8)for all ϕ ∈ C ∞ (Ω T ). Remark 2.2.
The extra integrability condition u ∈ L β +1 (Ω T ) is made to justify a testfunction containing u β . The condition is needed since we are considering the fast diffusioncase, in which βp < β + 1. By contrast, in the slow diffusion case m + p > Preliminaries
Here we introduce some notation and present auxiliary tools that will be useful in thecourse of the paper.3.1.
Notation.
With B ρ ( x o ) we denote the open ball in R n with radius ρ at center x o ,and the corresponding closed ball is denoted ¯ B ρ ( x o ). Furthermore, we use the notation Q ρ,θ ( z o ) := B ρ ( x o ) × ( t o − θ, t o ) for space-time cylinders, where z o := ( x o , t o ) ∈ Ω T . For w, v ≥ b [ v, w ] := β +1 ( v β +1 − w β +1 ) − w β ( v − w )(3.1) = ββ +1 ( w β +1 − v β +1 ) − v ( w β − v β ) , b [ v, w ] + := b [ v, w ] χ ( w, ∞ ) ( v ) , (3.2)where β is defined by (2.2). For any real-valued essentially bounded function g defined ona measurable set E ⊂ R n +1 we define its essential oscillation in E asess osc E g := ess sup E g − ess inf E g. The oscillation osc E g of a bounded function g is defined analogously, using the ordinarysupremum and infimum. The parameters C , C , m, n, p will collectively be referred to asthe data.3.2. Auxiliary tools.
We now recall some elementary lemmas that will be used later, andstart by defining a mollification in time as in [14], see also [1]. For
T > t ∈ [0 , T ], h ∈ (0 , T )and v ∈ L (Ω T ) we set v h ( x, t ) := 1 h ˆ t e s − th v ( x, s ) d s. (3.3)Moreover, we define the reversed analogue by v h ( x, t ) := 1 h ˆ Tt e t − sh v ( x, s ) d s. V. VESPRI AND M. VESTBERG
For details regarding the properties of the exponential mollification we refer to [14, Lemma2.2], [1, Lemma 2.2], [20, Lemma 2.9]. The properties of the mollification that we will usehave been collected for convenience into the following lemma:
Lemma 3.1.
Suppose that v ∈ L (Ω T ) , and let p ∈ [1 , ∞ ) . Then the mollification v h definedin (3.3) has the following properties: (i) If v ∈ L p (Ω T ) then v h ∈ L p (Ω T ) , k v h k L p (Ω T ) ≤ k v k L p (Ω T ) , and v h → v in L p (Ω T ) . (ii) In the above situation, v h has a weak time derivative ∂ t v h on Ω T given by ∂ t v h = h ( v − v h ) , whereas for v h we have ∂ t v h = h ( v h − v ) . (iii) If v ∈ L p (0 , T ; W ,p (Ω)) then v h → v in L p (0 , T ; W ,p (Ω)) as h → . (iv) If v ∈ L p (0 , T ; L p (Ω)) then v h ∈ C ([0 , T ]; L p (Ω)) . The next Lemma provides us with some useful estimates for the quantity b [ v, w ] that wasdefined in (3.1). The proof can be found in [2, Lemma 2.3]. Lemma 3.2.
Let v, w ≥ and β > . Then there exists a constant c depending only on β such that: (i) c (cid:12)(cid:12) w β +12 − v β +12 (cid:12)(cid:12) ≤ b [ v, w ] ≤ c (cid:12)(cid:12) w β +12 − v β +12 (cid:12)(cid:12) (ii) c | w β − v β | ≤ (cid:0) w β − + v β − (cid:1) b [ v, w ] ≤ c | w β − v β | (iii) b [ v, w ] ≤ c | v β − w β | β +1 β Next, we recall a well-known parabolic Sobolev inequality, which can be found for examplein [4]. For the proof, we refer to [17, Lemma 3.2].
Lemma 3.3.
Let z o = ( x o , t o ) ∈ R n +1 and θ > . Suppose that q > , p > . Then forevery u ∈ L ∞ ( t o − θ, t o ; L q ( B r ( x o ))) ∩ L p ( t o − θ, t o ; W ,p ( B r ( x o ))) we have ¨ Q r,θ ( z o ) | u | p (1+ qn ) d x d t ≤ c (cid:18) ess sup t ∈ ( t o − θ,t o ) ˆ B r ( x o ) ×{ t } | u | q d x (cid:19) pn ¨ Q r,θ ( z o ) |∇ u | p d x d t for a constant c = c ( n, p, q ) . The following lemma can be proven using an inductive argument, see for example [11,Lemma 7.1].
Lemma 3.4.
Let ( Y j ) ∞ j =0 be a positive sequence such that Y j +1 ≤ Cb j Y δj , where C, b > and δ > . If Y ≤ C − δ b − δ , then ( Y j ) converges to zero as j → ∞ . A form of the following lemma was originally proven by De Giorgi [3], see also [4].
EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 5
Lemma 3.5.
Let v ∈ W , ( B ρ ( x o )) for some ρ > and x o ∈ R n . Let k and l be realnumbers such that k < l . Then there exists a constant c depending only on n (and thusindependent of k, l, v, x o and ρ ) such that for any representative of v , we have ( l − k ) |{ x ∈ B ρ ( x o ) : v ( x ) > l }| ≤ cρ n +1 |{ x ∈ B ρ ( x o ) : v ( x ) < k }| ˆ { k Lemma 3.6. Let < p < and suppose that v ∈ L p (0 , T ; W ,p (Ω)) ∩ L ∞ (Ω T ) is a weaksolution to the equation ∂ t v − ∇ · (cid:0) ˜ A ( x, t, v, ∇ v ) (cid:1) = 0 , where ˜ A satisfies the structure conditions | ˜ A ( x, t, v, ξ ) | ≤ ˜ C | ξ | p − ˜ A ( x, t, v, ξ ) · ξ ≥ ˜ C | ξ | p . Then v is locally H¨older continuous in Ω T and there are constants c > and ν ∈ (0 , depending only on n, p, ˜ C , ˜ C such that for any subset K ⊂ Ω T , compactly contained in Ω × (0 , T ] , we have for all ( x, t ) , ( y, s ) ∈ K that | v ( x, t ) − v ( y, s ) | ≤ c k v k L ∞ (Ω T ) (cid:18) k v k − pp L ∞ (Ω T ) | x − y | + | t − s | p d p ( K ) (cid:19) ν , where d p ( K ) := inf ( x,t ) ∈ K ( y,s ) ∈ ∂ p Ω T (cid:16) k v k − pp L ∞ (Ω T ) | x − y | + | t − s | p (cid:17) . The next lemma shows that weak solutions to (2.4) which are bounded from below andabove by positive constants are in fact also solutions to an equation of parabolic p -Laplacetype (in the case M = 1). It also investigates how solutions are affected by re-scaling. Lemma 3.7. Let A satisfy the structure conditions (2.5) and (2.6) and suppose that u is aweak solution to (2.4) in the cylinder B R ( x o ) × (0 , M − m − p τ ) . Suppose furthermore that β M ≤ u ≤ β M, (3.4) for some positive constants β , β . Then the function v ( x, t ) = M − u ( x, M − m − p t ) , ( x, t ) ∈ B R ( x o ) × (0 , τ ) , has a weak p -integrable gradient, and is a weak solution in B R ( x o ) × (0 , τ ) to the equation ∂ t v − ∇ · (cid:0) ˜ A ( x, t, ∇ v ) (cid:1) = 0 , (3.5) where ˜ A ( x, t, ξ ) := M − m − p A (cid:0) x, M − m − p t, M v ( x, t ) , βM β v β − ( x, t ) ξ (cid:1) . The vector field ˜ A satisfies the structure conditions | ˜ A ( x, t, ξ ) | ≤ C β p − β ( β − p − | ξ | p − ˜ A ( x, t, ξ ) · ξ ≥ C β p − β ( β − p − | ξ | p , where C and C are the constants appearing in the structure conditions (2.5) and (2.6) . V. VESPRI AND M. VESTBERG Proof. The bounds on u show that the chain rule holds in the following form: ∇ u = ∇ ( u β ) β = β − u − β ∇ u β . (3.6)Note especially that the lower bound on u guarantees that u − β stays bounded despitethe negative exponent. From these observations it follows that also v has a weak gradientwhich is p -integrable. By a change of variables in the time variable in the weak formulation(2.8), and by taking note of (3.6), one can see that v satisfies (3.5) weakly. The structureconditions for ˜ A follow from the corresponding conditions satisfied by A , and the bounds(3.4). (cid:3) Continuity in time and mollified weak formulation. In this subsection we showthat weak solutions are continuous in time as maps into L β +1loc (Ω). The proof is adaptedfrom [20]. We start with a lemma. Lemma 3.8. Suppose that u is a weak solution in the sense of Definition 2.1 and define V := (cid:8) w ∈ L β +1 (Ω T ) | w β ∈ L p (0 , T ; W ,p (Ω)) , ∂ t w β ∈ L β +1 β (Ω T ) (cid:9) . Then, for every ζ ∈ C ∞ (Ω T , R ≥ ) and w ∈ V we have ¨ Ω T ∂ t ζ b [ u, w ] d x d t = ¨ Ω T A ( x, t, u, ∇ u β ) · ∇ [ ζ ( u β − w β )] + ζ∂ t w β ( u − w ) d x d t. (3.7) Proof. Let w ∈ V , ζ ∈ C ∞ (Ω T , R ≥ ) and choose ϕ = ζ (cid:0) w β − [ u β ] h (cid:1) as test function in (2.8). Our goal is to pass to the limit h → 0. It follows from Lemma 3.1(iii) that ¨ Ω T A ( x, t, u, ∇ u β ) · ∇ ϕ d t d t −−−→ h → ¨ Ω T A ( x, t, u, ∇ u β ) · ∇ [ ζ ( w β − u β )] d x d t. Note that Lemma 3.1 (ii) implies (cid:0) [ u β ] β h − u (cid:1) ∂ t [ u β ] h ≤ , which shows that we can treat the parabolic part as follows. ¨ Ω T u∂ t ϕ d x d t = ¨ Ω T ζu∂ t w β d x d t − ¨ Ω T ζ [ u β ] β h ∂ t [ u β ] h d x d t + ¨ Ω T ζ (cid:0) [ u β ] β h − u (cid:1) ∂ t [ u β ] h d x d t + ¨ Ω T ∂ t ζu (cid:0) w β − [ u β ] h (cid:1) d x d t ≤ ¨ Ω T ζu∂ t w β d x d t + ¨ Ω T ββ +1 ∂ t ζ [ u β ] β +1 β h d x d t + ¨ Ω T ∂ t ζu (cid:0) w β − [ u β ] h (cid:1) d x d t −−−→ h → ¨ Ω T ζu∂ t w β d x d t + ¨ Ω T ∂ t ζ (cid:0) ββ +1 u β +1 + u ( w β − u β ) (cid:1) d x d t = ¨ Ω T ζ∂ t w β ( u − w ) d x d t − ¨ Ω T ∂ t ζ b [ u, w ] d x d t, This shows “ ≤ ” in (3.7). The reverse inequality can be derived in the same way by taking ϕ = ζ (cid:0) w β − [ v β ] h (cid:1) as test function. (cid:3) EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 7 Theorem 3.9. Let u be a weak solution in the sense of Definition 2.1. Then u ∈ C ([0 , T ]; L β +1 loc (Ω)) . Proof. We prove continuity on the interval [0 , T ] and describe later how the argumentcan be modified to show continuity also on [ T, T ], thus completing the proof. We first notethat due to Lemma 3.1, w := ([ u β ] ¯ h ) β belongs to the set of admissible comparison functions V of Lemma 3.8. Furthermore, since Lemma 3.1 (iv) guarantees that w β is continuous[0 , T ] → L β +1 β (Ω) and since | w ( x, s ) − w ( x, t ) | β +1 ≤ | w β ( x, s ) − w β ( x, t ) | β +1 β = | [ u β ] ¯ h ( x, s ) − [ u β ] ¯ h ( x, t ) | β +1 β , we see that w is continuous [0 , T ] → L β +1 (Ω). We will show that u is essentially the uniformlimit on the time interval [0 , T ] of the functions w as h → 0, and the continuity will followfrom this. For a compact set K ⊂ Ω we take η ∈ C ∞ (Ω; [0 , η = 1 on K and |∇ η | ≤ C K . Furthermore, take ψ ∈ C ∞ ([0 , T ]; [0 , ψ = 1 on [ T, T ], ψ = 0 on [ T, T ]and | ψ ′ | ≤ T . For τ ∈ (0 , T ) and ε > τ + ε < T we define χ τε ( t ) = , t < τε − ( t − τ ) , t ∈ [ τ, τ + ε ]1 , t > τ + ε. We use (3.7) with ζ = ηχ τε ψ and w = ([ u β ] ¯ h ) β to obtain ε − ˆ τ + ετ ˆ Ω b [ u, w ] η d x d t = ¨ Ω T A ( x, t, u, ∇ u β ) · ∇ [ η ( u β − w β )] χ τε ψ d x d t + ¨ Ω T ηχ τε ψ∂ t w β ( u − w ) d x d t − ¨ Ω T b [ u, w ] ηψ ′ d x d t ≤ ¨ Ω T | A ( x, t, u, ∇ u β ) | ( |∇ u β − ∇ [ u β ] ¯ h | + |∇ η || u β − [ u β ] ¯ h | ) d x d t + 8 T ¨ Ω T b [ u, w ] d x d t. Here we were able to drop the term involving ∂ t w β since Lemma 3.1 (ii) shows that thefactors ∂ t w β and ( u − w ) are of opposite sign, and hence their product is nonpositive.Passing to the limit ε → ˆ K b [ u, w ]( x, τ ) d x ≤ C K ¨ Ω T | A ( x, t, u, ∇ u β ) | ( |∇ u β − ∇ [ u β ] ¯ h | + | u β − [ u β ] ¯ h | ) d x d t (3.8) + 8 T ¨ Ω T b [ u, w ] d x d t for all τ ∈ [0 , T ] \ N h , where N h is a set of measure zero. Note that the integrand on theleft-hand side can be estimated using Lemma 3.2 (ii) and the fact that β > | u − w | β +1 = ( | u − w | β +12 ) ≤ (cid:12)(cid:12) u β +12 − w β +12 (cid:12)(cid:12) ≤ c b [ u, w ] . For the term on the last line of (3.8) we can use Lemma 3.2 (iii) to make the estimate b [ u, w ] ≤ c | u β − [ u β ] ¯ h (cid:12)(cid:12) β +1 β = c | u β − [ u β ] ¯ h (cid:12)(cid:12) β | u β − [ u β ] ¯ h (cid:12)(cid:12) ≤ c ( u + ([ u β ] ¯ h ) β ) | u β − [ u β ] ¯ h | . The first factor stays bounded in L β +1 as h → L β +1 β as h → 0. The fact that | A ( u, ∇ u β ) | ∈ L p ′ (Ω T ) combined with Lemma 3.1 (iii)show that also the first integral on the right-hand side of (3.8) converges to zero as h → V. VESPRI AND M. VESTBERG Picking now a sequence h j → w j = ([ u β ] ¯ h j ) β and N := ∪ N h j (which has measurezero) we see that (3.8) combined with the previous observations implieslim j →∞ sup τ ∈ [0 , T ] \ N ˆ K | u − w j | β +1 ( x, τ ) d x = 0 . (3.9)As noted earlier, each w j is continuous as a map [0 , T ] → L β +1 ( K ) so the uniform limit (3.9)shows that u has a representative which is continuous on [0 , T ] \ N . By the completenessof L β +1 ( K ) we find a representative of u which is continuous [0 , T ] → L β +1 ( K ). Thecontinuity on [ T, T ] follows from a similar argument with w = ([ u β ] h ) β and with ψ and χ τε mirrored on the interval [0 , T ] under the map t T − t . (cid:3) Now that we have established the continuity in time it is possible to show that weaksolutions in the sense of Definition 2.1 satisfy a mollified weak formulation. Lemma 3.10. Let u be a weak solution in the sense of Definition 2.1. Then we have ¨ Ω T [ A ( x, · , u, ∇ u β )] h · ∇ φ + ∂ t u h φ d x d t − ˆ Ω u ( x, φ ¯ h ( x, 0) d x = 0(3.10) for all φ ∈ C ∞ (Ω × [0 , T ]) with support contained in K × [0 , τ ] ,where K ⊂ Ω is compactand τ ∈ (0 , T ) . Here u ( x, refers to the value at time zero of the continuous representativeof u as a map [0 , T ] → L β +1 ( K ) . Proof. Consider the piecewise smooth function η ε ( t ) := ( tε , t ∈ [0 , ε ]1 , t ∈ ( ε, T ] , and use (2.8) with the test function ϕ = η ε φ ¯ h . Taking the limit ε → A ( x, · , u, ∇ u β )] h · ∇ φ .Note now that ¨ Ω T u∂ t ( η ε φ ¯ h ) d x d t = ¨ Ω T uη ε φ ¯ h − φh d x d t + ε − ˆ ε ˆ Ω uφ ¯ h d x d t. In the first term we can pass to the limit ε → 0, use Fubini’s theorem and Lemma 3.1 (ii)to obtain the integral of ∂ t u h ϕ . It remains to investigate what happens to the last term inthe limit ε → 0. Note that we can write this term as ε − ˆ ε ˆ K uφ ¯ h d x d t = ε − ˆ ε ˆ K u ( x, t ) φ ¯ h (0) d x d t + ε − ˆ ε ˆ K u ( x, t )[ φ ¯ h ( t ) − φ ¯ h (0)] d x d t. The second term on the right-hand side converges to zero since φ ¯ h is uniformly continuousand k u ( t ) k L β +1 ( K ) is bounded independent of t . The first term on the right-hand sideconverges to the second integral on the left-hand side of (3.10) since u ∈ C ([0 , T ]; L β +1 ( K ))and φ ¯ h (0) ∈ L β +1 β (Ω). (cid:3) Energy Estimates Here we discuss various energy estimates. We begin by showing that the assumptionson u made in Definition 2.1 allow suitable choices of test functions in the mollified weakformulation. This is a crucial step in obtaining a rigourous proof for the energy estimates.We want to use test functions involving ( u β − k β ) ± for some k ≥ 0. Since these functionshave a p -integrable gradient, they automatically fit with the elliptic term in (3.10). The EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 9 minimal integrability of u which justifies the test function becomes apparent from the diffu-sive part of the mollified weak formulation: If u ∈ L q then ∂ t u h ∈ L q and ( u β − k β ) ± ∈ L qβ .These exponents should be at least dual exponents so we need1 q + 1 q/β ≤ , which is equivalent to q ≥ β + 1. This is exactly the integrability we required in Definition2.1.We now show the energy estimate for solutions according to Definition 2.1. Lemma 4.1. Let u be a weak solution in the sense of Definition 2.1. Then ¨ Ω T |∇ ( u β − k β ) ± | p ϕ p d x d t + ess sup τ ∈ [0 ,T ] ˆ Ω b [ u, k ] χ { ( u − k ) ± > } ϕ p ( x, τ ) d x (4.1) ≤ C ¨ Ω T ( u β − k β ) p ± |∇ ϕ | p d x d t + ¨ Ω T b [ u, k ] χ { ( u − k ) ± > } | ∂ t ϕ p | d x d t, for all smooth ϕ ≥ defined on ¯Ω T , vanishing for x outside a compact K ⊂ Ω and for alltimes less than some δ > . The constant C only depends on the data. Proof. We prove the case for the positive part. The case for the negative part is similar.We use the mollified weak formulation (3.10) with the test function φ = ( u β − k β ) + ϕ p ξ τ,ε where ϕ is as in the statement of the lemma and ξ τ,ε is defined as ξ τ,ε ( t ) := , t < τ − ε − ( τ − t ) , t ∈ [ τ, τ + ε ]0 , t > τ + ε. (4.2)Even though φ is nonsmooth, it is still an admisssible test function since we can find asequence of functions φ j ∈ C ∞ (Ω T ) converging to φ in L p (0 , T ; W ,p (Ω)) ∩ L β +1 β (Ω T ). Ourgoal is to make some estimates in (3.10) and pass to the limit h → ε → 0. Wefirst show that the term involving the initial value vanishes in this process. Taking intoaccount the support of φ we have ˆ Ω u ( x, φ ¯ h ( x, 0) d x = ¨ Ω T u ( x, h − e − th φ ( x, t ) d x d t = ˆ Tδ ˆ Ω u ( x, h − e − th φ ( x, t ) d x d t ≤ ˆ Tδ ˆ Ω u ( x, δ − ( δh e − δh ) φ ( x, t ) d x d t −−−→ h → , due to the dominated convergence theorem. The elliptic term can be treated using Lemma3.1 (i) as ¨ Ω T [ A ( x, · , u, ∇ u β )] h · ∇ φ d x d t −−−→ h → ¨ Ω T A ( x, t, u, ∇ u β ) · ∇ φ d x d t −−−→ ε → ¨ Ω τ A ( x, t, u, ∇ u β ) · ∇ [( u β − k β ) + ϕ p ] d x d t We now calculate ∇ φ = ϕ p ξ τ,ε χ { u>k } ∇ u β + p ( u β − k β ) + ξ τ,ε ϕ p − ∇ ϕ. From the properties of the vector field, here denoted only A ( u, ∇ u β ) for brevity, and Young’sinequality we obtain A ( u, ∇ u β ) · ∇ [( u β − k β ) + ϕ p ] = A ( u, ∇ u β ) · ∇ u β χ { u>k } ϕ p + A ( u, ∇ u β ) · ∇ ( ϕ p )( u β − k β ) +0 V. VESPRI AND M. VESTBERG ≥ c |∇ u β | p χ { u>k } ϕ p − | A ( u, ∇ u β ) ||∇ ϕ | pϕ p − ( u β − k β ) + ≥ c |∇ u β | p χ { u>k } ϕ p − c |∇ u β | p − ϕ p − ( u β − k β ) + |∇ ϕ |≥ c |∇ u β | p χ { u>k } ϕ p − c ( u β − k β ) p + |∇ ϕ | p . Using Lemma 3.1 (ii) and the fact that s ( s β − k β ) + is increasing we can treat the diffusionterm as ∂ t u h φ = (cid:0) u − u h h (cid:1) [( u β − k β ) + − ([ u h ] β − k β ) + ] ϕ p ξ τ,ε + ∂ t u h ([ u h ] β − k β ) + ϕ p ξ τ,ε ≥ ∂ t G ( u h ) ϕ p ξ τ,ε , where G ( u ) := ˆ u ( s β − k β ) + d s = b [ u, k ] χ { u>k } . (4.3)The chain rule works in our case since Lemma 3.1 guarantees that both u h and ∂ t u h are in L β +1 (Ω T ). Thus, we may estimate ¨ Ω T ∂ t u h φ d x d t ≥ ¨ Ω T ∂ t G ( u h ) ϕ p ξ τ,ε d x d t = − ¨ Ω T G ( u h ) ∂ t ( ϕ p ξ τ,ε ) d x d t −−−→ h → − ¨ Ω T G ( u ) ∂ t ( ϕ p ξ τ,ε ) d x d t = − ¨ Ω T G ( u ) ∂ t ϕ p ξ τ,ε d x d t + ε − ˆ τ + ετ ˆ Ω G ( u ) ϕ p d x d t −−−→ ε → − ¨ Ω τ G ( u ) ∂ t ϕ p d x d t + ˆ Ω G ( u ) ϕ p ( x, τ ) d x, for a.e. τ . Putting together the estimates for the elliptic and diffusion terms we have c ¨ Ω τ |∇ u β | p χ { u>k } ϕ p d x d t + ˆ Ω G ( u ) ϕ p ( x, τ ) d x ≤ c ¨ Ω τ ( u β − k β ) p + |∇ ϕ | p d x d t + ¨ Ω τ G ( u ) | ∂ t ϕ p | d x d t, for a.e. τ . We obtain the desired estimate by using (4.3) and noting that the right-handside can be estimated upwards by replacing τ by T . (cid:3) The following variant of the energy estimate will also be useful. Lemma 4.2. Let ϕ ∈ C ∞ (Ω; R ≥ ) and suppose that [ t , t ] ⊂ (0 , T ) . Then the time-continuous representative of u satisfies c − ˆ t t ˆ Ω |∇ ( u β − k β ) − | p d x d t + ˆ Ω b [ u, k ] χ { u Proof. We use the mollified weak formulation (3.10) with the test function φ = − ( u β − k β ) − ϕ p ( x ) ξ ε ( t ), where ξ ε ( t ) = , t ≤ t ,ε − ( t − t ) , t ∈ ( t , t + ε ) , , t ∈ [ t + ε, t ] ,ε − ( t + ε − t ) , t ∈ ( t , t + ε ) , t ∈ ( t , t + ε ) , , t ≥ t + ε. Reasoning as in the proof of Lemma 4.1 leads to (4.4). (cid:3) L -Harnack inequality In order to obtain the reduction of the oscillation we will use the fact that weak solutionssatisfy a local L -Harnack inequality. Such a result was already obtained in [8, Theorem 5.1]in a quite general setting, allowing for all m > u itself hasa gradient, whereas in our case we only know that u β has a gradient. It turns out that thesame strategy as in [8] works also in our case with some modifications. In this section wepresent the full proof in the case m > Theorem 5.1 (Harnack inequality) . Let u be a nonnegative weak solution to (2.4) where thevector field A ( x, t, u, ξ ) satisfies the structure conditions (2.5) and (2.6) , and the parameterssatisfy the conditions (1.2) . Then there exists a positive constant γ depending only on m, n, p, C , C such that for all cylinders ¯ B ρ ( y ) × [ s, t ] ⊂ Ω × [0 , T ) , ess sup τ ∈ [ s,t ] ˆ B ρ ( y ) u ( x, τ ) d x ≤ γ ess inf τ ∈ [ s,t ] ˆ B ρ ( y ) u ( x, τ ) d x + γ (cid:18) t − sρ λ (cid:19) − m − p , where λ = n ( p + m − 3) + p . Note that λ can have any sign. If we use the time continuous representative of u we canreplace the essential infimum and supremum by the actual infimum and supremum. Beforeproceeding we note that by translation we may assume that s = 0. All of the calculationswill be performed under this assumption, and the time interval [ s, t ] will henceforth belabelled [0 , τ ], where τ ∈ (0 , T ). The first step of the argument is a lemma correspondingto [8, Lemma 5.2]. Lemma 5.2. Let u be a weak solution, τ ∈ (0 , T ) , σ ∈ (0 , and B ρ ( x o ) ⊂ Ω . Then ¨ B σρ ( x o ) × (0 ,τ ) |∇ u β | p ( u β + ε β ) m + p − βp − t p d x d t + ¨ B σρ ( x o ) × (0 ,τ ) F ε ( u ) t p − d x d t (5.1) ≤ cρ (1 − σ ) p (cid:16) τρ λ (cid:17) p h sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x + ερ n i p + m − p , where λ = n ( p + m − 3) + p , ε = ( τρ p ) − m − p and F ε is defined in (5.2) below. The constant c depends only on m, n, p, C , C . Proof. Consider the mollified weak formulation (3.10) with the test function φ ( x, t ) = − ( u β + ε β ) m + p − βp t p ϕ p ( x ) ξ τ,δ ( t ) , where ε > ξ τ,δ is defined as in (4.2) and ϕ ∈ C ∞ ( B ρ ( x o ); [0 , ϕ = 1 on B σρ ( x o ).We may thus choose ϕ such that |∇ ϕ | ≤ − σ ) − ρ − . We have ∇ φ = (3 − m − p ) βp ( u β + ε β ) m + p − βp − t p ϕ p ( x ) ξ τ,δ ( t ) ∇ u β − ( u β + ε β ) m + p − βp t p ξ τ,δ ( t ) ∇ ϕ p ( x ) . We see that ¨ Ω T [ A ( x, · , u, ∇ u β )] h · ∇ φ d x d t −−−→ h → ¨ Ω T A ( x, t, u, ∇ u β ) · ∇ φ d x d t, and A ( x, t, u, ∇ u β ) · ∇ φ = (3 − m − p ) βp ( u β + ε β ) m + p − βp − t p ϕ p ξ τ,δ A ( x, t, u, ∇ u β ) · ∇ u β − p ( u β + ε β ) m + p − βp t p ξ τ,δ ϕ p − A ( x, t, u, ∇ u β ) · ∇ ϕ ≥ c |∇ u β | p ( u β + ε β ) m + p − βp − t p ϕ p ξ τ,δ − c ( u β + ε β ) m + p − βp t p ξ τ,δ ϕ p − |∇ u β | p − |∇ ϕ |≥ ˆ c |∇ u β | p ( u β + ε β ) m + p − βp − t p ϕ p ξ τ,δ − ˆ c ( u β + ε β ) p − m + p − βp t p ξ τ,δ |∇ ϕ | p . Here c , c , ˆ c , ˆ c are constants depending only on m, p, C , C . For the initial value term wenote that (cid:12)(cid:12)(cid:12) ˆ Ω u ( x, φ ¯ h ( x, 0) d x (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ¨ Ω T u ( x, h − e − th φ ( x, t ) d x d t (cid:12)(cid:12)(cid:12) ≤ c ¨ supp ϕ × [0 ,T ] u ( x, th e − th ) t p − d x d t −−−→ h → , by the dominated convergence theorem. The diffusion part is treated as follows: φ∂ t u h = (cid:0)(cid:2) ([ u h ] β + ε β ) m + p − βp − ( u β + ε β ) m + p − βp (cid:3) ( u − u h ) h − ([ u h ] β + ε β ) m + p − βp ∂ t u h (cid:1) t p ϕ p ξ τ,δ ≥ − ([ u h ] β + ε β ) m + p − βp ∂ t u h t p ϕ p ξ τ,δ = − ∂ t [ F ( u h )] t p ϕ p ξ τ,δ , where F ε ( s ) := ˆ s ( t β + ε β ) m + p − βp d t ≤ ˆ s t m + p − p d t = p p + m − s p + m − p . (5.2)From this we see that ¨ Ω T φ∂ t u h φ d x d t ≥ ¨ Ω T F ε ( u h ) ∂ t ( t p ϕ p ξ τ,δ ) d x d t −−−→ h → ¨ Ω T F ε ( u ) ∂ t ( t p ϕ p ξ τ,δ ) d x d t = 1 p ¨ Ω T F ε ( u ) ϕ p t p − ξ τ,δ d x d t − δ − ˆ τ + δτ ˆ Ω F ε ( u ) ϕ p t p d x d t −−−→ δ → p ¨ Ω τ F ε ( u ) ϕ p t p − d x d t − τ p ˆ Ω F ε ( u ) ϕ p ( x, τ ) d x. To conclude the limit in the last term we use the Lipschitz continuity of F and the time-continuity of u . Combining these estimates we have ¨ Ω τ |∇ u β | p ( u β + ε β ) m + p − βp − t p ϕ p d x d t + ¨ Ω τ F ε ( u ) ϕ p t p − d x d t (5.3) EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 13 ≤ c ¨ Ω τ ( u β + ε β ) p − m + p − βp t p |∇ ϕ | p d x d t + cτ p ˆ Ω F ε ( u ) ϕ p ( x, τ ) d x. Taking into account the estimate in (5.2) and the support of ϕ , and applying H¨older’sinequality we see that τ p ˆ Ω F ε ( u ) ϕ p ( x, τ ) d x ≤ cτ p ˆ B ρ ( x o ) u p + m − p ( x, τ ) ϕ p ( x ) d x ≤ τ p h ˆ B ρ ( x o ) u ( x, τ ) d x i p + m − p | B ρ ( x o ) | − m − pp ≤ cτ p h sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x i p + m − p ρ n (3 − m − p ) p = cρ (cid:16) τρ λ (cid:17) p h sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x i p + m − p . Using the bound on the gradient of ϕ we may now estimate the other term on the right-handside of (5.3) as ¨ Ω τ ( u β + ε β ) p − m + p − βp t p |∇ ϕ | p d x d t (5.4) ≤ c (1 − σ ) p ρ p ¨ B ρ ( x o ) × (0 ,τ ) ( u β + ε β ) m + p − β ( u β + ε β ) p + m − βp t p d x d t ≤ c (1 − σ ) p ρ p ε m + p − ˆ τ ˆ B ρ ( x o ) ( u β + ε β ) p + m − βp d x t p d t ≤ c (1 − σ ) p ρ p ε m + p − τ p +1 sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) ( u β + ε β ) p + m − βp ( x, t ) d x, In the second step we use the fact that the exponent β − ( m + p − 3) is negative. In the laststep we estimate the integral over the ball by the supremum in time of such integrals, leavingonly an integral in time of the factor t p . The integral appearing in the last expression maybe estimated further using H¨older’s inequality and the definition of λ as ˆ B ρ ( x o ) ( u β + ε β ) p + m − βp d x ≤ h ˆ B ρ ( x o ) ( u β + ε β ) β d x i p + m − p | B ρ ( x o ) | − m − pp (5.5) ≤ c h ˆ B ρ ( x o ) u d x + ερ n i p + m − p ρ − λp . Since the exponent p − (2 p + m − 3) is positive, we can combine (5.4) and (5.5) taking thesupremum inside the square brackets to obtain ¨ Ω τ ( u β + ε β ) p − m + p − βp t p |∇ ϕ | p d x d t ≤ cρ (1 − σ ) p (cid:16) τρ p (cid:17) ε m + p − (cid:16) τρ λ (cid:17) p × h sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x + ερ n i p + m − p Combining the estimate for the two terms on the right-hand side of (5.3) we end up with ¨ Ω τ |∇ u β | p ( u β + ε β ) m + p − βp − t p ϕ p d x d t + ¨ Ω τ F ε ( u ) ϕ p t p − d x d t (5.6) ≤ cρ (1 − σ ) p (cid:16) τρ λ (cid:17) p h ε m + p − (cid:16) τρ p (cid:17)ih sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x + ερ n i p + m − p . Choosing now ε = ( τρ p ) − m − p confirms (5.1). (cid:3) Because of the somewhat more complicated calculations in our setting, we also need thefollowing result, which does not appear in [8]. Lemma 5.3. Let F ε be defined by (5.2) and let ε > . Then there is a constant c = c ( m, p ) such that ( u β + ε β ) [ − m − pβp +1]( p − ≤ cF ε ( u ) + cε m + p − p +1 , (5.7) for all u ≥ . Proof. Assume first u > ε . Then since m + p − < F ε ( u ) = ˆ u ( t β + ε β ) m + p − βp d t ≥ ˆ uε ( t β + ε β ) m + p − βp d t ≥ ˆ uε (2 t β ) m + p − βp d t = c (cid:0) u m + p − p +1 − ε m + p − p +1 (cid:1) ≥ ˜ cu m + p − p +1 , where in the last step we used the assumption u > ε and the fact that the exponent of ε ispositive. On the other hand, since u > ε we also have( u β + ε β ) [ − m − pβp +1]( p − ≤ cu [ − m − pp + β ]( p − = cu m + p − p +1 , and combining the two estimates we have verified the claim in the case u > ε . Supposenow u ≤ ε . Then F ε ( u ) = ˆ u ( t β + ε β ) m + p − βp d t ≥ ˆ u ((1 + 2 β ) ε β ) m + p − βp d t = cε m + p − p u ≥ cu m + p − p +1 = c ( u β ) [ − m − pβp +1]( p − ≥ c ( u β + ε β ) [ − m − pβp +1]( p − − c ε m + p − p +1 , where in the last step we used the fact that for positive α and nonnegative a, b we have a α ≥ − α ( a + b ) α − b α . Thus, we have verified the claim also in the case u ≤ ε . (cid:3) The next lemma corresponds to [8, Lemma 5.3]. A formal application of the chain ruleshows that the the integrands on the left-hand side in both lemmas are essentially the same,although in our case the gradient of u need not exist. The proof in our case is somewhatmore complicated as we need also to use Lemma 5.3. Lemma 5.4. Let u be a weak solution and δ ∈ (0 , . Then there is a constant c dependingonly on m, n, p, C , C such that ρ ˆ τ ˆ B σρ ( x o ) |∇ u β | p − d x d t ≤ δ sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x + cδ − p − m − m − p (1 − σ ) p − m − p (cid:16) τρ λ (cid:17) − m − p . Proof. Choose ε as in Lemma 5.2. By H¨older’s inequality and the previous lemma, we have ˆ τ ˆ B σρ ( x o ) |∇ u β | p − d x d t (5.8)= ˆ τ ˆ B σρ ( x o ) h |∇ u β | p − ( u β + ε β ) [ m + p − βp − ( p − p t p − p ih ( u β + ε β ) [ − m − pβp +1] ( p − p t − pp i d x d t ≤ h ˆ τ ˆ B σρ ( x o ) |∇ u β | p ( u β + ε β ) m + p − βp − t p d x d t i p − p × h ˆ τ ˆ B σρ ( x o ) ( u β + ε β ) [ − m − pβp +1]( p − t − pp d x d t i p . EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 15 The second integral in the last expression can be estimated by combining (5.7) and (5.1): ˆ τ ˆ B σρ ( x o ) ( u β + ε β ) [ − m − pβp +1]( p − t − pp d x d t ≤ c ˆ τ ˆ B σρ ( x o ) ( F ε ( u ) + ε m + p − p +1 ) t − pp d x d t ≤ c ˆ τ ˆ B σρ ( x o ) F ε ( u ) t − pp d x d t + cρ n ε m + p − p +1 τ p ≤ cρ (1 − σ ) p (cid:16) τρ λ (cid:17) p h sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x + ερ n i p + m − p + cρ (cid:16) τρ λ (cid:17) p ( ερ n ) p + m − p ≤ cρ (1 − σ ) p (cid:16) τρ λ (cid:17) p h sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x + ερ n i p + m − p . Since also the other integral appearing in the last expression of (5.8) can be estimated using(5.1), we have ˆ τ ˆ B σρ ( x o ) |∇ u β | p − d x d t ≤ cρ (1 − σ ) p (cid:16) τρ λ (cid:17) p h sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x + ερ n i p + m − p . Dividing by ρ and applying Young’s inequality to the right-hand side yields the claim. (cid:3) Now we can finally prove the Harnack inequality. Proof of Theorem 5.1. . For j ∈ N we choose ρ j := 2(1 − − j ) ρ, ˜ ρ j := 12 ( ρ j + ρ j +1 ) B j := B ρ j ( x o ) , ˜ B j := B ˜ ρ j ( x o )Pick ζ j ∈ C ∞ ( B ˜ ρ j ( x o ); [0 , ζ j = 1 on B ρ j ( x o ) and We use the weak formulation(2.8) with the test function ϕ = ζ j ξ rτ ,τ where r > τ < τ < τ and ξ rτ ,τ ( t ) = , t < τ ,r − ( t − τ ) , t ∈ [ τ , τ + r ]1 , t ∈ ( τ + r, τ ) r − ( τ + r − t ) , t ∈ [ τ , τ + r ] , , t > τ + r. This implies1 r ˆ τ τ ˆ Ω uζ j d x d t = ¨ Ω T A ( u, ∇ u β ) · ∇ ζ j ξ rτ ,τ d x d t + 1 r ˆ τ τ ˆ Ω uζ j d x d t. Passing to the limit r → ζ j we have ˆ B j u ( x, τ ) d x ≤ ˆ Ω uζ j ( x, τ ) d x = ˆ τ τ ˆ Ω A ( u, ∇ u β ) · ∇ ζ j d x d t + ˆ Ω uζ j ( x, τ ) d x (5.9) ≤ ˆ τ τ ˆ Ω | A ( u, ∇ u β ) ||∇ ζ j | d x d t + ˆ Ω uζ j ( x, τ ) d x ≤ c j ρ ˆ τ ˆ ˜ B j |∇ u β | p − d x d t + ˆ B j +1 u ( x, τ ) d x, for all τ , τ due to the time-continuity of u . Although we assumed τ < τ , we see bya similar calculation that the estimate remains valid for τ ≥ τ . We want to estimate the double integral in the last expression using Lemma 5.4 with ρ replaced by ρ j +1 , andconsequently with σ := ˜ ρ j /ρ j +1 . Directly from the definition it follows that11 − σ < j +2 . Taking this into account, Lemma 5.4 shows that ˆ B j u ( x, τ ) d x ≤ c j δ sup t ∈ [0 ,τ ] ˆ B j +1 u ( x, t ) d x + c jp − m − p + j δ p + m − − m − p (cid:16) τρ λ (cid:17) − m − p + ˆ B j +1 u ( x, τ ) d x, for all δ ∈ (0 , ρ j )are comparable in size to ρ . Taking now δ = c − − ε o where ε o ∈ (0 , 1) and c ≥ ˆ B j u ( x, τ ) d x ≤ ε o sup t ∈ [0 ,τ ] ˆ B j +1 u ( x, t ) d x + cb j (cid:16) τρ λ (cid:17) − m − p + ˆ B ρ ( x o ) u ( x, τ ) d x, where b = b ( m, n, p, C , C ) and c = c ( m, n, p, C , C , ε o ). We also used the fact that B j +1 ⊂ B ρ ( x o ). Recalling that the inequality holds for a.e. τ , τ ∈ (0 , τ ) we see that itimplies S j ≤ ε o S j +1 + cb j (cid:16) τρ λ (cid:17) − m − p + I, where S j := sup t ∈ [0 ,τ ] ˆ B j u ( x, t ) d x, I := inf t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x. (5.10)Iterating (5.10) we havesup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x = S ≤ ε Mo S M +1 + cb (cid:16) τρ λ (cid:17) − m − p M − X j =0 ( bε o ) j + I M − X j =0 ε jo . (5.11)choose now for example ε o = b so that both of the sums in (5.11) converge in the limit M → ∞ . Then, since S M +1 ≤ sup t ∈ [0 ,τ ] ˆ B ρ ( x o ) u ( x, t ) d x, where the right-hand side finite due to the time-continuity of u , we see that we can pass tothe limit M → ∞ which yields the claim. (cid:3) Expansion of Positivity In this section we show that weak solutions exhibit expansion of positivity. This typeof result was already obtained in [9], but the calculations were made under the assumptionthat u has a gradient, which is not necessarily true in our case. We demonstrate that thesame strategy as in [9] can nevertheless be applied with some modifications. For the reader’sconvenience detailed proofs are provided. We start with a lemma corresponding to Lemma3.1 of [8]. Lemma 6.1 (General De Giorgi type lemma) . Suppose that v : Ω T → R ≥ satisfies v β ∈ L p (0 , T ; W ,p (Ω)) and the energy estimate c g ¨ Ω T |∇ v β | p ϕ p χ { v Define ρ j := ρ ρ j +1 , k βj := (cid:16) a β + (1 − a β )2 j (cid:17) K β , B j := B ρ j ( y o ) , T j := ( t o − θρ pj , t o ) ,Q j := B j × T j = Q ρ j ,θρ pj ( y o , t o ) , A j := Q j ∩ { v < k j } , Y j := | A j | / | Q j | . Pick ϕ j ∈ C ∞ ( Q j ; [0 , ϕ j = 1 on Q j +1 and ϕ j = 0 in a neighborhood of ∂ p Q j ,and |∇ ϕ j | ≤ j +3 ρ − , | ∂ t ϕ | ≤ c p θ − j ρ − p . In the set where v < k j +1 we have( v β − k βj ) − ≥ k βj − k βj +1 = (1 − a β )2 j +1 K β , so (1 − a β ) p ( j +1) p K βp | A j +1 | ≤ ¨ A j +1 ( v β − k βj ) p − d x d t (6.3) ≤ (cid:16) ¨ A j +1 ( v β − k βj ) p ( n + p ) n − d x d t (cid:17) nn + p | A j +1 | pn + p . We treat the integral inside the brackets by applying H¨older’s inequality to the integral overthe space variables. One of the resulting integrals is then estimated by taking the essentialsupremum over the time interval, and the Gagliardo-Nirenberg inequality provides an upperbound for the other integral. All in all, we have ¨ A j +1 ( v β − k βj ) p ( n + p ) n − d x d t = ˆ T j +1 ˆ B j +1 ( v β − k βj ) p pn − χ A j +1 ( v β − k βj ) p − d x d t ≤ ˆ T j +1 h ˆ B j +1 ( v β − k βj ) p − χ A j +1 d x i pn h ˆ B j +1 ( v β − k βj ) p ∗ − d x i pp ∗ d t ≤ h ess sup T j +1 ˆ B j +1 ( v β − k βj ) p − χ A j +1 d x i pn ˆ T j h ˆ B j (cid:0) ( v β − k βj ) − ϕ j (cid:1) p ∗ d x i pp ∗ d t ≤ c h ess sup T j +1 ˆ B j +1 ( v β − k βj ) p − − χ A j +1 ( v β − k βj ) − d x i pn ¨ Q j |∇ (cid:0) ( v β − k βj ) − ϕ j (cid:1) | p d x d t ≤ c (1 − a β ) ( p − pn K β ( p − pn j (2 − p ) pn k ( β − pn j h ess sup T j +1 ˆ B j +1 k − βj ( v β − k βj ) − d x i pn × ¨ Q j |∇ (cid:0) ( v β − k βj ) − ϕ j (cid:1) | p d x d t ≤ c (1 − a β ) ( p − pn K pn ( m + p − j (2 − p ) pn h ess sup T j ˆ B j b [ v, k j ] χ { v 0, which means that v ≥ aK in Q ρ ,θ ( ρ ) p ( z o ). (cid:3) The following variant of the De Giorgi lemma will also be useful. The extra assumption(6.4), regarding the values of u at the initial time of the space-time cylinder, allows us toget a lower bound which holds on a cylinder which has only been reduced in the spatialdimensions. It is understood that we consider the time-continuous representative of u , sothat (6.4) makes sense. Lemma 6.2 (Variant of the general De Giorgi type lemma) . Let u be a weak solution inthe sense of Definition 2.1. Suppose that Q ρ,θρ p ( z o ) ⊂ Ω T and that u ( x, t o − θρ p ) ≥ K, (6.4) for a.e. x ∈ B ρ ( x o ) . Then there is a constant c depending only on m, n, p, C , C such thatif | Q ρ,θρ p ( z o ) ∩ { u < K }| ≤ c (1 − a β ) n +2 θK m + p − | Q ρ,θρ p ( z o ) | , (6.5) then u ≥ aK a.e. in Q ρ ,θρ p ( z o ) . Proof. Define k j , ρ j and B j as in Lemma 6.1, but choose Q j := B j × ∆ = B j × ( t o − θρ p , t o ) = Q ρ j ,θρ p . EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 19 As before, we denote A j = Q j ∩ { u < k j } and Y j = | A j | / | Q j | . Choose ϕ j ∈ C ∞ ( B j ; [0 , ϕ j = 1 on B j +1 and |∇ ϕ j | ≤ ρ − j +3 . We use the energy estimate (4.4) of Lemma 4.2 with ϕ = ϕ j , k = k j , t = t o − θρ p and t ∈ ∆. The assumption (6.4) guarantees that the second term on the right-hand side of(4.4) vanishes and we end up with ¨ Q j |∇ ( u β − k βj ) − | p ϕ pj d x d t + ess sup ∆ ˆ B j b [ u, k j ] χ { u Let u be a weak solution on Ω T . Suppose that B ρ ( y ) × { s } ⊂ Ω T and that | B ρ ( y ) ∩ { u ( · , s ) ≥ M }| ≥ α | B ρ ( y ) | . (6.7) Then there are δ = δ ( m, n, p, C , C , α ) and ǫ = ǫ ( α ) such that | B ρ ( y ) ∩ { u ( · , t ) ≥ ǫM }| ≥ α | B ρ ( y ) | , (6.8) for all t ∈ ( s, min { T, s + δM − m − p ρ p } ) . Proof. Let τ < min { T, s + δM − m − p ρ p } , where δ is a positive number which is yet to bechosen and consider (4.4) of Lemma 4.2 with t = s , t = τ and k = M . Discarding the firstterm on the left-hand side, which is non-negative we end up with ˆ Ω (cid:2) b [ u, M ] χ { u 1) we can estimate the last integral as ˆ B ρ ( y ) b [ u, M ] χ { u Combining all the estimates we have | B ρ ( y ) ∩ { u ( · , τ ) < ǫM }| ≤ | B ρ ( y ) | (1 − ǫ ) [˜ c βn σ + (1 − α ) + cσ − p δ ] . (6.11)where ˜ c βn = ˜ c βn ( β, n ) and c = c ( m, n, p, C , C ). Choose σ = σ ( α, n, m, p ) so small that˜ c βn σ < α/ 8. With this choice of σ , choose δ = δ ( m, n, p, C , C , α ) so small that cσ − p δ < α/ c denotes the constant in (6.11). This leads to | B ρ ( y ) ∩ { u ( · , τ ) < ǫM }| ≤ | B ρ ( y ) | (1 − ǫ ) (1 − α/ . From this it follows that (6.8) is true for any0 < ǫ ≤ α − α ) . (cid:3) We are now ready to prove the main result of this section. Theorem 6.4 (Expansion of Positivity) . Suppose that ( x o , s ) ∈ Ω T and u is a weak solutionsatisfying | B ρ ( x o ) ∩ { u ( · , s ) ≥ M }| ≥ α | B ρ ( x o ) | , (6.12) for some M > and α ∈ (0 , . Then there exist ε, δ, η ∈ (0 , depending only on m, p, n, C , C , α such that if B ρ ( x o ) × ( s, s + δM − m − p ρ p ) ⊂ Ω T then u ≥ ηM in B ρ ( x o ) × ( s + (1 − ε ) δM − m − p ρ p , s + δM − m − p ρ p ) . Proof. The proof is divided into several steps. Step 1: Change of variables, transformed equation and energy estimates. Let δ = δ ( m, n, p, C , C , α ) ∈ (0 , 1) be the constant from Lemma (6.3). By translation we mayassume that ( y, s ) = (¯0 , B ρ (¯0) × (0 , δM − m − p ρ p ) ⊂ Ω T ,since otherwise there is nothing to prove. Introduce the new variables ( y, τ ) defined by theequations y = xρ , − e − τ = t − δM − m − p ρ p δM − m − p ρ p . These coordinates transform the cylinder B ρ (¯0) × (0 , δM − m − p ρ p ) into B (¯0) × (0 , ∞ ),preserving the direction of time. Define the function v : B (¯0) × (0 , ∞ ) → R , v ( y, τ ) = e τ − m − p M u ( x, t ) = e τ − m − p M u (cid:0) ρy, δM − m − p ρ p (1 − e − τ ) (cid:1) . A routine calculation confirms that v β ∈ L p (0 , S ; W ,p ( B (¯0))), for all S > 0, and that v isa weak solution to the equation ∂ τ v − ∇ · ˜ A ( y, τ, v, ∇ v β ) = − m − p v, where˜ A ( y, τ, v, ξ ) = δρ p − e ( m + p − − m − p ) τ M m + p − A (cid:0) ρy, δM − m − p ρ p (1 − e − τ ) , M e − τ − m − p v, ρ − M β e − βτ − m − p ξ (cid:1) satisfies the structure conditions˜ A ( y, τ, v, ξ ) · ξ ≥ δC | ξ | p , (6.13) | ˜ A ( y, τ, v, ξ ) | ≤ δC | ξ | p − , (6.14) where C and C are the constants appearing in the structure conditions (2.5) and (2.6).The time continuity of u obtained in Subsection 3.3 implies that v ∈ C ([0 , ∞ ); L β +1loc ( B (¯0)).This allows us to reason as in the proof of Lemma 3.10, to conclude that v satisfies themollified weak formulation ˆ ∞ ˆ B (¯0) [ ˜ A ( y, · , v, ∇ v β )] h · ∇ φ + ∂ τ v h φ d y d τ − ˆ B (¯0) ( vφ ˜ h )( y, 0) d y (6.15) = − m − p ˆ ∞ ˆ B (¯0) v h φ d y d τ, for all φ ∈ C ∞ ( B (¯0) × (0 , ∞ )). The only difference is that we have replaced φ ¯ h by φ ˜ h ( y, τ ) := 1 h ˆ ∞ τ e τ − sh φ ( y, s ) d s, which in practice always can be written as a finite integral due to the support of φ . Thisenables us to prove an energy estimate for v . Namely, we use (6.15) with the test function φ = − ( v β − k β ) − ϕ p ξ r ( τ ), where ϕ is a smooth function vanishing near ∂ p ( B (¯0) × (0 , ∞ )),and ξ r ( τ ) = , τ ≤ ˜ τ ,r − (˜ τ + r − t ) , τ ∈ [˜ τ , ˜ τ + r ] , , τ > ˜ τ + r. Here ˜ τ > 0. We see that ˆ ∞ ˆ B (¯0) [ ˜ A ( y, τ, v, ∇ v β )] h · ∇ φ d y d τ −−−→ h → ˆ ∞ ˆ B (¯0) ˜ A ( y, τ, v, ∇ v β ) · ∇ φ d y d τ −−−→ r → − ˆ ˜ τ ˆ B (¯0) ˜ A ( y, τ, v, ∇ v β ) · ∇ [( v β − k β ) − ϕ p ] d y d τ = ˆ ˜ τ ˆ B (¯0) ˜ A ( y, τ, v, ∇ v β ) · ∇ v β χ { v 0. Combining the estimates for all terms we end up with δ C ˆ ˜ τ ˆ B (¯0) |∇ v β | p ϕ p χ { v 1) we finally obtain the desired energy estimate cδ ˆ ˜ τ ˆ B (¯0) |∇ v β | p ϕ p χ { v From the assumption (6.12) and Lemma6.3 it follows that there is an ǫ = ǫ ( α ) such that | B (¯0) ∩ { v ( · , τ ) > ǫe τ − m − p }| ≥ α | B (¯0) | , (6.18)for all τ ∈ [0 , ∞ ). Pick τ o > k o := ǫe τo − m − p , k j := k o (2 β ) j , j ∈ N . (6.19)With these definitions, (6.18) implies that | B (¯0) ∩ { v ( · , τ ) > k j }| ≥ α − n | B (¯0) | , (6.20)for all τ ∈ [ τ o , ∞ ) and j ∈ N . We introduce the cylinders Q τ o := B (¯0) × ( τ o + k − m − po , τ o + 2 k − m − po ) , Q ′ τ o := B (¯0) × ( τ o , τ o + 2 k − m − po )Pick ζ ∈ C ∞ ( B (¯0)) such that ζ = 1 on B (¯0) and |∇ ζ | ≤ . Pick ζ ∈ C ∞ ( R ) suchthat ζ ( τ ) = 0 for τ < τ o , ζ ( τ ) = 1 for τ ≥ τ o + k − m − po and 0 ≤ ζ ′ ( τ ) ≤ k − m − po . Usingthe energy estimate (6.17) with ϕ ( y, τ ) = ζ ( y ) ζ ( τ ), k = k j and τ = τ o + 2 k − m − po yields ¨ Q τo |∇ v β | p χ { v For i belonging to { , . . . (2 j ∗ ) − m − pβ − } We definethe subcylinders Q i = B (¯0) × (cid:0) τ o + k − m − po + ik − m − pj ∗ , τ o + k − m − po + ( i + 1) k − m − pj ∗ (cid:1) , which is a parition of Q τ o (discarding only a set of measure zero). Thus, (6.22) implies thatfor at least one of the subcylinders we must have | Q i ∩ { v < k j ∗ }| ≤ ν | Q i | . Since v satisfies the energy estimates (6.17), we may apply Lemma 6.1 to Q i with ρ = 8, θ = 8 − p k − m − pj ∗ , K = k j ∗ and a = . Now c g = cδ for a c only depending on m, n, p, C , C and also c e only depends on these parameters. Plugging in everything into (6.2) we see thatthere is a constant c depending only on m, n, p, C , C , such that if ν ≤ ν o := cδ np then v ≥ k j ∗ in B (¯0) × (cid:0) τ o + k − m − po + ( i + 1 − − p ) k − m − pj ∗ , τ o + k − m − po + ( i + 1) k − m − pj ∗ (cid:1) . (6.23) EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 25 Fixing j := j ( m, n, p, C , C , α, ν o ), we obtain by the definitions of ν o and δ that thecorresponding j ∗ ultimately depends only on m, n, p, C , C , α , and that (6.23) is indeedvalid. Hence, there is a τ ∈ ( τ o + k − m − po , τ o + 2 k − m − po ) such that for a.e. y ∈ B (¯0), v ( y, τ ) ≥ k j ∗ = k o j ∗ β +1 = ǫ j ∗ β +1 e τo − m − p = σ o e τo − m − p , (6.24)where σ o = σ o ( m, n, p, C , C , α ). Step 4: Returning to the original coordinates. By the definition of v , (6.24) says thatfor a.e. x ∈ B ρ (¯0) u ( x, t ) ≥ σ o M e τo − τ − m − p =: M o , where t := δM − m − p ρ p (1 − e − τ ). We want to apply Lemma 6.2 with K = M o , a = and θ = c − n − M − m − po , where c is the constant from the assumption (6.5). With thesechoices the assumption in Lemma 6.2 is automatically true since it becomes the statement | Q ∩ { u < M o }| ≤ | Q | for a certain cylinder Q . As a consequence, Lemma 6.2 implies that u ≥ M o , (6.25)in B ρ (¯0) × ( t , t + c − n − M − m − po (4 ρ ) p ). In order to complete the proof, it is sufficientthat t + c − n − M − m − po (4 ρ ) p ) = δM − m − p ρ p . Using the definition of t we see that this is equivalent to τ o = ln (cid:16) n +2 δc p σ − m − po (cid:17) , where c is the constant from assumption (6.5). The right hand side depends only on m, n, p, C , C , α . Hence, with this choice of τ o , (6.25) and the upper bound for τ implythat u ≥ M o = σ o e τo − τ − m − p M > σ o e − k − m − po − m − p M =: ηM. in B ρ (¯0) × ( t , δM − m − p ρ p ). Note that η only depends on m, n, p, C , C , α . From theupper bound for τ it also follows that t = δM − m − p ρ p (1 − e − τ ) < δM − m − p ρ p (1 − e − τ − k − m − po ) , so the claim of the theorem is true if we take ε = e − τ − k − m − po , and the right-hand side clearly only depends only on m, n, p, C , C , α . (cid:3) Local Boundedness We prove that in the range (1.3) all weak solutions are locally bounded. We use a DeGiorgi iteration combining the energy estimates obtained in Lemma 4.1 with a Sobolevembedding. Theorem 7.1. Let u be a weak solution in the sense of Definition 2.1 and suppose that theparameters m and p satisfy (1.3) . Then u is locally bounded and for any cylinder of theform Q ρ, τ ( z o ) contained in Ω T and any σ ∈ (0 , we have the explicit bound ess sup Q σρ,στ ( z o ) u ≤ c h(cid:0) (1 − σ ) p τ (cid:1) − n + pp ¨ Q ρ,τ ( z o ) u β +1 d x d t i ppβn +( β +1)( p − n ) + (cid:16) τρ p (cid:17) − m − p , where c is a constant depending only on m, n, p, C , C . Proof. Suppose that Q ρ,τ ( z o ) ⊂ Ω T . Define sequences ρ j := σρ + (1 − σ )2 j ρ, τ j := στ + (1 − σ )2 j τ, k j := k (1 − − j ) β +1 , where k > Q j := Q ρ j ,τ j ( z o ) = B j × T j . Choose functions ϕ j ∈ C ∞ ( Q j ; [0 , Q j and satisfying φ j = 1 on Q j +1 and for which |∇ ϕ j | ≤ j +2 (1 − σ ) ρ , | ∂ t ϕ j | ≤ j +2 (1 − σ ) τ . Furthermore, we define the sequence Y j := ¨ Q j (cid:0) u β +12 − k β +12 j (cid:1) d x d t. Note that Y j is finite for every j since u ∈ L β +1 (Ω T ). Define the auxiliary parameters M := β +1 β , q := p ( M + n ) = pM ( n + Mn ) . A straightforward calculation shows that (2.3) (and hence (1.3)) guarantees that q > Y j +1 ≤ h ¨ Q j +1 (cid:0) u β +12 − k β +12 j +1 (cid:1) q + d x d t i q | Q j +1 ∩ { u > k j +1 }| q ′ . (7.1)We will use the shorthand notation φ := (cid:0) u β +12 − k β +12 j +1 (cid:1) M + ≤ ( u β − k βj +1 ) + . The upper bound, is a consequence of the definition of M and the fact that M > 1. In thefollowing calculation we express the integral on the right-hand side of (7.1) in terms of φ and split the integral into space and time variables. We apply H¨older’s inequality to theintegral over the space variables, and then estimate one of the resulting factors upwards bythe essential supremum over time. After this, we introduce the cut-off function ϕ j whichallows us to apply the Gagliardo-Nirenberg inequality. We also apply Lemma 3.2 (i). Thus,we obtain two factors which both are bounded by the right-hand side of the energy estimate(4.1). All in all, we have ¨ Q j +1 φ p ( n + Mn ) d x d t = ˆ T j +1 ˆ B j +1 φ p φ pMn d x d t (7.2) ≤ ˆ T j +1 h ˆ B j +1 φ p ∗ d x i pp ∗ h ˆ B j +1 φ M d x i pn d t ≤ h ess sup T j +1 ˆ B j +1 φ M d x i pn ˆ T j +1 h ˆ B j +1 φ p ∗ d x i pp ∗ d t ≤ h ess sup T j +1 ˆ B j +1 (cid:0) u β +12 − k β +12 j +1 (cid:1) d x i pn ˆ T j +1 h ˆ B j +1 ( u β − k βj +1 ) p ∗ + d x i pp ∗ d t ≤ h ess sup T j ˆ B j (cid:0) u β +12 − k β +12 j +1 (cid:1) ϕ pj d x i pn ˆ T j h ˆ B j (( u β − k βj +1 ) + ϕ j ) p ∗ d x i pp ∗ d t ≤ c h ess sup T j ˆ B j b [ u, k j ] χ { u>k j } ϕ pj d x i pn ˆ T j ˆ B j |∇ [( u β − k βj +1 ) + ϕ j ] | p d x d t EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 27 ≤ c h ¨ Q j ( u β − k βj +1 ) p + |∇ ϕ j | p + (cid:0) u β +12 − k β +12 j +1 (cid:1) ϕ p − j | ∂ t ϕ j | d x d t i n + pn . The constant c depends only on m, n, p, C , C . In the set where u > k j +1 we can estimate( u β − k βj +1 ) p + (cid:0) u β +12 − k β +12 j (cid:1) = u m + p − (1 − ( k j +1 /u ) β ) p + (cid:0) − ( k j /u ) β +12 (cid:1) ≤ u m + p − (cid:0) − ( k j /k j +1 ) β +12 (cid:1) ≤ ck m + p − (cid:0) − ( k j /k j +1 ) β +12 (cid:1) < c j k m + p − , where the constant c only depends on m, p . In the second last step we used m + p < 3, andthe fact that k j +1 is comparable in size to k . Applying the previous estimate to the firstterm to the last line of (7.2) and noting that in the second term we can replace k j +1 by k j we obtain ¨ Q j +1 φ p ( n + Mn ) d x d t ≤ c h ¨ Q j (cid:2) j k m + p − |∇ ϕ j | p + ϕ p − j | ∂ t ϕ j | (cid:3)(cid:0) u β +12 − k β +12 j (cid:1) d x d t i n + pn . Combining this estimate with the bounds for ϕ j and its derivatives leads to ¨ Q j +1 φ p ( n + Mn ) d x d t ≤ c (cid:16) ( p +1) j (1 − σ ) p τ h(cid:16) τρ p (cid:17) k m + p − + 1 i Y j (cid:17) n + pn . From the last expression we see that if k ≥ (cid:0) τρ p (cid:1) − m − p then ¨ Q j +1 φ p ( n + Mn ) d x d t ≤ c (cid:16) ( p +1) j (1 − σ ) p τ Y j (cid:17) n + pn . (7.3)Observe now that | Q j ∩ { u > k j +1 }| k β +1 − j +1) = | Q j ∩ { u > k j +1 }| ( k β +12 j +1 − k β +12 j ) (7.4) ≤ ¨ Q j ∩{ u>k j +1 } ( u β +12 − k β +12 j ) d x d t ≤ Y j . Using (7.3) and (7.4) in (7.1) we end up with Y j +1 ≤ Cb j Y δj , (7.5)where b = 2 ( n + pn )( p +1) q + q ′ , C = ck − ( β +1) q ′ (cid:0) (1 − σ ) p τ (cid:1) n + pnq , δ = pnq = Mn + M . and c only depends on m, n, p, C , C . We want to show that Y j → 0. According to Lemma3.4 this is true provided that Y ≤ C − δ b − δ . Using the definition of Y and the parameters we see that this is equivalent to k ≥ c h(cid:0) (1 − σ ) p τ (cid:1) − n + pp ¨ Q ρ,τ ( z o ) u β +1 d x d t i ppβn +( β +1)( p − n ) , (7.6)where c is a constant depending only on m, n, p, C , C . Since ¨ Q σρ,στ ( z o ) ( u β +12 − k β +12 (cid:1) d x d t ≤ Y j → , this means that u ≤ k almost everywhere in Q σρ, (1+ σ ) τ ( z o ). The only lower bounds for k required in this argument were k ≥ (cid:0) τρ p (cid:1) − m − p and (7.6), so we have verified the estimatefor the essential supremum. (cid:3) We end this section by proving that the estimate of Theorem 7.1 can be somewhatimproved. This result will also be used in the reasoning leading to the Harnack estimate inSection 9. Note first that (2.3) can be rephrased as( β + 1) p + n ( m + p − > . Thus there exists r ∈ (0 , β + 1) such that λ r := rp + n ( m + p − > . (7.7)The next theorem shows that there is an upper bound in terms of the L r -norm of u . Theorem 7.2. Let r ∈ (0 , β + 1) be such that (7.7) is valid. Then for any cylinder Q ρ, τ ( z o ) ⊂ Ω T , ess sup Q ρ,τ ( z o ) u ≤ c h τ − n + pp ¨ Q ρ, τ ( z o ) u r d x d t i pλr + c (cid:16) τρ p (cid:17) − m − p (7.8) where the constant c depends only on r and the data. Proof. Define the increasing sequences ρ j := (2 − − j ) ρ, τ j := (2 − − j ) τ. Define cylinders Q j = Q ρ j ,τ j ( z o ). Applying Theorem 7.1 to the cylinder Q j +1 with σ = ρ j /ρ j +1 = τ j /τ j +1 and noting that 1 − σ > − ( j +2) we end up withess sup Q j u ≤ c h j ( n + p ) τ − n + pp j ¨ Q j +1 u β +1 d x d t i ppβn +( β +1)( p − n ) + (cid:16) τ j ρ pj (cid:17) − m − p ≤ c h j ( n + p ) τ − n + pp ¨ Q j +1 u β +1 d x d t i ppβn +( β +1)( p − n ) + (cid:16) τρ p (cid:17) − m − p , where in the second step we used the fact that ρ j ≥ ρ and τ ≤ τ j < τ . Denoting now M j := ess sup Q j u and noting that u ≤ M j +1 a.e. in Q j +1 we see that M j ≤ cM p ( β +1 − r ) pβn +( β +1)( p − n ) j +1 h j ( n + p ) τ − n + pp ¨ Q ρ, τ ( z o ) u r d x d t i ppβn +( β +1)( p − n ) + (cid:16) τρ p (cid:17) − m − p . Due to (7.7), the exponent of M j +1 lies in the interval (0 , M j +1 to 1 we end up with M j ≤ εM j +1 + c ( ε ) h j ( n + p ) τ − n + pp ¨ Q ρ, τ ( z o ) u r d x d t i pλr + (cid:16) τρ p (cid:17) − m − p = εM j +1 + c ( ε ) b j h τ − n + pp ¨ Q ρ, τ ( z o ) u r d x d t i pλr + (cid:16) τρ p (cid:17) − m − p , where b = 2 p ( n + p ) λr and the constant ε > M ≤ ε N M N + c ( ε ) h τ − n + pp ¨ Q ρ, τ ( z o ) u r d x d t i pλr N − X j =0 ( εb ) j + (cid:16) τρ p (cid:17) − m − p N − X j =0 ε j , EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 29 for N ≥ 1. Choosing ε = b we see that both sums on the right-hand side converge as N → ∞ . Since M N is bounded from above by the essential supremum of u over Q ρ, τ ( z o ),the term ε N M N vanishes in the limit and we end up with (7.8). (cid:3) H¨older continuity In this setion we consider only m and p in the supercritical range (1.4). We show that inthis case weak solutions are locally H¨older continuous. The starting point of the argumentis a De Giorgi type lemma providing a sufficient condition for the reduction of the oscillationfrom above. First we introduce some notation. For 0 < µ + < ∞ we denote θ = εµ − m − p + , (8.1)where ε ∈ (0 , ε will be chosen later in this section. Initiallyit is important that our results work for all ε ∈ (0 , Lemma 8.1. Let u be a weak solution to (2.4) in the sense of Definition 2.1. Suppose thatwe are given a number < µ + < ∞ , and let θ be chosen as in (8.1) . Moreover, suppose Q ρ,θρ p ( z o ) ⊂ Ω T is a parabolic cylinder satisfying ess sup Q ρ,θρp ( z o ) u ≤ µ + . then there exists a constant ν o depending only on m, n, p, C , C such that if | Q ρ,θρ p ( z o ) ∩ { u β > µ β + / }| ≤ ν o ε np | Q ρ,θρ p ( z o ) | (8.2) then u β ≤ µ β + a.e. in Q ρ/ ,θ ( ρ/ p ( z o ) . Proof. Define sequences of numbers and sets as follows: ρ j := 12 (cid:0) j (cid:1) ρ, k βj := (cid:0) − − j +2 (cid:1) µ β + , Q j := Q ρ j ,θρ pj ( z o ) ,A j := Q j ∩ { u > k j } , Y j := | A j || Q j | . We can now choose functions ϕ j ∈ C ∞ ( Q j ; [0 , Q j and satisfying φ j = 1 on Q j and for which |∇ ϕ j | ≤ ρ − j +2 , | ∂ t ϕ j | ≤ c p θ − ρ − p jp . Note that in the set where u > k j +1 we have u β − k βj > k βj +1 − k βj = µ β + j +3 . (8.3)This observation and H¨older’s inequality show that µ βp + ( j +3) p | A j +1 | ≤ ¨ A j +1 ( u β − k βj ) p + d x d t (8.4) ≤ h ¨ A j +1 ( u β − k βj ) p ( n + p ) n + d x d t i nn + p | A j +1 | pn + p . (8.5)The integral in the last expression can be estimated using H¨older’s inequality and (8.3) as ¨ A j +1 ( u β − k βj ) p ( n + p ) n + d x d t ≤ ˆ T j +1 ˆ B j +1 ( u β − k βj ) p pn + χ A j +1 ( u β − k βj ) p + d x d t ≤ ˆ T j +1 h ˆ B j +1 ( u β − k βj ) p + χ A j +1 d x i pn h ˆ B j +1 ( u β − k βj ) p ∗ + d x i pp ∗ d t ≤ c j (2 − p ) pn µ β ( p − pn + ˆ T j +1 h ˆ B j +1 ( u β − k βj ) d x i pn h ˆ B j +1 ( u β − k βj ) p ∗ + d x i pp ∗ d t ≤ c j (2 − p ) pn µ β ( p − pn + h ess sup T j +1 ˆ B j +1 ( u β − k βj ) d x i pn ˆ T j +1 h ˆ B j +1 ( u β − k βj ) p ∗ + d x i pp ∗ d t, where in the last step we have estimated one of the integrals over space upwards by takingthe essential supremum in time. Note that by Lemma (3.2) (ii) we have c ( u β − k βj ) ≤ ( u β − + k β − j ) b [ u, k j ] ≤ µ β − b [ u, k j ] . Using this observation and introducing the cut-off functions ϕ j puts us into a position toapply Sobolev inequality and the energy estimate (4.1) as follows. ¨ A j +1 ( u β − k βj ) p ( n + p ) n + d x d t ≤ c j (2 − p ) pn µ pn ( m + p − h ess sup T j ˆ B j b [ u, k j ] + ϕ pj d x i pn ˆ T j h ˆ B j [( u β − k βj ) + ϕ j ] p ∗ d x i pp ∗ d t ≤ c j (2 − p ) pn µ pn ( m + p − h ess sup T j ˆ B j b [ u, k j ] + ϕ pj d x i pn ˆ T j ˆ B j |∇ [( u β − k βj ) + ϕ j ] | p d x d t ≤ c j (2 − p ) pn µ pn ( m + p − h ¨ A j ( u β − k βj ) p + |∇ ϕ j | p + b [ u, k j ] ϕ p − j | ∂ t ϕ j | d x d t i n + pn . The second term in the last integral can be estimated using Lemma 3.2 (iii) and the boundfor | ∂ t ϕ j | as b [ u, k j ] ϕ p − j | ∂ t ϕ j | ≤ cµ β +1+ θ − ρ − p jp = cε − µ βp + ρ − p jp . Using this and the bound for |∇ ϕ j | and u we see that ¨ A j +1 ( u β − k βj ) p ( n + p ) n + d x d t ≤ c j pn ( n +2) µ pn ( m + p − ( ε − ρ − p µ βp + | A j | ) n + pn . Combining this estimate with (8.4) and the observation that | A j +1 | ≤ | A j | we have | A j +1 | ≤ cε − jp [1+ n +2 n + p ] µ pn + p ( m + p − ρ − p | A j | pn + p = cε − nn + p b j θ − pn + p ρ − p | A j | pn + p , where c and b only depend on m, n, p, C , C . Dividing the last expression by | Q j | and notingthat | Q j | is proportional to θρ n + p we obtain Y j +1 ≤ cε − nn + p b j Y pn + p j . Setting δ := pn + p we see that Lemma 3.4 guarantees that Y j → | Q ρ,θρ p ( z o ) ∩ { u β > µ β + / }|| Q ρ,θρ p ( z o ) | = Y ≤ ( cε − nn + p ) − δ b − δ = ε np ν o , where ν o = c − δ b − δ only depends on m, n, p, C , C . Since | Q j | is bounded from above, thisalso means that | A j | → 0. Furthermore, since Q ρ/ ,θ ( ρ/ p ( z o ) ∩ { u β > µ β + } ⊂ A j , for all j , the measure of the set on the left hand side must be zero. (cid:3) EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 31 Reduction of the oscillation. We are now ready to prove the reduction of the oscil-lation in the case µ − = 0. If the condition of the De Giorgi lemma holds, then we have areduction of the oscillation from above. Suppose now that the condition in the De Giorgilemma fails, i.e. | Q ρ,θρ p ( z o ) ∩ { u β > µ β + / }| > ν o ε np | Q ρ,θρ p ( z o ) | . Then there is a set ∆ ⊂ ( t o − θρ p , t o ) of positive measure such that |{ x ∈ B ρ ( x o ) | u ( x, τ ) > − β µ + }| > ν o ε np | B ρ ( x o ) | , for all τ ∈ ∆. Provided that ¯ Q ρ,θρ p ⊂ Ω × [0 , T ), the L -Harnack inequality of Theorem5.1 for the time-continuous representative of u shows that for τ ∈ ∆, ν o ε np c n ρ n − β µ + < − β µ + |{ x ∈ B ρ ( x o ) | u ( x, τ ) > − β µ + }| ≤ ˆ B ρ ( x o ) u ( x, τ ) d x (8.6) ≤ γ inf t ∈ ( t o − θρ p ,t o ) ˆ B ρ ( x o ) u ( x, t ) d x + γ (cid:16) θρ p ρ λ (cid:17) − m − p (8.7)By the definition of λ and θ we see that (cid:16) θρ p ρ λ (cid:17) − m − p = ε − m − p µ + ρ n . Moving this term to the right-hand side of (8.6) we obtain ε np ( c − γε κ ) µ + ρ n ≤ γ inf t ∈ ( t o − θρ p ,t o ) ˆ B ρ ( x o ) u ( x, t ) d x, (8.8)where c = c ( m, n, p, C , C ) and κ = 13 − m − p − np , is a positive number by (1.4). If we now choose ε := (cid:16) c γ (cid:17) κ , which clearly only depends on m, n, p, C , C we also see from (8.8) that Cµ + ρ n ≤ inf t ∈ ( t o − θρ p ,t o ) ˆ B ρ ( x o ) u ( x, t ) d x, (8.9)for a constant C = C ( m, n, p, C , C ) ≤ 1. Take now ζ > ˆ B ρ ( x o ) u ( x, t ) d x = ˆ B ρ ( x o ) ∩{ u ( x,t ) ≥ ζµ + } u ( x, t ) d x + ˆ B ρ ( x o ) ∩{ u ( x,t ) <ζµ + } u ( x, t ) d x ≤ µ + | B ρ ( x o ) ∩ { u ( x, t ) ≥ ζµ + }| + ζµ + | B ρ ( x o ) | . With the choice ζ := C − ( n +1) /c n where C is the constant from (8.9), the last estimate and(8.9) show that | B ρ ( x o ) ∩ { u ( x, t ) ≥ ζµ + }| ≥ α | B ρ ( x o ) | (8.10)for all t ∈ ( t o − θρ p , t o ) for a constant α depending only on m, n, p, C , C . Suppose nowthat Q ρ,θρ p ( z o ) ⊂ Ω T . This puts us in a position to apply Theorem 6.4 for a sufficientlysmall M . Namely, taking M = min { ζ, ( ε/ p ) − m − p } µ + , we see that (8.10) is still valid with ζµ + replaced by M and furthermore that B ρ ( x o ) × ( t o − δM − m − p (2 ρ ) p , t o ) ⊂ Q ρ,θρ p ( z o ) ⊂ Ω T , where δ ∈ (0 , 1) is the constant from Theorem 6.4. Hence, we may apply Theorem 6.4 with s = t o − δM − m − p (2 ρ ) p and ρ replaced by 2 ρ to conclude that there is a ξ ∈ (0 , 1) and ˜ ε < ε depending only on m, n, p, C , C such that u ≥ ξµ + in B ρ ( x o ) × ( t o − ˜ εµ − m − p + ρ p , t o ) , (8.11)which is the reduction of the oscillation from below. Combining the previous reasoning andLemma 8.1, we have shown the following. Lemma 8.2. There are constants ε, γ, η ∈ (0 , depending only on m, n, p, C , C such thatfor any weak solution u and number µ + > satisfying the conditions Q ρ,εµ − m − p + ρ p ( z o ) ⊂ Ω T and u ≤ µ + on Q ρ,εµ − m − p + ρ p ( z o ) , we have ess osc Q ρ ,γεµ − m − p + ρp ( z o ) u ≤ ηµ + . (8.12) Furthermore, one of the following condition must hold in the cylinder Q ρ ,γεµ − m − p + ρ p ( z o ) : ( i ) ess sup Q ρ ,γεµ − m − p + ρp ( z o ) u ≤ (cid:0) η (cid:1) µ + , or (8.13) ( ii ) ess inf Q ρ ,γεµ − m − p + ρp ( z o ) u ≥ (cid:0) − η (cid:1) µ + . Proof. By Lemma 8.1 and the previous reasoning, (8.12) is valid with γ = min { − p , ˜ εε } and η = max { (3 / β , − ξ } , where ˜ ε and ξ are the constants appearing in (8.11). Furthermore,if (8.13) ( i ) fails, (8.12) shows that we must haveess inf Q ρ ,γεµ − m − p + ρp ( z o ) u ≥ (cid:0) η (cid:1) µ + − ηµ + = (cid:0) − η (cid:1) µ + , so that (8.13) (ii) holds. (cid:3) Lemma 8.3. There are constants c and ν depending only on m, n, p, C , C such that forany weak solution u and number µ + > for which Q ρ,εµ − m − p + ρ p ( z o ) ⊂ Ω T and u ≤ µ + on Q ρ,εµ − m − p + ρ p ( z o ) , we have ess osc Q r,εµ − m − p + rp ( z o ) u ≤ cµ + (cid:16) rρ (cid:17) ν , (8.14) for all < r ≤ ρ . Here, ε is the constant from Lemma 8.2. Proof. Denote C := 2 max { , γ − p } where γ is the constant from Lemma 8.2 and define µ j + := (cid:16) η (cid:17) j µ + , ρ j := ρ/C j . With these choices, Q ρ ,ε ( µ ) − m − p ρ p ( z o ) ⊂ Q ρ ,γεµ − m − p + ρ p ( z o ) , and Lemma 8.2 guarantees that ess osc Q ρ ,ε ( µ − m − pρp ( z o ) u ≤ ηµ +EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 33 Furthermore, if we are in the case (8.13) (i), we have u ≤ µ on Q ρ ,ε ( µ ) − m − p ρ p ( z o ) andwe may apply Lemma 8.2 to this subcylinder instead to conclude thatess osc Q ρ ,ε ( µ − m − pρp ( z o ) u ≤ ηµ Also, Lemma 8.2 guarantees that one of the conditions of (8.13) holds with ρ replaced by ρ and µ + replaced by µ . If condition (i) is true, we are again in a position to continue theiteration. Continuing in this way, we see that as long as we stay in case (i) at every step ofthe iteration we have ess osc Q ρj,ε ( µj +)3 − m − pρpj ( z o ) u ≤ ηµ j − , (8.15) ess sup Q ρj − ,ε ( µj − 1+ )3 − m − pρpj − ( z o ) u ≤ µ j − (8.16)Either this estimate holds for every j ∈ N , or there is a k ∈ N such that (8.15) holds for all j ∈ { , . . . , k } and ess inf Q ρk − ,γε ( µk − 1+ )3 − m − pρpk − ( z o ) u ≥ (cid:0) − η (cid:1) µ k − = (cid:0) − η η (cid:1) µ k + . (8.17)We assume for now the existence of such a k and investigate its consequences. In the endwe will show that the estimate (8.14) holds whether k exists or not. Since ρ k − C ρ k ≥ ρ k , γε ( µ k − ) − m − p ρ pk − = γC p (cid:0) η (cid:1) − m − p ε ( µ k + ) − m − p ρ pk ≥ p (cid:0) η (cid:1) − m − p ε ( µ k + ) − m − p ρ pk > ε ( µ k + ) − m − p (2 ρ k ) p , it follows from (8.17) and (8.16) with j = k that (cid:0) − η η (cid:1) µ k + ≤ u ≤ 21 + η µ k + , in Q ρ k ,ε ( µ k + ) − m − p (2 ρ k ) p ( z o ) . Up to a translation in the time variable this is exactly the situation of Lemma 3.7 with M = µ k + . By translation we may assume that t o = 0. Lemma 3.7 shows that the function v ( x, t ) = ( µ k + ) − u (cid:0) x, ( µ k + ) − m − p t (cid:1) , ( x, t ) ∈ Q ρ k ,ε (2 ρ k ) p ( x o , . solves an equation of parabolic p -Laplace type, where the constants in the structure con-ditions only depend on m, n, p, C , C . Applying Lemma 3.6 to v then shows that for all( x, t ) , ( y, s ) ∈ Q ρ k ,ε ( ρ k ) p ( x o , | v ( x, t ) − v ( y, s ) | ≤ c h | x − y | + | t − s | p ρ k i ν o , where the constants c and ν o only depend on m, n, p, C , C . Since the fraction in the lastestimate is bounded from above by 2 + ε p we see that for any 0 < ν ≤ ν o we have | v ( x, t ) − v ( y, s ) | ≤ c (2 + ε p ) ν o − ν h | x − y | + | t − s | p ρ k i ν < c (2 + ε p ) ν o h | x − y | + | t − s | p ρ k i ν , for all ( x, t ) , ( y, s ) ∈ Q ρ k ,ε ( ρ k ) p ( x o , u this translates into | u ( x, t ) − u ( y, s ) | ≤ cµ k + h | x − y | + ( µ k + ) m + p − p | t − s | p ρ k i ν , (8.18) for all ( x, t ) , ( y, s ) ∈ Q ρ k ,ε ( µ k + ) − m − p ρ pk ( z o ) and 0 < ν ≤ ν o . The constant c still depends onlyon m, n, p, C , C . Now we are ready to prove (8.14). For this, take 0 < r ≤ ρ . Pick j ∈ N such that (cid:16) η (cid:17) ( j +1) (3 − m − p ) p ρC j +1 < r ≤ (cid:16) η (cid:17) j (3 − m − p ) p ρC j . From the left inequality we can deduce thatln h rρ i > ln h C (cid:16) η (cid:17) − m − pp i + j ln h C (cid:16) η (cid:17) − m − pp i , and with some further manipulations that j > − − b ln h rρ i , (8.19)for some b > m, n, C , C . Note that r < ρ j and εµ − m − p + r p ≤ ε ( µ j + ) − m − p ρ pj , so Q r,εµ − m − p + r p ( z o ) ⊂ Q ρ j ,ε ( µ j + ) − m − p ρ pj ( z o ). If j ≤ k (or if k does not exist, which meansthat (8.15) is valid for all j ) then (8.15) and (8.19) imply thatess osc Q r,εµ − m − p + rp ( z o ) u ≤ ηµ j − = 2 η η µ + h η i j < ηµ + h 21 + η i h 21 + η i b ln[ r/ρ ] = cµ + (cid:16) rρ (cid:17) ν , for some positive constants c and ν depending only on m, n, p, C , C . Suppose now insteadthat j > k . Then Q r,εµ − m − p + r p ( z o ) ⊂ Q ρ k ,ε ( µ k + ) − m − p ρ pk ( z o ) so from (8.18) we see thatess osc Q r,εµ − m − p + rp ( z o ) u ≤ cµ k + h r + ( µ k + ) m + p − p ε p µ (3 − m − p ) p + rρ k i ν = c (cid:16) η (cid:17) k µ + h r + ( η ) k (3 − m − p ) p ε p rρ/C k i ν ≤ c h(cid:16) 21 + η (cid:17) ν (3 − m − p ) p − C ν i k µ + (cid:16) rρ (cid:17) ν . Observe now that the expression inside the square brackets can be made smaller than orequal to one by taking ν ≤ ν where the upper bound ν depends only on C and η andhence only on m, n, p, C , C . Taking now ν := min { ν , ν } we finally have verified that(8.14) holds in all cases. (cid:3) H¨older continuity. Using Lemma 8.12 we can now easily prove the local H¨oldercontinuity. Theorem 8.4. Let u be a weak solution in the sense of Definition 2.1. Let m and p bein the supercritical range (1.4) . Then u is locally H¨older continuous in Ω T and the H¨olderexponent depends only on m, n, p, C , C . Proof. Let z o ∈ Ω T . Pick R > n + 1)-dimensional closed ball ¯ B n +12 R ( z o )centered at z o is contained in Ω T and define µ + = ess sup ¯ B n +12 R ( z o ) u < ∞ . The number µ + is finite since the range (1.4) is contained in the range (1.3) which accordingto Theorem 7.1 guarantees local boundedness. By picking a suitable representative of u , we EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 35 may assume that µ + is the actual supremum of u on the ball ¯ B n +12 R ( z o ). We can now choose ρ > z ∈ ¯ B n +1 R ( z o ), we have Q ρ,εµ − m − p + ρ p ( z ) ⊂ ¯ B n +12 R ( z o ) , Q ρ,εµ − m − p + ρ p ( z ) ⊂ Ω T . From the first condition it follows that u ≤ µ + in every cylinder Q ρ,εµ − m − p + ρ p ( z ) where z ∈ ¯ B n +1 R ( z o ). Thus, according to Lemma 8.3,ess osc Q r,εµ − m − p + rp ( z ) u ≤ cµ + (cid:16) rρ (cid:17) ν , (8.20)for every r ∈ (0 , ρ ) and z ∈ ¯ B n +1 R ( z o ). If in the above estimate we had the oscillationrather than the essential oscillation we could now apply (8.20) to any pair of points thatare sufficiently close to each other. Since this is not case, we must first exclude a set ofmeasure zero so that the different types of oscillation coincide. In order to ensure thatwe are only disregarding a set of measure zero, this should be done only for a countablenumber of cylinders. We now make this idea precise. For every ( z, r ) in the countable set[ ¯ B n +1 R ( z o ) ∩ Q n +1 ] × [(0 , ρ ) ∩ Q ], there is a set N zr ⊂ Q r,εµ − m − p + r p ( z ) of measure zero suchthat for all ( y, s ) ∈ Q r,εµ − m − p + r p ( z ) \ N zr ,ess inf Q r,εµ − m − p + rp ( z ) u ≤ u ( y, s ) ≤ ess sup Q r,εµ − m − p + rp ( z ) u. Define N = ∪ ( z,r ) N zr , and suppose that z , z ∈ B n +1 R ( z o ) \ N . We may also assume that t ≤ t . Suppose first that z ∈ Q ρ,εµ − m − p + ρ p ( z ) ∪ ( B ρ ( x ) × { t } ). Then there is a sequenceof numbers ( z j ) ⊂ B n +1 R ( z o ) ∩ Q n +1 such that z j → z , t j ≥ t , and z ∈ Q ρ,εµ − m − p + ρ p ( z j )for all j ∈ N . Define ˆ r j := | x − x j | < ρ, ˜ r j := (cid:16) | t − t j | εµ − m − p + (cid:17) p < ρ, ˆ r := | x − x | < ρ, ˜ r := (cid:16) | t − t | εµ − m − p + (cid:17) p < ρ. Take now r j ∈ (max { ˆ r j , ˜ r j } , max { ˆ r j , ˜ r j } + j ) ∩ Q such that r j < ρ . Then r j converges tomax { ˆ r, ˜ r } =: r . Moreover, z ∈ Q r j ,εµ − m − p + r pj ( z j ) \ N and also z belongs to this set forlarge j so | u ( z ) − u ( z ) | ≤ ess osc Q rj,εµ − m − p + rpj ( z j ) u ≤ cµ + (cid:16) r j ρ (cid:17) ν −−−→ j →∞ cµ + (cid:16) rρ (cid:17) ν ≤ cµ + ρ − ν ( | x − x | + (cid:16) | t − t | εµ − m − p + (cid:17) p ) ν ≤ C | z − z | νp , where the constant C depends on the data and ρ, µ + . Suppose now instead that z does notbelong to the set Q ρ,εµ − m − p + ρ p ( z ) ∪ ( B ρ ( x ) × { t } ). Then | u ( z ) − u ( z ) | ≤ max { u ( z ) , u ( z ) }| z − z | νp | z − z | νp ≤ µ + min { ρ, εµ − m − p + ρ p } − νp | z − z | νp . Thus, we have verified that for all z , z ∈ ¯ B n +1 R ( z o ) \ N , | u ( z ) − u ( z ) | ≤ C | z − z | νp , (8.21)for a constant C = C ( m, n, p, C , C , µ + , R ). (Note that ρ depends only on R , the data and µ + .) Since the set N has measure zero, we can re-define u at every point of N as the uniquelimit guaranteed by (8.21) when approaching the point through the set ¯ B n +1 R ( z o ) \ N . In thisway we obtain a representative of u which satisfies (8.21) for all points z , z ∈ ¯ B n +1 R ( z o ). (cid:3) Harnack estimates We conclude this paper considering the Harnack inequality for solutions of parabolicsingular supercritical equations. Such results were proved in [6] for equations of parabolic p -Laplace and porous medium type. For doubly nonlinear equations see [10] under morerestrictive assumptions. Our method is based on the pattern scheme of [7].Let us state and prove some lemmata. Lemma 9.1 (Measure-to-point estimate) . Let u ≥ be a weak solution of (2.4) . Supposethat B ρ ( x o ) × [ t o , t o + M − m − p ρ p ] ⊂ Ω T . Let µ ∈ (0 , and suppose that | B ρ ( x o ) ∩ { u ( · , t o ) ≥ M }| ≥ µ | B ρ ( x o ) | . (9.1) Then there exist constants ξ, τ ∈ (0 , depending only on the data and µ , such that u ≥ ξM, in B ρ ( x o ) × [ t o + τ M − m − p ρ p , t o + τ M − m − p ρ p ] . Moreover, τ can be chosen arbitrarily small by decreasing ξ . Proof. Assumption (9.1) and the fact that B ρ ( x o ) × [ t o , t o + M − m − p ρ p ] is contained inthe domain Ω T allow us to apply Lemma 6.3 to conclude that there exists ǫ ( µ ) such that | B ρ ( x o ) ∩ { u ( · , t ) ≥ ǫM }| ≥ µ | B ρ ( x o ) | , (9.2)for all t ∈ ( t o , t o + δM − m − p ρ p ). Here, δ = δ ( data, µ ) ∈ (0 , 1) is the constant from Lemma6.3. In order to facilitate the latter part of the proof we note that we may instead use δ = δ ( data, µ ) which by the construction in the proof of Lemma 6.3 is a smaller number.Note that (9.2) remains valid if we replace M by any θM , where θ ∈ (0 , B ρ ( x o ) × [ t o , t o + M − m − p ρ p ] is contained in the domain, we may apply Theorem 6.4 with M replacedby ǫθM , α = µ and considering all s in ( t o , t o + δM − m − p ρ p ) for which s + δ ( ǫθM ) − m − p ρ p ) ≤ t o + M − m − p ρ p . Thus, we obtain u ≥ ηǫθM in B ρ ( x o ) × ( t o + (1 − ε ) δ ( ǫθM ) − m − p ρ p , t o + δM − m − p ρ p ) . Here, η and ε only depend on the data and µ . For any τ ∈ (0 , δ ) we may thus first choose θ so small that (1 − ε ) δ ( ǫθ ) − m − p < τ / ξ = ηǫθ . (cid:3) We now prove an alternative form of the reduction of the oscillation which will be conve-nient in the sequel. Lemma 9.2 (Estimates of H¨older regularity) . Let u be a weak solution of (2.4) in Ω T inthe supercritical range. Then for any S > there exist constants ¯ C > and ¯ α > dependingonly on S and the data, such that if Q R,k − m − p R p ( z o ) ⊂ Ω T for some k, R > then (9.3) sup Q R,k − m − pRp ( z o ) u ≤ S k ⇒ osc Q r,k − m − prp u ≤ ¯ C k (cid:0) rR (cid:1) ¯ α , r ≤ R. EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 37 Proof. Let ε be the constant from Lemma 8.2 and define the re-scaled function v ( x, t ) = S − u ( x, t o + ε − t ) , ( x, t ) ∈ Q R,εk − m − p R p ( x o , . Then v satisfies an equation of type (2.4), where the constants appearing in the structureconditions depend only on S and the data from the original problem. Furthermore,sup Q R,εk − m − pRp ( x o , ≤ k, so Lemma 8.3 implies that for all r ∈ (0 , R ],osc Q r,εk − m − prp ( x o , v ≤ ˜ ck (cid:0) rR (cid:1) ¯ α , where ˜ c and ¯ α only depend on S and the data of the original problem. Expressing thisestimate in terms of u and the original coordinates we obtain the desired estimate with¯ C = ˜ cS . (cid:3) We will also use the following version of the expansion of positivity. Lemma 9.3 (Expansion of positivity) . There exists ¯ λ > p/ (3 − m − p ) and, for any µ > , c ( µ ) , γ ( µ ) , γ ( µ ) ∈ (0 , depending only on µ and the data, such that if u ≥ is a solutionin B R (¯0) × [0 , k − m − p R p ] then | B r (¯0) ∩ { u ( · , ≥ k }| ≥ µ | B r (¯0) | (9.4) ⇒ inf B ρ u (cid:0) · , k − m − p r p (cid:0) γ ( µ ) + γ ( µ ) (cid:0) − ( r/ρ ) ¯ λ (3 − m − p ) − p (cid:1)(cid:1) ≥ c ( µ ) k (cid:16) rρ (cid:17) ¯ λ , whenever r < ρ ≤ R . Here, γ ( µ ) and γ ( µ ) are so small that γ ( µ ) + γ ( µ ) ≤ , whichguarantees that the time level is contained in the interval k − m − p R p . Moreover, the γ i ( µ ) can be chosen arbitrarily small by lowering c ( µ ) . Proof. Suppose that the measure condition of (9.4) holds. Then, by Lemma 9.1, we have u ≥ ξ ( µ ) k, in B r (¯0) × [ τ ( µ )2 k − m − p r p , + τ ( µ ) k − m − p r p ] . (9.5)Denote ξ := ξ (1) and note that, since m + p < 3, we can suppose that b := 2 p ξ − m − p ≤ . (9.6)Consider first the case 2 r ≤ R . We may now define ρ j := 2 j r, for all j ∈ N such that ρ j ≤ R,τ := τ (1) ≤ . Note that we are considering the case where at least ρ is defined. The bound on τ can beobtained due to Lemma 9.1. This might require shrinking ξ , but this does not violate thebound on b . We define recursively(9.7) t = τ ( µ )2 k − m − p r p , t j +1 = t j + τ ξ ( µ ) kξ j ) − m − p ρ pj +1 . From (9.5) it follows that | B r (¯0) ∩ { u ( · , t ) ≥ ξ ( µ ) k }| = | B r (¯0) | . Hence, we may applyLemma 9.1 with µ = 1 repeatedly and obtain u ≥ ξ ( µ ) ξ j k in B ρ j +1 × (cid:2) t j , t j + τ ξ ( µ ) k ξ j − ) − m − p ρ pj (cid:3) (9.8) for all integers j ≥ ρ j ≤ R , provided that the end time of the cylinder in (9.8)does not exceed k − m − p R p . In fact, this cannot happen, since an explicit calculation showsthat for all integers N ≥ t N = τ ( µ )2 k − m − p r p + τ k − m − p ξ ( µ ) − m − p r p p N − X j =0 b j (9.9) ≤ k − m − p R p τ (cid:16) p − b N − b (cid:17) ≤ k − m − p R p τ , where in the first step we used the fact that ξ ( µ ) ≤ ξ < τ ( µ ) ≤ τ . Thus, we have t N + τ ξ ( µ ) k ξ N − ) − m − p ρ pN ≤ t N + k − m − p R p τ ≤ k − m − p R p τ ≤ k − m − p R p , which means that the cylinders are all contained in the domain of u . From (9.6) we infer t j + τ ( ξ ( µ ) kξ j − ) − m − p ρ pj ≥ t j +1 , and thus (9.8) implies that u ≥ ξ ( µ ) (cid:16) rρ j (cid:17) ¯ λ k in B ρ j +1 × [ t j , t j +1 ] , where ¯ λ = − log ξ > p/ (3 − m − p ). Using the first line of (9.9) we can re-write t N as t N = k − m − p r p h τ ( µ )2 + 2 p − τ ξ ( µ ) − m − p (1 − b ) (1 − b N ) i = k − m − p r p h γ ( µ ) + γ ( µ ) (cid:16) − (cid:16) rρ N (cid:17) ¯ λ (3 − m − p ) − p (cid:17)i . For an arbitrary ρ ∈ [ r, R ] we now choose the smallest integer N such that ρ ≤ N +1 r . Butthis means that ρ N = 2 N r ≤ ρ ≤ R. Thus, we may conclude that u ≥ ξ ( µ ) (cid:16) rρ N (cid:17) ¯ λ k ≥ ξ ( µ ) (cid:16) rρ (cid:17) ¯ λ k = c ( µ ) (cid:16) rρ (cid:17) ¯ λ k in B ρ × [ t N , t N +1 ] . It now suffices to note that since ρ N ≤ ρ ≤ ρ N +1 ,[ t N , t N +1 ] ∋ k − m − p r p h γ ( µ ) + γ ( µ ) (cid:16) − (cid:16) rρ (cid:17) ¯ λ (3 − m − p ) − p (cid:17)i . By the definitions it is clear that γ ( µ ) and γ ( µ ) can be made arbitrarily small by lowering c ( µ ). It only remains to consider the case that 2 r > R . But in this case a bound of thecorrect form follows already from (9.5) since r < ρ < r . (cid:3) Since we are considering the super-critical range, Theorem 7.2 holds with r = 1. Com-bining this result with the L - Harnack estimate of Theorem 5.1, we immediately obtain thefollowing lemma. Lemma 9.4. Let u be a solution to (2.4) for some m, p satisfying (1.4) and suppose that ¯ Q ρ, τ ( z o ) ⊂ Ω × [0 , T ) . Then (9.10) sup Q ρ,τ ( z o ) u ≤ cτ − nλ h inf t ∈ [ t o − τ,t o ] ˆ B ρ ( x o ) u ( x, t ) d x i pλ + c (cid:16) τρ p (cid:17) − m − p , where λ = p + n ( m + p − and the constant c only depends on m, n, p, C , C . EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 39 Here we are able to use the actual infimum and supremum rather than their essentialequivalents, since we are considering the continuous representative of u . Similar results havebeen shown previously in [6, Appendix A] for the p -Laplacian with p < B R/ (¯0), so thatthe supremum and infimum are taken over the same ball. Theorem 9.5 (Harnack inequality) . Let u ≥ solve (2.4) for some m, p satisfying (1.4) ,in a domain containing B R (¯0) × [ − T, T ] . Suppose that u (0 , > and (9.11) 4 R p sup B R (¯0) u ( · , − m − p ≤ T. Then there exist constants ¯ C ≥ , ¯ θ > depending only on the data such that ¯ C − sup B R/ (¯0) u ( · , s ) ≤ u (0 , ≤ ¯ C inf B R (¯0) u ( · , t ) , (9.12) for − ¯ θ u (0 , − m − p R p ≤ s, t ≤ ¯ θ u (0 , − m − p R p . Proof. In the cylinder B (¯0) × [ − T ′ , T ′ ], where T ′ = T R − p u (0 , m + p − , the function v ( x, t ) = u (0 , − u ( R x, R p u (0 , − m − p t ) , satisfies a doubly singular equation with the same structure conditions as the original equa-tion. With these definitions, (9.11) implies(9.13) 1 ≤ M − m − p := sup B (¯0) v ( · , − m − p ≤ T ′ / , where the left inequality follows from the fact that v (¯0 , 0) = 1. We first prove the infbound in (9.12). Let ¯ λ > p/ (3 − m − p ) be the expansion of positivity exponent, define ψ ( ρ ) = (1 − ρ ) ¯ λ sup ¯ B ρ v ( · , 0) for ρ ∈ [0 , 1] and choose ρ ∈ [0 , x o ∈ ¯ B ρ (¯0) such thatmax [0 , ψ = ψ ( ρ ) = (1 − ρ ) ¯ λ v , v = v ( x o , ≥ . Let ¯ ξ ∈ [0 , 1) be the unique number such that (1 − ¯ ξ ) − ¯ λ = 2. Setting r = ¯ ξ (1 − ρ ) we have v r ¯ λ = ψ ( ρ ) ¯ ξ ¯ λ ≥ ¯ ξ ¯ λ , (9.14)where we used the fact that ψ ( ρ ) ≥ ψ (0) = 1. Furthermore, we may estimatesup ¯ B r ( x ) v ( · , ≤ (1 − [ ¯ ξ (1 − ρ ) + ρ ]) − ¯ λ (1 − [ ¯ ξ (1 − ρ ) + ρ ]) ¯ λ sup ¯ B ξ (1 − ρ ρ (¯0) v ( · , − ¯ ξ ) − ¯ λ (1 − ρ ) − ¯ λ ψ ( ¯ ξ (1 − ρ ) + ρ ) ≤ (1 − ¯ ξ ) − ¯ λ (1 − ρ ) − ¯ λ ψ ( ρ )= (1 − ¯ ξ ) − ¯ λ v = 2 v . Let a := v − m − p r p . By construction v ≤ M and by (9.13), B r ( x o ) × [ − a, a ] is containedin the domain of v . Thus we can apply Lemma 9.4 to conclude thatsup B r ( x o ) × [ − a,a ] v ≤ ca nn ( m + p − p (cid:16) ˆ B r ( x o ) v ( x, dx (cid:17) pn ( m + p − p + c a − m − p r pm + p − (9.16) ≤ c (2 v r n ) pn ( m + p − p ( v − m − p r p ) nn ( m + p − p + c v ≤ c v , where we used (9.15) to bound the integral. The constant c depends only on the data.Since a = v − m − p r p , we can apply (9.3) with k = v , and taking S to be the constant c from the last line of the previous estimate, in both B r/ ( x o ) × [ − v − m − p ( r/ p , 0] and B r/ ( x o ) × [ v − m − p ρ p − v − m − p ( r/ p , v − m − p ρ p ] for any ρ ≤ r/ v, B ρ ( x o ) × [ − v − m − p ρ p , v − m − p ρ p ]) ≤ ¯ c v ( ρ/r ) ¯ α , ρ ≤ r/ , where the constants ¯ c and ¯ α only depend on the data. This estimate also relies on the factthat B r ( x o ) × [ − a, a ] is contained in B × [ − T ′ , T ′ ], and hence in the domain of v . As v ( x o ) = v o we infer that v ≥ v o / B ¯ ηr ( x o ) × [ − ¯ η p a, ¯ η p a ] , for some suitable ¯ η ∈ (0 , / 4) depending only on the data. Thus, | B r ( x o ) ∩ { v ( · , t ) ≥ v / }| ≥ ¯ η n | B r ( x o ) | , for all | t | ≤ v − m − p ¯ η p r p . For any such time, the cylinder B ( x o ) × [ t, t + ( v / − m − p p ]is contained in the domain of v , so we may apply Lemma 9.3 with k = v / R = ρ = 2.Choosing the γ i (¯ η n ) so small that γ (¯ η n ) + γ (¯ η n ) < ¯ η p / 2, its conclusion implies, thanks to B ( x ) ⊇ B ,inf B v ( · , t + γ r v − m − p r p ) ≥ ¯ c v r ¯ λ , γ r := γ (¯ η n ) + γ (¯ η n ) (cid:0) − ( r/ ¯ λ (3 − m − p ) − p (cid:1) < ¯ η p / | t | ≤ ¯ η p v − m − p r p . The latter readily gives v ( x, t ) ≥ ¯ c v r ¯ λ for x ∈ B and | t | ≤ ¯ η p v − m − p r p / 2. Finally, observe that since r ≤ λ ≥ p/ (3 − m − p ), it holds v − m − p r p ≥ ( v r ¯ λ ) − m − p , so that (9.14) yields v ( x, t ) ≥ ¯ c ¯ ξ ¯ λ =: 1 / ¯ C for x ∈ B and | t | ≤ ¯ η p ¯ ξ ¯ λ (3 − m − p ) / θ . Expressing this in terms of u , we obtain the estimate for theinfimum in (9.12).To prove the bound for the supremum we proceed similarly. Indeed, let x ∗ ∈ ¯ B R (¯0) besuch that u ( x ∗ , 0) = sup ¯ B R (¯0) u ( · , 0) and define the rescaled translated function w ( x, t ) = u ( x ∗ , − u (cid:0) x ∗ + Rx, R p u ( x ∗ , − m − p t (cid:1) , ( x, t ) ∈ B (¯0) × [ − ˜ T , ˜ T ] , where ˜ T = R − p u ( x ∗ , m + p − T . Proceeding as before, we obtain that w ( x, t ) ≥ C for x ∈ ¯ B (¯0) and | t | ≤ ¯ θ . Writing this estimate in terms of u we see that u ( x, t ) ≥ u ( x ∗ , C , x ∈ ¯ B R ( x ∗ ) , | t | ≤ R p u ( x ∗ , − m − p ¯ θ. Noting that ¯0 ∈ ¯ B R ( x ∗ ), and taking into account the definition of x ∗ we obtain¯ C − sup ¯ B R (¯0) u ( · , ≤ u (¯0 , . (9.17)Since u is a solution on ¯ B R (¯0) × [ − H, H ] with H = 4 R p u (¯0 , − m − p , we can combine (9.17)and Lemma 9.4 (with t o = H/ τ = H/ 2) to conclude similarly as in (9.16) thatsup B R/ (¯0) × [ − H/ ,H/ u ≤ c u (¯0 , , which concludes the proof. (cid:3) EGULARITY OF SOLUTIONS TO DOUBLY SINGULAR EQUATIONS 41 References [1] V. B¨ogelein, F. Duzaar, P. Marcellini: Existence of evolutionary variational solutions via the calculusof variations , J. Differential Equations 256, no. 12, 3912–3942, 2014. 3, 4[2] V. B¨ogelein, F. Duzaar, R. Korte and C. Scheven: The higher integrability of weak solutions of porousmedium systems , Adv. Nonlinear Anal, 8(1), 10041034, 2018. 4[3] E. De Giorgi: Sulla differenziabilit`a e l’analiticit`a delle estremali degli integrali multipli regolari , Mem.Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., P.I. III. Ser. 3, 25–43, 1957. 4[4] E. DiBenedetto: Degenerate Parabolic Equations , Springer Verlag, 1993. 2, 4, 5[5] E. DiBenedetto, U. Gianazza and V. Vespri: Harnack’s Inequality for Degenerate and Singular ParabolicEquations , Springer Science+Business Media, 2012. 19[6] E. DiBenedetto, U. Gianazza and V. Vespri: Forward, backward and elliptic Harnack inequalitiesfor non-negative solutions to certain singular parabolic partial differential equations. , Ann. Sc. Norm.Super. Pisa Cl. Sci. (5) 9 385-422, 2010. 2, 36, 39[7] F.G. D¨uzg¨un, S. Mosconi and V. Vespri: Harnack and Pointwise Estimates for Degenerate or SingularParabolic Equations , Contemporary Research in Elliptic PDEs and Related Topics. Dipierro Editor.Springer-INdAM series 301-368, 2019. 36[8] S. Fornaro, M. Sosio and V. Vespri: Energy estimates and integral Harnack inequality for some dou-bly nonlinear singular parabolic equations . Recent trends in nonlinear partial differential equations. I.Evolution problems, 179–199, Contemp. Math., 594, Amer. Math. Soc., Providence, RI, 2013. 1, 11, 14,16[9] S. Fornaro, M. Sosio and V. Vespri: L r loc - L ∞ loc Estimates and Expansion of Positivity for a class ofDoubly Non Linear Singular Parabolic equations . Discrete and Continuous Dynamical Systems SeriesS, 7(4), 737–760, 2013. 1, 16[10] S. Fornaro, M. Sosio and V. Vespri: Harnack type inequalities for some doubly nonlinear singularparabolic equations. Discrete and Continuous Dynamical Systems Series 35, 12, 5909–5926, 2015. 36, 39[11] E. Giusti: Direct Methods in the Calculus of Variations , World Scientific, 2003. 4[12] A.V. Ivanov: Regularity for doubly nonlinear parabolic equations , Journal of Mathematical Sciences,Vol. 83, no. 1, 22 – 37 1997. 3[13] A.S. Kalashnikov: Some problems of the qualitative theory of nonlinear degenerate second order equa-tions , Russian Math. Surveys, 42, 169–222, 1987. 1[14] J. Kinnunen and P. Lindqvist: Pointwise behaviour of semicontinuous supersolutions to a quasilinearparabolic equation , Ann. Mat. Pura Appl. (4) 185(3): 411–435, 2006. 3, 4[15] J.L. Lions: Quelques m´ethodes de r´esolutiondes probl´emes aux limites nonlin´eaires. Dunod, Paris,1969. 1[16] M.M. Porzio and V. Vespri: Holder Estimates for Local Solutions of Some Doubly Nonlinear DegenerateParabolic Equations , Journal of Differential Equations, 103, 146–178, 1993. 3[17] C. Scheven: Regularity for subquadratic parabolic systems: higher integrability and dimension estimates ,Proceedings of the Royal Society of Edinburgh, 140A, 12691308, 2010. 4[18] T. Singer and M. Vestberg: Local Boundedness of Weak Solutions to the Diffusive Wave Approximationof the Shallow Water Equations , Journal of Differential Equations, Vol. 266 no. 6, 3014-3033, 2019. 3[19] T. Singer and M. Vestberg: Local H¨older Continuity of Weak Solutions to a Diffusive Shallow Mediumequation , Nonlinear Analysis, Vol. 185, 306-335, 2019. 3[20] S. Sturm: Existence of weak solutions of doubly nonlinear parabolic equations , J. Math. Anal. Appl.455, no. 1, 842–863, 2017. 4, 6 Vincenzo Vespri, Universit`a degli Studi di Firenze, Dipartimento di Matematica ed Informatica”Ulisse Dini”, Viale Morgagni 67/a, 50134 Firenze, Italy, Member of G.N.A.M.P.A. (I.N.d.A.M.) E-mail address : [email protected] Matias Vestberg, Department of Mathematics and Systems Analysis, Aalto University, P. O. Box11100, FI-00076 Aalto University, Finland E-mail address ::