0 weakly inET, (1.1) whereA:ET×RN+1→RN is measurable and subject to the structure conditions (A(x, t, z, ξ)·ξ≥C0m|z|m−1|ξ|2 |A(x, t, z, ξ)| ≤C1m|z|m−1|ξ| (1.2) for a.e

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

H ¨OLDER REGULARITY FOR SIGNED SOLUTIONS TO SINGULAR POROUS MEDIUM TYPE EQUATIONS

SIMONA PUGLISI

Abstract. We prove H¨older regularity for bounded signed solution to singular porous medium type equations, whose prototype is

ut−divm|u|^m−1Du= 0 weakly inET, withm∈(0,1).

1. Introduction and statement of main result LetE be an open set inR^N, forT >0 denote the cylindrical domain

ET =E×(0, T]

and let Γ = ∂ET \E¯ × {T} be its parabolic boundary. We consider quasi-linear homogeneous singular parabolic partial differential equation

u_t−divA(x, t, u, Du) = 0 weakly inE_T, (1.1) whereA:ET×R^N+1→R^N is measurable and subject to the structure conditions

(A(x, t, z, ξ)·ξ≥C₀m|z|^m−1|ξ|²

|A(x, t, z, ξ)| ≤C₁m|z|^m−1|ξ| (1.2) for a.e. (x, t) ∈ ET, for every z ∈ R, ξ ∈ R^N, where C0, C1 are given positive constants and 0< m <1.

The prototype of this class of parabolic equations is the porous medium equation ut−divm|u|^m−1Du= 0 weakly inET.

The modulus of ellipticity of this class of parabolic equations ism|u|^m−1. Whenever m >1, such a modulus vanishes whenuvanishes, and for this reason we say that the equation (1.1)-(1.2) is degenerate. Whenever 0 < m < 1, such a modulus approaches infinity as u→0, and for this reason we say that the equation (1.1)- (1.2) issingular. One also speaks aboutslow, when m >1, orfast diffusion, when 0< m <1 (see the monograph [8]).

2000Mathematics Subject Classification. 35K67, 35B65, 35B45.

Key words and phrases. Singular parabolic equations; H¨older continuity.

c

2012 Texas State University - San Marcos.

Submitted July 17, 2012. Published November 15, 2012.

1

We are interested only inlocal solutions tosingular porous medium type equa- tion. The parameters{N, m, C0, C1} are the data, and we say that a generic con- stantγ=γ(N, m, C0, C1) depends upon the data, if it can be quantitatively deter- mined a priori only in terms of the indicated parameters. As usual, in the following the constantγ may change from line to line.

Let us give the notion of weak solution for this kind of equations as follows.

A functionu∈ C_loc(0, T;L²_loc(E)) with |u|^m ∈L²_loc(0, T;H_loc¹ (E)) is a local weak sub(super)-solution to (1.1) if for every compact setK ⊂E and every subinterval [t1, t2]⊂(0, T]

Z

K

uϕ dx

t₂ t₁+

Z t₂

t1

Z

K

[−uϕt+A(x, t, u, Du)·Dϕ]dx dt≤(≥) 0, for all non-negative test functionsϕ∈H_loc¹ (0, T;L²(K))∩L²_loc(0, T;H₀¹(K)).

Our aim is to show that locally bounded, local, weak solutions of variable sign to our problem (1.1)-(1.2), with 0< m <1, are locally H¨older continuous.

Let us introduce the parabolic m-distance of a compact set K ⊂ E_T from the parabolic boundary Γ in the following way

m-dist(K,Γ) = inf

(x,t)∈K,(y,s)∈Γ

kuk

1−m

∞, E2 _T|x−y|+|t−s|^1/2 . We can state the main result of this paper as follows.

Theorem 1.1. Let u be a bounded, local, weak solution to (1.1)-(1.2). Then uis locally H¨older continuous inET and there exist constantsc >1 andα∈(0,1)such that for every compact setK ⊂ET

|u(x1, t1)−u(x2, t2)| ≤ckuk∞,ET

kuk_∞,E^1−m2

T|x₁−x₂|+|t₁−t₂|^1/2 m-dist(K,Γ)

^α , for every pair of points (x1, t1),(x2, t2)∈ K.

The constantcdepends only upon the data, the normkuk∞,K and m-dist(K,Γ);

the constantαdepends only upon the data and the normkuk∞,K.

In some physical applications it is natural to consider positive solutions to quasi- linear parabolic equations of the form (1.1), and it is also a very useful simplification from the mathematical point of view. Therefore, most of the papers directly deal with positive solutions.

A H¨older regularity result for signed solutions was obtained first by DiBenedetto in [3] for degenerate (p > 2) p-laplacian type equations and then by Chen and DiBenedetto in [1] for singular (1 < p < 2) p-laplacian type ones (see also [4]).

Later on, in 1993 Porzio and Vespri [7] considered the case of a degenerate doubly non-linear equation, whose prototype is

u_t−div |u|^m−1|Du|^p−2Du

= 0,

for p ≥2 and m ≥ 1. Notice that this kind of equations admits as a particular case both the degeneratep-laplacian type equations (form= 1 andp >2) and the degenerate porous medium type equations (forp= 2 andm >1). As a consequence, it only remained open the case of the singular porous medium type equations.

We want to point out that the difficulty in our case is due to the presence of the term|u|^m−1in the modulus of continuity; indeed, the fact thatuchanges sign plays a crucial role here. In thep-laplacian case, the modulus of continuity is |Du|^p−2,

thus the proof does not change ifuis positive or if it changes sign. One could think to follow the lines of [1] with minor modification, but at some point it will appear

|u|^m−1that one cannot control from above in a sublevel of the modulus ofu, being 0< m <1.

An important point of our strategy is to work with a different equation, ap- parently more complicated, but instead easier to handle, to which we can reduce, thanks to a change of variables introduced by Vespri in [9]. We will apply a tech- nique due to DiBenedetto [3, 4] via an alternative argument; we will write energy estimates for super(sub)-solutions and logarithmic estimates. We notice that, due to the change of variables, our logarithmic function has to be different by the usual one (see for instance [4]). Then we will use the so-called reduction of oscillation procedure: the H¨older continuity of a solutionuto the transformed equation (2.2) will be heuristically a consequence of the following fact: for every (x0, t0) ∈ ET

there exists a family of nested and shrinking cylinders in which the essential oscil- lation of u goes to zero in a way that can be quantitatively determined in terms of the data. Since this result is well known for non-negative solutions (see [4, 5]), it will suffice to consider the case in which the infimum of our solution is negative and the supremum is positive.

2. Change of variables

To justify some of the following calculations, we assumeuto be smooth. In no way this is a restrictive assumption: indeed the modulus of continuity ofuwill play no role in the forthcoming calculations.

Let us considern∈Nsuch that

n > 1 m, and define

|v|ⁿ⁻¹v=u, which is equivalent to

v=|u|¹ⁿ⁻¹u.

Notice that

Du=n|v|ⁿ⁻¹Dv, Dv= 1

n|u|ⁿ¹⁻¹Du.

With this substitution equation (1.1) becomes

|v|ⁿ⁻¹v

t−divA(x, t, v, Dv) = 0e weakly inET, where

A(x, t, v, Dv) =e A(x, t, u, Du)

_u=|v|n−1v. Now, let us see what the structure conditions become. We have

A(x, t, v, Dv)e ·Dv= 1

n|u|ⁿ¹⁻¹A(x, t, u, Du)·Du

≥m

n C₀|u|^1n+m−2|Du|²

=nmC0|v|^1+nm−2n|v|²⁽ⁿ⁻¹⁾|Dv|²

=nmC0|v|^nm−1|Dv|²;

since the exponent isnm−1>0, the equation is “degenerate”.

In the same way

|A(x, t, v, Dv)|e =|A(x, t, u, Du)| ≤mC1|u|^m−1|Du|

=mC1|v|^n(m−1)n|v|ⁿ⁻¹|Dv|=nmC1|v|^nm−1|Dv|.

If we denote our variable withuagain, we are then led to consider equations of the type

(|u|ⁿ⁻¹u)_t−divA(x, t, u, Du) = 0e weakly inE_T, with structure conditions

(

A(x, t, z, ξ)e ·ξ≥nmC0|z|^nm−1|ξ|²

|A(x, t, z, ξ)| ≤e nmC1|z|^nm−1|ξ|, (2.1) for a.e. (x, t)∈ET and for everyz∈R,ξ∈R^N.

Without loss of generality, we can assumento be odd; in this case

|u|ⁿ⁻¹u=uⁿ, and we can rewrite the equation as

(uⁿ)t−divA(x, t, u, Du) = 0e weakly inET. (2.2) Hence we have reduced problem (1.1)-(1.2) to (2.2) with structure conditions (2.1).

Let us now see what the notion of weak solution becomes in this new setting. A functionusuch thatuⁿ ∈Cloc(0, T;L²_loc(E)) with|u|^nm∈L²_loc(0, T;H_loc¹ (E)) is a local weak sub(super)-solution to (2.2) if for every compact set K ⊂E and every subinterval [t1, t2]⊂(0, T]

Z

K

uⁿϕ dx

t2

t1+ Z t₂

t₁

Z

K

[−uⁿϕt+A(x, t, u, Du)e ·Dϕ]dx dt≤(≥) 0, for all non-negative test functionsϕ∈H_loc¹ (0, T;L²(K))∩L²_loc(0, T;H₀¹(K)).

3. Preliminaries Letr, s≥1 and let us consider the Banach spaces

V^r,s(E_T) =L^∞ 0, T;L^r(E)

∩L^s 0, T;W^1,s(E) , V₀^r,s(E_T) =L^∞ 0, T;L^r(E)

∩L^s 0, T;W₀^1,s(E) , both equipped with the norm

kvk_Vr,s(ET)= ess sup

0<t<T

kv(·, t)kr,E+kDvks,E_T;

when r = s, let V^r,r(E_T) = V^r(E_T) and V₀^r,r(E_T) = V₀^r(E_T). Both spaces are embedded inL^q(E_T), for someq > s(for a proof one can see [4]).

Proposition 3.1. Ifv∈V₀^r,s(E_T), then there exists a positive constantγ, depend- ing only uponN, r,ands, such that

Z Z

E_T

|v|^qdx dt≤γ^qZ Z

E_T

|Dv|^sdx dt ess sup

0<t<T

Z

E

|v|^rdxs/N

withq=s^N_N^+r. In particular

kvkq,ET ≤γkvkV^r,s(E_T).

Note that, taking r = s in the previous proposition, and applying H¨older in- equality, one obtains the following result.

Proposition 3.2. Ifv∈V₀^r(ET), then there exists a positive constant γdepending only upon N andr, such that

kvk^r_r,ET ≤γ

{|v|>0}

r

N+rkvk^r_Vr(ET).

Given (y, s)∈ET, andλ, R >0, we will denote byKR(y) the cube centered at y with edge 2R; i.e.,

K_R(y) =n

x∈R^N : max

1≤i≤N|x_i−y_i|< Ro ,

and let∂K_R(y) be its boundary. Let (y, s) +Q_R(λ) be the generic cylinder (y, s) +Qρ(λ) =Kρ(y)×[s−λ, s].

Ifk∈R, introduce the truncated functions

(u−k)_±= max{±(u−k),0}.

The following lemma, proved in [2], will be very useful in the sequel.

Lemma 3.3. Let v ∈ W^1,1(Kρ(y)) and let k, l ∈ R, with k < l. There exists a constant γ=γ(N, p) independent ofk, l, v, y, ρsuch that

(l−k)|{v > l}| ≤γ ρ^N+1

|{v < k}|

Z

{k<v<l}

|Dv|dx. (3.1)

Let us state now a lemma on fast geometric convergence one can find in [2]; for a simple proof see again [4] and [6].

Lemma 3.4. Let {Yn}_n∈N be a sequence of positive numbers satisfying Y_n+1≤CbⁿY_n^1+α,

beingC, b >1 andα >0. If

Y0≤C^−α1b^−α12, thenYn converges to 0, as ntends to+∞.

Let us prove energy estimates we will need later. We start with estimates for super-solutions, then we will state the analogous ones for sub-solutions.

Proposition 3.5 (Energy estimates for super-solutions). Let u be a local, weak super-solution to (2.1)-(2.2) in ET. There exists a positive constant γ, depending only upon the data, such that for every cylinder (y, s) +QR(λ)⊂ET, every level k ∈ R and every non-negative, piecewise smooth cutoff function ζ vanishing on

∂K_R(y),

ess sup

s−λ<t≤s

Z

KR(y)

Z k

u

(k−s)+sⁿ⁻¹ds

ζ²(x, t)dx +

Z Z

(y,s)+Q_R(λ)

|u|^nm−1

D[(u−k)₋ζ]

2dx dτ

≤γnZ

K_R(y)

Z k

u

(k−s)+sⁿ⁻¹ds

ζ²(x, s−λ)dx +

Z Z

(y,s)+Q_R(λ)

Z k

u

(k−s)+sⁿ⁻¹ds

ζ|ζτ|dx dτ +

Z Z

(y,s)+Q_R(λ)

|u|^nm−1(u−k)²₋|Dζ|²dx dτo .

(3.2)

Proof. After a translation we may assume that (y, s) coincides with the origin and it suffices to prove (3.2) for the cylinder QR(λ). In the weak formulation of (2.2), take the test function

ϕ=−(u−k)₋ζ² overQ_t=K_R×(−λ , t],where−λ < t≤0.

Taking into account that

∂

∂τ Z k

u

(k−s)+sⁿ⁻¹ds

=−uⁿ⁻¹(u−k)−uτ, and estimating the various terms separately, we have first

− Z Z

Q_t

(uⁿ)τ(u−k)−ζ²dx dτ =n Z Z

Q_t

∂

∂τ Z k

u

(k−s)+sⁿ⁻¹ds ζ²dx dτ

≥n Z

K_R

Z k

u

(k−s)₊sⁿ⁻¹ds

ζ²(x, t)dx

−n Z

K_R

Z k

u

(k−s)₊sⁿ⁻¹ds

ζ²(x,−λ)dx

−2n Z Z

Q_t

Z k

u

(k−s)₊sⁿ⁻¹ds

ζ|ζ_τ|dx dτ.

From the structure conditions (2.1) and Young’s inequality it follows that

− Z Z

Q_t

A(x, τ, u, Du)De

(u−k)₋ζ² dx dτ

=− Z Z

Q_t

A(x, τ, u, Du)D(ue −k)₋ζ²dx dτ

−2 Z Z

Q_t

A(x, τ, u, Du)(ue −k)₋ζDζ dx dτ

≥nmC₀ Z Z

Q_t

|u|^nm−1|D(u−k)₋|²ζ²dx dτ

−2nmC₁ Z Z

Q_t

|u|^nm−1|D(u−k)₋|(u−k)₋ζ|Dζ|dx dτ

≥nmC₀ 2

Z Z

Q_t

|u|^nm−1

D[(u−k)₋ζ]

2dx dτ

−2nmC₁² C₀

Z Z

Qt

|u|^nm−1(u−k)²₋|Dζ|²dx dτ.

Combining these estimates and taking the supremum over t ∈(−λ,0], completes

the proof.

Proposition 3.6(Energy estimates for sub-solutions). Letube a local, weak sub- solution to (2.1)-(2.2) in E_T. There exists a positive constant γ, depending only upon the data, such that for every cylinder(y, s) +Q_R(λ)⊂E_T, every levelk∈R

and every non-negative, piecewise smooth cutoff functionζ vanishing on ∂KR(y), ess sup

s−λ<t≤s

Z

KR(y)

Z u

k

(s−k)+sⁿ⁻¹ds

ζ²(x, t)dx +

Z Z

(y,s)+Q_R(λ)

|u|^nm−1

D[(u−k)+ζ]

2dx dτ

≤γnZ

KR(y)

Z u

k

(s−k)+sⁿ⁻¹ds

ζ²(x, s−λ)dx +

Z Z

(y,s)+Q_R(λ)

Z u

k

(s−k)₊sⁿ⁻¹ds

|ζτ|dx dτ +

Z Z

(y,s)+QR(λ)

|u|^nm−1(u−k)²₊|Dζ|²dx dτo .

(3.3)

Proof. The proof is analogous to the previous one; we just need to take the test functionϕ= (u−k)+ζ² and observe that

∂

∂τ Z u

k

(s−k)+sⁿ⁻¹ds

=uⁿ⁻¹(u−k)+uτ.

Let us introduce the logarithmic function

ψ(Hⁿ,(uⁿ−kⁿ)₊, νⁿ) = log⁺ Hⁿ

Hⁿ−(uⁿ−kⁿ)++νⁿ

, where

Hⁿ = ess sup

(y,s)+Q_R(λ)

(uⁿ−kⁿ)+, 0< νⁿ<min{1, Hⁿ}, and fors >0

log⁺s= max{logs,0}.

Proposition 3.7(Logarithmic estimates). Let ube a local, weak solution to (2.1)- (2.2) in E_T. There exists a positive constant γ, depending only upon the data, such that for every cylinder (y, s) +Q_R(λ) ⊂ E_T, every level k ∈ R and every non-negative, piecewise smooth cutoff function ζ=ζ(x)

ess sup

s−λ<t≤s

Z

KR(y)

ψ² Hⁿ,(uⁿ−kⁿ)+, νⁿ

(x, t)ζ²(x)dx

≤ Z

K_R(y)

ψ²(Hⁿ,(uⁿ−kⁿ)₊, νⁿ) (x, s−λ)ζ²(x)dx +γ

Z Z

(y,s)+QR(λ)

|u|^n(m−1)ψ(Hⁿ,(uⁿ−kⁿ)+, νⁿ)|Dζ|²dx dτ.

(3.4)

Proof. Again we assume that (y, s) coincides with the origin. Putv =uⁿ and, in the weak formulation of (2.2), take the test function

ϕ=∂ψ²

∂v ζ²= 2ψψ⁰ζ², overQt=KR×(−λ, t], where−λ < t≤0.

By direct calculation

ψ²00

= 2(1 +ψ)(ψ⁰)²∈L^∞_loc(E_T), (3.5)

which implies that such a ϕ is an admissible testing function. Estimating the various terms separately, we have

Z Z

Q_t

vτ

∂ψ²

∂v ζ²dx dτ = Z Z

Q_t

∂

∂τψ²ζ²dx dτ

= Z

K_R

ψ²(x, t)ζ²(x)dx− Z

K_R

ψ²(x,−λ)ζ²(x)dx;

using (3.5) and the structure conditions (2.1) Z Z

Q_t

A(x, τ, u, Du)De ∂ψ²

∂v ζ² dx dτ

= Z Z

Q_t

A(x, τ, u, Du)Dv(ψe ²)⁰⁰ζ²dx dτ + 2 Z Z

Q_t

(ψ²)⁰ζA(x, τ, u, Du)Dζ dx dτe

= 2n Z Z

Q_t

uⁿ⁻¹A(x, τ, u, Du)Du(1 +e ψ)(ψ⁰)²ζ²dx dτ + 4

Z Z

Q_t

ψψ⁰ζA(x, τ, u, Du)Dζ dx dτe

≥2n²mC0

Z Z

Q_t

uⁿ⁻¹|u|^nm−1|D(u−k)+|²(1 +ψ)(ψ⁰)²ζ²dx dτ

−4nmC1

Z Z

Q_t

|u|^nm−1|D(u−k)+|ζ|Dζ|ψψ⁰dx dτ.

Applying Young’s inequality, we obtain Z Z

Q_t

A(x, τ, u, Du)De ∂ψ²

∂v ζ² dx dτ

≥2nm(nC₀−C₁ε²) Z Z

Q_t

|u|^nm−1|u|ⁿ⁻¹|D(u−k)₊|²ψ(ψ⁰)²ζ²dx dτ

−2nmC₁ ε²

Z Z

Q_t

|u|^n(m−1)|Dζ|²ψ dx dτ.

Combining these estimates, discarding the term with the gradient on the left-hand side, and taking the supremum overt∈(−λ ,0], proves the proposition.

4. Reduction of the oscillation

To obtain the H¨older regularity, we argue as usual with this kind of estimate by a reduction-of-oscillation procedure. Let us state the basic result.

Theorem 4.1. Let (y, s)∈ET, andρ, ω >0 such that (y, s) +Q2θρ

(2ρ)² ω^nm−1

⊂ET, ess osc

(y,s)+Q_2θρ

(2ρ)2 ωnm−1

u≤ω , where

θ=ω¹⁻ⁿ² .

Then, there exist η_∗, c0∈(0,1), depending only upon data, such that ess osc

Q^∗ u≤η_∗ω ,

where

Q^∗= (y, s) +Qθρ θ_∗ρ²

, θ_∗=c0

2 ω^1−nm.

As we show at the end, the local H¨older continuity of locally bounded solutions is a straightforward consequence of Theorem 4.1. The proof of this theorem splits into two alternatives.

Let∈(0,1), R >0, and (y, s)∈ET. Consider the cylinder Q:=K

R¹⁻

n−1

2 (y)×(s−R^2−(nm−1), s]⊂ET, and set

µ₊≥ ess sup

(y,s)+Q

u , µ₋≤ ess inf

(y,s)+Q

u , ω=µ₊−µ₋.

Let us recall that, without loss of generality, we can assumeµ₊>0,µ₋<0 and µ+≥ |µ₋|.

Indeed, otherwise just change the sign ofuand work with the new function.

If we take 2ρ < R, and assume without loss of generality

ω > R, (4.1)

then we guarantee that

(y, s) +Q_2θρ (2ρ)² ω^nm−1

⊂Q. 5. The first alternative

We distinguish two alternatives; the first of them consists in assuming

n

u < µ₋+ω 2

o∩n

(y, s) +Q_2θρ (2ρ)² ω^nm−1

o ≤c₀

Q_2θρ (2ρ)² ω^nm−1

, (5.1) beingc0∈(0,1) a constant to be determined later.

Let us prove now the following De Giorgi type lemma.

Lemma 5.1. There exists a number c0 ∈(0,1), depending only upon data, such that if (5.1)holds, then

u≥µ−+ω

4 a.e. in(y, s) +Qθρ

ρ² ω^nm−1

. (5.2)

Proof. Without loss of generality we may assume (y, s) = (0,0) and fork= 0,1, . . ., set

ρk =ρ+ ρ

2^k, Kek =Kθρ_k, Qek=Kek× − ρ²_k ω^nm−1,0

.

Let ζ_k be a piecewise smooth cutoff function in Qe_k vanishing on the parabolic boundary ofQe_k such that 0≤ζ_k ≤1,ζ_k= 1 inQe_k+1 and

|Dζk| ≤ 2^k+2

ρ ωⁿ⁻¹² , 0≤ζk,t≤ 2^k

ρ²ω^nm−1. We consider the following levels

h_k =µ₋+ω 4 + ω

2^k+2 ifµ₋≥ −ω 8, h_k =µ₋+ ω

2⁵ + ω

2^k+5 ifµ₋<−ω 8.

(5.3)

We first treat the least favorable case in which umight be close to zero; i.e., we assume first that

µ₋≥ −ω

8. (5.4)

Write down the energy estimates (3.2) for (u−hk)−over the cylinderQek, to obtain ess sup

− ^ρ2k

ωnm−1<t≤0

Z

Ke_k

Z hk

u

(h_k−s)₊sⁿ⁻¹ds

ζ_k²(x, t)dx

+ Z Z

Qe_k

|u|^nm−1

D[(u−hk)−ζk]

2dx dτ

≤γnZ Z

Qek

Z hk

u

(hk−s)+sⁿ⁻¹ds

|ζk,τ|dx dτ +

Z Z

Qek

|u|^nm−1(u−hk)²₋|Dζk|²dx dτo . Let us introduce the truncation

v= max u, ω

2⁴

,

in order to estimate the terms with the integral over [u, hk]; we have Z hk

u

(h_k−s)₊sⁿ⁻¹ds≥ Z hk

v

(h_k−s)₊sⁿ⁻¹ds

≥vⁿ⁻¹(v−hk)²₋

2 ≥ω

2⁴

n−1(v−hk)²₋

2 .

(5.5)

On the other hand, as (u−h_k)₋≤ω and−^ω₈ ≤µ₋<0, we have Z hk

u

(h_k−s)₊sⁿ⁻¹ds≤hⁿ⁻¹_k (u−hk)²₋

2 ≤ωⁿ⁺¹

2 . (5.6)

By the definition ofv, we obtain Z Z

Qe_k

v^nm−1

D[(v−hk)−ζk]

2dx dτ

= Z Z

Qe_k∩{^u>₂₄^ω}|u|^nm−1

D[(u−hk)−ζk]

2dx dτ

+ Z Z

Qe_k∩{^u≤₂₄^ω} ω

2⁴

nm−1ω 2⁴−hk

²

−|Dζk|²dx dτ

≤ Z Z

Qe_k

|u|^nm−1

D[(u−hk)₋ζk]

2dx dτ+2^2(k+1)

ρ² ω^n(m+1)|Ak|,

(5.7)

where

Ak ={u < hk} ∩Qek. Let us observe that

A_k =Ae_k:={v < hk} ∩Qe_k. (5.8) Indeed, the inclusionAk ⊇Aek follows by the definition of v; let us now prove the other one: ifv=uthere is nothing to prove; ifv=₂^ω₄, by (5.4) we have

h_k=µ₋+ω 4 + ω

2^k+2 ≥ω 8 + ω

2^k+2 ≥ ω 2⁴.

Taking into account that|u| ≤ω, (5.5)-(5.8) yield ω

2⁴ n−1

ess sup

− ^ρ

2 k ωnm−1<t≤0

Z

Kek

(v−hk)²₋ζ_k²(x, t)dx+ Z Z

Qek

v^nm−1

D[(v−hk)₋ζk]

2dx dτ

≤γ2^2k

ρ² ω^n(m+1)|Aek|,

and again, thanks to the definition ofv, it follows that ess sup

− ^ρ2k

ωnm−1<t≤0

Z

Ke_k

(v−hk)²₋ζ_k²(x, t)dx+ω 2⁴

n(m−1)Z Z

Qe_k

D[(v−hk)₋ζk]

2dx dτ

≤γ2^2k

ρ² ω^nm+1|Aek|.

(5.9) The change of variables

¯

x=x θ⁻¹, ¯t=ω^nm−1τ

maps the cube Kek intoKρ_k, and the cylinder Qek intoQk =Kρ_k×(−ρ²_k,0]. With (¯x,¯t) →u(¯x,¯t) denoting again the transformed function, the assumption (5.1) of the lemma implies

n

v < µ−+ω 2

o∩Q0

≤c0|Q0|. (5.10)

Performing such a change of variables in (5.9), we have ess sup

−ρ²_k<t≤0

Z

K_ρk

(v−h_k)²₋ζ_k²(¯x, t)d¯x+ Z Z

Q_k

D[(v−h_k)₋ζ_k]

2d¯xdt¯

≤γ2^2k

ρ² ω²|A¯k|, where

A¯k={v < hk} ∩Qk. This implies

(v−hk)₋ζk

2

V²(Q_k)≤γ2^2k

ρ² ω²|A¯k|. (5.11) Then from Proposition 3.2 withr= 2 and (5.11), one obtains

Z Z

Q_k+1

(v−hk)²₋d¯xdt¯≤ Z Z

Q_k

(v−hk)²₋ζ_k²d¯xdt¯

≤γ|{v < hk} ∩Qk|^N+22

(v−hk)₋ζk

2 V²(Q_k)

≤γ2^2k

ρ² ω²|A¯k|^1+N+22 ; the left-hand side is estimated by

Z Z

Q_k+1

(v−h_k)²₋d¯xd¯t= Z Z

Q_k+1∩{v<h_k}

(h_k−v)²d¯xd¯t

≥ Z Z

Qk+1∩{v<hk+1}

(hk−v)²d¯xd¯t

≥(h_k−h_k+1)²|A¯_k+1|

= ω 2^k+3

2

|A¯k+1|.

Combining the previous estimates yields

|A¯_k+1| ≤γ2^4k

ρ²|A¯_k|^1+N+22 , and setting

Yk= |A¯k|

|Qk|, it follows that

Yk+1≤γ2^4kY¹⁺

2 N+2

k .

Thanks to Lemma 3.4, we deduce thatYk tends to zero ask→ ∞, provided Y₀= |{v < h0} ∩Q0|

|Q0| = |

v < µ₋+^ω₂ ∩Q0|

|Q0| ≤γ^−N+22 2^−(N+2)2, that is (5.10), withc0:=γ^−N+22 2^−(N+2)2.

Therefore,

v≥µ−+ω

4 a.e. inKρ×(−ρ²,0].

Returning to the variablesx, t, we have v≥µ₋+ω

4 a.e. inQ_θρ ρ² ω^nm−1

; (5.12)

this implies thatu=vin Q_{θρ ω}nm−1^ρ2

and, consequently, (5.2). In fact, by contra- diction, if there were a point (x, t)∈Qθρ ρ²

ω^nm−1

such thatv(x, t) = ₂^ω4, by (5.12) and (5.4), we would obtain

ω

2⁴ ≥µ−+ω 4 ≥ ω

8 .

Assume now that (5.4) is violated; that is, µ₋ < −^ω₈. Choosing the levels hk

according to (5.3), we have hk=µ−+ ω

2⁵ + ω

2^k+5 <−ω 8 + ω

2⁵+ ω

2^k+5 ≤ −ω 2⁵. Thus on the set{u≤hk}, one has

|u|^nm−1≥ω 2⁵

nm−1

.

It follows that|u|^nm−1can be estimate above and below byω^nm−1up to a constant, depending only upon the data; the proof can be repeated as before, but in this case there is no need to introduce the truncated functionv.

Therefore under assumption (5.1),

− ess inf

Q_θρ( ^ρ2

ωnm−1)

u≤ −µ₋−ω 4; adding ess sup

Q_θρ( ^ρ2

ωnm−1)

u, gives

ess osc

Qθρ( ^ρ2

ωnm−1)

u≤ ess sup

Qθρ( ^ρ2

ωnm−1)

u−µ₋−ω 4 ≤ 3

4ω.

6. The second alternative

Let us recall the two fundamental hypotheses we assume, namely µ₊>0, µ₋<0, µ₊≥ |µ₋|.

Throughout this new section, let us assume that (5.1) does not hold; i.e.,

u≥µ₋+ω 2 ∩n

(y, s) +Q2θρ

(2ρ)² ω^nm−1

o

<(1−c0) Q2θρ

(2ρ)² ω^nm−1

. For simplicity in the following we assume (y, s) = (0,0).

Lemma 6.1. There exists a time level t^∗ in the interval − _ω^(2ρ)nm−1² ,−^c₂⁰_ω^(2ρ)nm−1²

such that

u(·, t^∗)< µ₋+ω

2 ∩K2θρ

>c0

2|K2θρ|. (6.1)

This in turn implies

u(·, t^∗)≥µ+−ω

4 ∩K2θρ

≤

1−c0

2

|K2θρ|. (6.2) Proof. By contradiction, suppose that (6.1) does not hold for anyt^∗in the indicated range; then

u < µ₋+ω

2 ∩Q2θρ

(2ρ)² ω^nm−1

=

Z −^c₂^{0 (2ρ)2}

ωnm−1

− ^(2ρ)2

ωnm−1

u(·, t^∗)< µ₋+ω

2 ∩K2θρ

dt^∗ +

Z 0

−^c₂^{0 (2ρ)2}

ωnm−1

u(·, t^∗)< µ₋+ω

2 ∩K_2θρ dt^∗

≤c₀

2|K2θρ| 1−c₀

2

(2ρ)²

ω^nm−1 +|K2θρ|c₀ 2

(2ρ)² ω^nm−1

< c0

Q2θρ

(2ρ)² ω^nm−1

.

This proves (6.1); (6.2) follows by the fact that (6.1) is equivalent to

u(·, t^∗)≥µ−+ω

2 ∩K2θρ

<

1−c0

2

|K2θρ|,

andµ−+^ω₂ ≤µ+−^ω₄.

The next lemma asserts that a property similar to (6.2) continues to hold for all time levels fromt^∗ up to zero.

Lemma 6.2. There exists a positive integer j^∗, depending upon the data andc₀, such that

u(·, t)> µ₊− ω

2^j∗ ∩K_2θρ

< 1−c²₀ 4

|K2θρ|, for all timest^∗< t <0.

Proof. Consider the logarithmic estimates (3.4) written over the cylinder K_2θρ× (t^∗,0) for the function (uⁿ−kⁿ)+ and for the level k = µⁿ₊−(^ω₄)^n1/n

. Notice that, thanks to our assumptions,µ+>^ω₄, sok >0. The numberν in the definition of the logarithmic function is taken asν = ₂j+2^ω , where j is a positive number to be chosen. Thus we have

ψ(Hⁿ,(uⁿ−kⁿ)₊, νⁿ) = log⁺ Hⁿ

Hⁿ−(uⁿ−kⁿ)++_2(j+2)n^ωn

,

where

Hⁿ= ess sup

K_2θρ×(t^∗,0)

h

uⁿ− µⁿ₊−(ω 4)ⁿi

+

. The cutoff functionx→ζ(x) is taken such that

ζ= 1 onK_{(1−σ)2θρ} forσ∈(0,1), |Dζ| ≤ 1 σθρ. With these choices, inequality (3.4) yields

Z

K_{(1−σ)2θρ}

ψ²(x, t)dx

≤ Z

K2θρ

ψ²(x, t^∗)dx+γ Z 0

t^∗

Z

K2θρ

|u|^n(m−1)ψ|Dζ|²dx dτ,

(6.3)

for allt^∗≤t≤0. Let us observe that ψ≤log

ωⁿ 2²ⁿ ωⁿ 2^(j+2)n

=jnlog 2.

To estimate the first integral on the right-hand side of (6.3), observe thatψvanishes on the set{uⁿ< kⁿ} and thatµⁿ₊− ^ω₄n

≥ µ+−^ω₄n

; therefore by (6.2) Z

K_2θρ

ψ²(x, t^∗)dx≤j²n²log²2 1−c₀

2

|K_2θρ|.

The remaining integral is estimated as follows γ

Z 0

t^∗

Z

K_2θρ

|u|^n(m−1)ψ|Dζ|²dx dτ

≤ γ

(σθρ)²jnlog 2 (2ρ)²

ω^nm−1ω^n(m−1)|K2θρ|= γ

σ²jn|K2θρ|.

Combining the previous estimates, Z

K(1−σ)2θρ

ψ²(x, t)dx≤n

j²n²log²2 1−c0

2

+ γ σ²jno

|K2θρ| (6.4) for allt^∗≤t≤0. The left-hand side of (6.4) is estimated below by integrating over the smaller set

n

uⁿ> µⁿ₊− ωⁿ 2^(j+2)n

o

;

on such a set, sinceψis a decreasing function of Hⁿ, we have ψ²≥log^{2 ω}

n

2²ⁿ ωⁿ 2^(j+1)n

= (j−1)²n²log²2;

hence, for allt^∗≤t≤0, we obtain

n

uⁿ(·, t)> µⁿ₊− ωⁿ 2^(j+2)n

o∩K_{(1−σ)2θρ}

≤n j j−1

² 1−c0

2

+ γ σ²j

o|K2θρ|.

On the other hand,

nuⁿ(·, t)> µⁿ₊− ωⁿ 2^(j+2)n

o∩K_2θρ

≤ n

uⁿ(·, t)> µⁿ₊− ωⁿ 2^(j+2)n

o∩K_{(1−σ)2θρ}

+|K2θρ\K_{(1−σ)2θρ}|

≤ n

uⁿ(·, t)> µⁿ₊− ωⁿ 2^(j+2)n

o∩K_{(1−σ)2θρ}

+N σ|K2θρ|.

Then n

uⁿ(·, t)> µⁿ₊− ωⁿ 2^(j+2)n

o∩K2θρ

≤n j j−1

2 1−c0

2

+ γ

σ²j +N σo

|K2θρ|.

for allt^∗≤t≤0. Now chooseσso small and thenj so large as to obtain

u(·, t)>

µⁿ₊− ωⁿ 2^(j+2)n

^1/n

∩K2θρ

≤ 1−c²₀ 4

|K2θρ| ∀t^∗≤t≤0.

Notice that our hypotheses implyµ+≥ ^ω₂, µ+ < ω; therefore,

µⁿ₊− ωⁿ 2^(j+2)n

1/n

<

µⁿ₊− µⁿ₊ 2^(j+2)n

1/n

=µ₊

1− 1

2^(j+2)n 1/n

≤µ₊

1− 1

2^(j+2)nn

≤µ₊− ω 2^(j+2)n+1n. The proof is completed once we choosej^∗ as the smallest integer such that

µ+− ω

2^(j+2)n+1n ≤µ+− ω

2^j∗.

Corollary 6.3. For allj≥j^∗ and for all times −^c₂⁰_ω^(2ρ)nm−1² < t <0,

nu(·, t)> µ₊− ω 2^j

o∩K_2θρ <

1−c²₀ 4

|K_2θρ|. (6.5) Motivated by Corollary 6.3, introduce the cylinder

Q_∗=K2θρ× −θ_∗(2ρ)²,0

, withθ_∗= c0

2ω^1−nm.

Lemma 6.4. For everyν_∗∈(0,1), there exists a positive integer q_∗=q_∗(data, ν_∗) such that

n

u≥µ+− ω 2^j∗+q∗

o∩Q_∗

≤ν_∗|Q_∗|.

Proof. Write down the energy estimates (3.3) for the truncated functions (u−k_j)₊, withk_j=µ₊−₂^ω_j, forj =j_∗, . . . , j_∗+q_∗ over the cylinder

Qe=K4θρ×

−c0

(2ρ)² ω^nm−1,0i

⊃Q_∗;

the cutoff functionζis taken to be one onQ_∗, vanishing on the parabolic boundary ofQe and such that

|Dζ| ≤ 1

θρ, 0≤ζt≤ ω^nm−1 c0ρ² .

Thanks to these choices, the energy estimates (3.3) take the form Z Z

Q˜

|u|^nm−1|D(u−kj)+|²ζ²dx dτ

≤γnω^nm−1 c₀ρ²

Z Z

Q˜

Z u

kj

(s−kj)+sⁿ⁻¹ds dx dτ

+ωⁿ⁻¹ ρ²

Z Z

Q˜

|u|^nm−1(u−kj)²₊dx dτo .

Estimating Z u

k_j

(s−kj)+sⁿ⁻¹ds≤uⁿ⁻¹(u−kj)²₊

2 ≤ωⁿ⁻¹(u−kj)²₊

2 ,

and taking into account that (u−k_j)₊≤₂^ωj, yields Z Z

Q˜

|u|^nm−1|D(u−kj)+|²ζ²dx dτ ≤γω 2^j

2

ωⁿ⁻¹ω^nm−1 c₀ρ² |Q_∗|.

Note thatu > kj≥ ^ω₄: indeed the second inequality is equivalent to µ+≥ |µ−|1

4 + 1 2^j

3 4− 1

2^j −1

, and this is implied by our assumptions. Thus we can estimate

Z Z

Qe

|u|^nm−1|D(u−kj)+|²ζ²dx dτ ≥ Z Z

Q∗

|u|^nm−1|D(u−kj)+|²dx dτ

≥ω 4

nm−1Z Z

Q∗

|D(u−kj)+|²dx dτ; it follows that

Z Z

Q∗

|D(u−kj)+|²dx dτ ≤γω 2^j

2

ωⁿ⁻¹ 1

c₀ρ²|Q_∗|. (6.6) Next, apply the isoperimetric inequality (3.1) to the function u(·, t), for t in the range (−θ_∗(2ρ)²,0], over the cubeK2θρ, and for the levels

k=k_j< l=k_j+1; in this way (l−k) =₂j+1^ω .

Taking into account (6.5), this gives ω

2^j+1|{u(·, t)> k_j+1} ∩K_2θρ|

≤ (2θρ)^N+1

|{u(·, t)< k_j} ∩K_2θρ| Z

{kj<u(·,t)<kj+1}∩K_2θρ

|Du|dx

≤ 8θρ c²₀

Z

{k_j<u(·,t)<k_j+1}∩K_2θρ

|Du|dx;

integrating indtover the indicated interval and applying the H¨older inequality, one gets

ω

2^j+1|Aj+1| ≤ 8θρ c²₀

Z Z

Q∗

|D(u−kj)+|²dx dt^1/2

(|Aj| − |Aj+1|)^1/2, where

Aj={u > kj} ∩Q_∗.

Square both sides of this inequality and estimate above the term containing|D(u−

k_j)₊|by inequality (6.6), to obtain

|A_j+1|²≤ γ

c⁵₀|Q_∗|(|A_j| − |A_j+1|).

Add these recursive inequalities for j = j∗+ 1, . . . , j∗+q∗−1, whereq∗ is to be chosen. Majorizing the right-hand side with the corresponding telescopic series, gives

(q_∗−2)|Aj∗+q∗|²≤

j∗+q∗−1

X

j=j∗+1

|Aj+1|²≤ γ c⁵₀|Q∗|². From this

|Aj∗+q∗| ≤ 1

√q_∗−2 r γ

c⁵₀ |Q∗|.

The numberν_∗being fixed, chooseq_∗ from

√ 1 q_∗−2

r γ

c⁵₀ =ν_∗.

Now letξ∈(0,¹₂),a∈(0,1) be fixed numbers.

Lemma 6.5. There exists a number c∗ ∈(0,1), depending upon the data, ξ, and a, such that if

| {u≥µ+−ξω} ∩Q_∗| ≤c_∗|Q_∗|, (6.7) then

u≤µ₊−aξω a.e. inQ_θρ(θ_∗ρ²).

Proof. Fork= 0,1, . . ., set ρk =ρ+ ρ

2^k, Kk =Kθρ_k, Qk =Kk×(−θ∗ρ²_k,0].

Letζ(x, t) =ζ₁(x)ζ₂(t) be a piecewise smooth cutoff function in Q_k such that ζ1=

(1 in K_k+1

0 in R^N\Kk, |Dζ1| ≤2^k+2 θρ ,

ζ2= (

1 ift≥ − ^ρ

2 k+1

ω^nm−1

0 ift <−_ωnm−1^ρ2k ,

0≤ζ2,t≤ 2^k θ_∗ρ². Choose the sequence of truncating levels

h_k=µ₊−ξ_kω, where ξ_k =aξ+1−a 2^k ξ,

and write down the energy estimates (3.3) for (u−hk)+ over the cylinderQk, ess sup

− ^ρ2k

ωnm−1<t≤0

Z

K_k

Z u

h_k

(s−h_k)₊sⁿ⁻¹ds

ζ²(x, t)dx

+ Z Z

Qk

|u|^nm−1

D[(u−hk)+ζ]

2dx dτ

≤γnZ Z

Q_k

Z u

h_k

(s−hk)+sⁿ⁻¹ds

|ζt|dx dτ +

Z Z

Q_k

|u|^nm−1(u−hk)²₊|Dζ|²dx dτo . Let us estimate

Z u

h_k

(s−h_k)₊sⁿ⁻¹ds≥hⁿ⁻¹_k (u−h_k)²₊

2 ,