Remarks on Hölder continuity for parabolic equations and convergence to global attractors

Nonlinear Analysis 41 (2000) 921–941www.elsevier.nl/locate/na

Remarks on H�older continuity for parabolicequations and convergence to global attractors

Le DungCenter for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology,

Atlanta, GA 30332-0190, USA

Received 5 March 1998; accepted 28 August 1998

Keywords: Degenerate parabolic equations; Weak solutions; H�older continuity; Global attractors

1. Introduction

In this paper we revisit the H�older continuity problem for bounded solutions ofquasilinear parabolic equations of the form

@u@t=div(a(x; t; u; Du)) + b(x; t; u; Du); (x; t)∈T ; (1.1)

where T = [0; T ]× with is a bounded open set in RN . The di�usion terms canbe nondegenerate or degenerate.First of all, the problem and related results are very old and well known. There is

already a considerable mathematical literature in the study of regularity of solutions tothis problem. First, the regularity result for nondegenerate linear parabolic equationswas proven by Nash [17] in 1957, and DeGiorgi in 1958 for elliptic equations. Later,Moser [16] extended his technique for elliptic equations to prove the Harnack inequalityfor bounded solutions of linear parabolic equations and thus obtain regularity results.Moser’s method had been then generalized to quasilinear cases by Aronson and Serrin[1] and Trudinger [19] (still for scalar nondegenerate case, see also [10]). This methodbases on the Harnack principle which is itself very important theoretically. However,the derivation is quite complicated and di�cult to cover degenerate cases (there areexamples where this principle fails, see [2]).

E-mail address: [email protected] (L. Dung)

0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(98)00319 -8

922 L. Dung / Nonlinear Analysis 41 (2000) 921–941

On the other hand, the method of level sets or truncation technique of DeGiorgi hadbeen generalized by Ladyzhenskaya et al. [13] and later by other authors (DiBenedetto[2, 3], Ivanov [12], Lieberman [14], Porzio and Vespri [18, 21]) to obtain H�olderestimates for weak solutions to quasilinear equations (nondegenerate or degenerate).The main ingredient in the above methods is to derive certain controllable decay

estimates for the oscillation of the solution in a sequence of nested cylinders whichimplies the H�older continuity. This decay estimate is a byproduct of the Harnackinequality in Moser’s method or the result of an analysis of two alternatives of the levelsets of the solution in DeGiorgi’s. The latter technique has been applied to degeneratecases (e.g. [2,18]) provided that one can show a dichotomy for the degenerate equation.Roughly speaking, if the solution is away from zero then the equation is nondegenerateand then the oscillation of the solution will decay in a controllable way. Otherwise,if the solution fails to be away from zero then it will decay in H�older fashion. Thelatter situation is easy to handle but the argument to obtain the decay estimate foroscillations has been very involved.Meanwhile, there is a less-well-known technique of using auxiliary logarithmic func-

tions to derive decay estimates of oscillations. This method had been used in [10] fornondegenerate elliptic equations and in [5, 7, 20] for degenerate and singular equations.The method seems to be more direct and less technical than the above ones. The mainingredients can be summarized as follows:1. Construct suitable logarithmic functions, denoted by w, whose upper bound impliesthe decay estimates.

2. Estimate the local suppremum of w in terms of its L2 norm. This can be done byobserving that w is a subsolution to certain equation.

3. Estimate the L2 norm of w and complete the proof, thanks to step 1.In this notes, we want to demonstrate that such a procedure for elliptic equations

can be carried over to the parabolic case. Although the results are not new as wementioned before but the approach is fairly basic and the extension is not trivialdue to the presence of the time derivative. Furthermore, it is interesting to see thatthe arguments can be applied to both nondegenerate and degenerate situations with-out any di�culty or major changes (in contrast to those mentioned before, where theproofs are very much di�erent). Steps 1 and 2 are simple and similar to those for theelliptic cases (see Lemma 2.3). Step 3 requires a little more new arguments based onelementary di�erential inequalities (see Lemma 2.4).Actually, the idea of using logarithmic functions is also very old. However, di�erent

forms of logarithmic functions had been used by other authors for di�erent purposes.Mainly, they were devised to obtain only certain auxiliary estimates as in Moser’s workto prove Harnack inequalities, or as in [13, Section III.10] to study the measure of thelevel sets. However, their uses did not play a direct role in obtaining decay estimatesas in the method presented here.In addition, we should also mention that the technique developed in this paper can

be extended to certain cases of systems with strong coupling (or cross-di�usion). Thekey ideas are similar (see [8]).In this note, we would like to emphasize the simplicity of the technique over the

more general setting of function spaces for the parameters of the equations so that our

L. Dung / Nonlinear Analysis 41 (2000) 921–941 923

assumptions are usually stronger than needed. However, one can see that the proofcan be carried over to the case when the parameters can be in Lp (see [20]) or evenCampanato–Morrey spaces as in [5, 7].On the other hand, the H�older estimates so obtained also imply the compactness of

the trajectories of solutions. The latter is very important when a dynamical system’sview point is taken into consideration (e.g. see [11]). In recent papers [6, 9], weconsidered a system of quasilinear parabolic equations of the form

@ui

@t=Aiui + fi(x; u; Dui); t¿0; x∈; i=1; : : : ; m;

Biui= v0i on @; t¿0;

ui(0; x)= u0i (x) in ;

where Ai’s are nondegenerate or degenerate elliptic operators of divergence form andBi’s are nonlinear boundary operators. Under certain conditions, we showed the L∞

dissipativity and therefore the existence of global attractors for the semi- ows generatedby the PDE systems. However, we were only able to obtain uniform L2 estimates forthe gradients of solutions and therefore the compactness of the trajectories in the L2

metric. Hence, we can assert that the solutions converge to the global attractors onlyin the L2 norm.Our regularity results here show that bounded solutions of nonlinear di�usion prob-

lems are H�older continuous and, more importantly, their H�older norms are uniformlybounded if so are the L∞ norms. This gives the compactness of trajectories in theL∞ metric and, therefore, the convergence of solutions to the global attractors in thisstronger norm.The paper is structured as follows: In Section 2 we demonstrate the main ideas

by considering the nondegerate case of Eq. (1.1). Then we extend the technique todegenerate equations in Section 3. Section 4 devotes to the study of regularity near theboundary and global estimates of H�older norms. Finally, we apply these results to thesystems considered in [6, 9] to improve the results on the convergence of solutions tothe global attractors.

2. H�older continuity for the nondegenerate case

In this section we use logarithmic functions to show that a bounded weak solutionu to a nondegenerate quasilinear equation of the form (1:1) is H�older continuous inthe interior of T . The result, Theorem 2.1, is not new but our approach is very basicand more elementary than those mentioned in the introduction.We impose the following structure condition on Eq. (1.1)

a(x; t; u; Du)Du≥ �0|Du|2 − 0(x; t);

|a(x; t; u; Du)| ≤ �1|Du|+ 1(x; t); (2.1)

|b(x; t; u; Du)| ≤ �2|Du|2 + 2(x; t):


The functions i’s on the right-hand side of Eq. (2.1) can belong to some Lp spacesor even certain Campanato–Morrey spaces as in [5,7]. We will not impose such a gen-erality here but simply assume that i’s are nonnegative bounded functions. However,by using certain weighted Sobolev inequalities as in [5, 7], one can see that the resultsstill hold under much more general assumptions.The following interior regularity result is well known (e.g. [13, Theorem 7.1]).

Theorem 2.1. Let u be a locally bounded weak solution of Eq. (1.1). AssumeEq. (2.1) and that �2 sup|u|¡�0. Then (x; t)→ u(x; t) is locally H�older continuousin the interior of T . That is, for every compact subset K of T , there exists aconstant C =C(‖u‖∞; K) and �= �(‖u‖∞; K) in (0; 1) such that

|u(x1; t1)− u(x2; t2)| ≤C(|x1 − x2|� + |t1 − t2|�=2)for every pair of points (x1; t1); (x2; t2)∈K .

Remark 2.2. The assumption �2 sup|u|¡�0 is actually inessential but makes our pre-sentation less technical. One can remove it easily by modifying the logarithmic func-tions de�ned below.

Let R; t0¿0 and x0 ∈. For i=1; 2; : : : we denote QiR=Bx0 (iR)× [t0 − iR2; t0] and

Mi= supQiR

u; mi= infQiR

u and !i=Mi − mi:

In Q4R, we consider the following logarithmic functions:

w1(x; t)= log(!4 + RN1(u)

); w2(x; t)= log

(!4 + RN2(u)

); (2.2)

where N1(u)= 2(M4 − u) + R; N2(u)= 2(u−m4) + R. Suppose that we can �nd some�nite constant C which is independent of R such that

w1(x; t)≤C or w2(x; t)≤C; ∀(x; t)∈Q4R: (2.3)

Then it is easy to see that either of the above inequality implies

!4≤C{!4 − !1 + R} or; equivalently; !1≤ �!4 + CR;

with �=(C − 1)=C¡1 and C are positive constants independent of R. This decayestimate for the oscillation of u in such nested cylinders and an elementary lemma in[10, lemma 8.23] give immediately the uniform H�older estimate for u. Hence, we needonly to show Eq. (2.3).First, we show that w1; w2 are subsolutions of some parabolic equations.

Lemma 2.3. For R small and for any nonnegative test function �, the functions w1; w2satisfy an inequality of the form∫

@w@t

� dx +∫�a(x; t; Dw)D� dx≤

∫

�b(x; t; Dw)� dx: (2.4)


The functions �a; �b satisfy the following structure conditions:

| �a(x; t; Dw)| ≤ �1|Dw|+ � 1;

�a(x; t; Dw)Dw≥ �0|Dw|2 − � 0; (2.5)

�b(x; t; Dw)≤ � 2;

where we can take � 0 = 4 0=N2(u), � 1 = 2 1=N (u) and � 2 = 2(2 0=N

2(u)+ 2=N (u)).

Proof. If w=w1, we denote N (u)=N1(u) and observe that

Dxw=2DxuN (u)

;@w@t=

2N (u)

@u@t

:

We multiply the equation of u by �= �=N (u) and integrate over to get

12

∫

@w@t

� dx +∫

(a(: : :)N (u)

D�+2a(: : :)�Du

N 2(u)

)dx=

∫

b(: : :)N (u)

� dx: (2.6)

Hence, w satis�es inequality (2:4) with

�a(x; t; Dw) :=2a(x; t; u(x; t); Du)

N (u(x; t));

�b(x; t; Dw) := 2(b(x; t; u(x; t); Du)

N (u(x; t))− 2a(: : :)Du

N 2(u)

):

From Eq. (2.1), we see that �a satis�es

| �a(x; t; Dw)| ≤ �1|Dw|+ 2 1N (u)

;

�a(x; t; Dw)Dw=4a(x; t; u; Du)Du

N 2(u)≥ �0|Dw|2 − 4 0

N 2(u):

Moreover, since

2a(: : :)DuN 2(u)

≥ 2�0|Du|2N 2(u)

− 2 0N 2(u)

(2.7)

and since N (u)≤ 2!+ R≤ 2 supQ4R |u|+ R, we also have

b(x; t; u(x; t); Du)N (u(x; t))

≤ �2

(2 sup

Q4R|u|+ R

)|Du|2N 2(u)

+ 2

N (u): (2.8)

So, if R is small, the fact that �2 sup |u|¡�0 implies that �b satis�es Eq. (2.5).Similar arguments show that w=w2 also satis�es an inequality of the same form asEq. (2.4).


Since w is a subsolution of a parabolic equation, it is well known that (using standarditeration technique of Moser as in [1, 15, 19], see also Lemma 3.5)

supBx0 (R)× [t0−R2 ; t0]

w≤Const( 1

Rn+2

∫∫Bx0 (2R)× [t0−2R2 ; t0]

w2 dx dt

)1=2+ 1

: (2.9)

We would like to remind the reader that, for simplicity, we are assuming that i arebounded functions.As mentioned before, to complete the proof of Holder continuity for u we need only

to estimate the quantity supBx0 (R)× [t−R2 ; t] w, where w is either w1 or w2, by a constantindependent of R. The following lemma gives an estimate for the right-hand side ofEq. (2.9) and therefore conclude the proof of the theorem.

Lemma 2.4. There exists a positive constant C independent of R such that either w1or w2 satis�es the following:

1Rn+2

∫∫Bx0 (2R)× [t0−2R2 ; t0]

w2 dx dt≤C:

Proof. Let �(x) be a cut-o� function for Bx0 (2R), i.e. �(x)≡ 1 in Bx0 (2R), �(x)≡ 0outside Bx0 (4R) and |Dx�| ≤ 1=2R. We go back to Eq. (2.6) and replace � by �2. UsingEqs. (2.7) and (2.8), we obtain in a standard way that

ddt

∫w�2 dx +

∫�2|Dw|2 dx≤C

∫

(|Dw|�|D�|+ �2

N 2(u)

)dx:

The Young inequality applies to the �rst integrand on the right-hand side and thefact that |Dx�| ≤ 1=2R and N (u)≥R (assuming also that R≤ 1) give

ddt

∫w�2 dx +

∫�2|Dw|2 dx≤ C

R2

∫�2 dx: (2.10)

Set I2 := [t0 − 4R2; t0 − 2R2], Q=Bx0 (2R)× I2, Qu := {(x; t)∈Q | u≤m4 + !4=2}.Obviously, w1≤ 0 on Qu and w2≤ 0 on Q\Qu. Therefore, the positive part of oneof these functions must vanish on a subset Q0 of Q with meas(Q0)≥ 1=2meas(Q). Wedenote such function by w and set

0t = {x∈Bx0 (2R): w+(x; t)= 0} and m(t)=meas(0t ); t ∈ I2:

Let V (t)=∫ w(x; t)�2 dx=

∫ �2 dx, then V (t)≥−log 2, for all t ∈ [t0−4R2; t0]. Using

the Poincar�e-type inequality due to Moser [16, Lemma 3], we have∫�2(w − V )2 dx≤ 16R2

∫�2|Dw|2 dx: (2.11)

By reducing the above integral to the smaller set 0t we have from Eq. (2.10) that∫�2 dx× d

dtV (t) +

1R2

V 2(t)m(t)≤CRn−2:


Since∫I2m(t) dt=meas(Q0)≥C1Rn+2 for some universal constant C1, the above

inequality and Lemma 2.5 (with d=1; V =V ) following this proof show that V (t1)≤Afor some t1 ∈ I2 and some positive constant A independent of R.Integrating Eq. (2.10) over [t1; t2] for t2 ∈ I1 := [t0 − 2R2; t0] (thus, |t2 − t1| ≤ 4R2),

we get

V (t2)∫�2 dx +

1R2

∫ t2

t1

∫�2|Dw|2 dx dt≤CRn + V (t1)

∫�2 dx: (2.12)

This and the fact that V (t1)≤A imply

− log 2≤V (t)≤C; ∀t ∈ [t0 − 2R2; t0] and∫ t0

t0−2R2

∫�2|Dw|2 dx dt≤CRn

for some universal constant C depending only on A. The above and Eq. (2.11) give

− log 2≤V (t)≤C; ∀t ∈ [t0 − 2R2; t0] and∫ t0

t0−2R2

∫�2(w − V (t))2 dx dt≤CRn+2:

Obviously, we have from these two estimates

1Rn+2

∫∫Bx0 (2R)× [t0−2R2 ; t0]

w2 dx dt≤C

for some universal constant C independent of R. As we already showed, this givesestimate (2:3) and completes the proof of Theorem 2.1.

We state here an elementary lemma which has been used in the above proof. Wewould like to present it more general than needed for a later use in the next section.

Lemma 2.5. Let R; d¿0 be given constants and V (t); V (t); m(t) be functions on aninterval I := [a; b] such that V (t)≥V (t) on I . Assume that∫

�2 dx× d

dtV (t) +

1dR2

V2(t)m(t)≤ CRn−2

d; ∀t ∈ I:

If∫I m(t) dt≥C1Rn+2 for some universal constant C1 and |b − a| ∼dR2 then there

exists t1 ∈ I and a positive constant A depending only on C; C1 such that V (t1)≤A.

Here and in the sequel, we use the notation A∼B to mean that the quantities (usuallynonnegative) A and B are comparable. That is, there exist positive constants C1; C2independent of the quantities in question such that C1B≤A≤C2B.

Proof. Assume that V (t)≥A¿0 for all t in I := [a; b] so that V2(t)≥V 2(t). We have∫

�2 dx

V ′(t)V 2(t)

+1

dR2m(t)≤ CRn−2

dA2:


Integrating over I and making use of the assumption on m(t) and the fact that|b− a| ∼ dR2, we get

C1Rn ≤ 1R2

∫Im(t) dt≤

(1

V (a)− 1

V (b)

)∫�2 dx +

2CRn

A2≤Rn

(1A+2CA2

):

By choosing A large enough (independent of R; d), we see that the above inequalitygives a contradiction. Hence, there must exist t1 ∈ I such that V (t1)≤A.

Remark 2.6. Parts of the proof of Lemmas 2.4 and 2.5 were inspired by some ar-guments in Moser’s work [16, Section 4] where he based on an inequality similar toEq. (2.10) to derive an estimate for

∫√log u (see [16, Eqs. (4:10) and (4:12)]). His

use of the function log u is to obtain certain estimates which connect the estimatesfrom above and below for the solution u and therefore give the Harnack inequality.As we discussed before; our use of logarithmic functions is di�erent and more directlyrelated to the H�older estimates.

3. Holder estimates for degenerate case

In this section we show that the same procedure can be used to obtain local Holderestimate for bounded weak solutions to Eq. (1.1) which satis�es the following structurecondition:

a(x; t; u; Du)Du≥ �0�(u)|Du|2 − 0(x; t);

|a(x; t; u; Du)| ≤ �1�(u)|Du|+�(u) 1(x; t); (3.1)

|b(x; t; u; Du)| ≤ �1�(u)|Du|2 + 2(x; t)

and the following condition on �(u),

�1|u|m ≤�(u)≤�2|u|m; (3.2)

where �0; �1;�1;�2; m¿0. In fact, one can see that the proof still applies to moregeneral structure for �. In particular, one may also consider the singular case wherem∈ (−1; 0). The result for systems of degenerate equations with general structure for� will appear elsewhere. We have not yet tried this method on the parabolic equationsof p-Laplacian type but the elliptic case has been treated in [7].Again, for simplicity, we assume that the functions i’s on the right-hand side of

Eq. (3.1) are bounded nonnegative functions and, in addition, we will consider onlynonnegative solutions to Eq. (1.1) although the proof for the case of signed solutionsis similar modulo some minor modi�cations.The ideas of Section 2 will be used to prove the following result ([3, 18], see

also [2]).

Theorem 3.1. Let u be a locally bounded weak solution of Eq. (1.1), and let Eq.(3.1) hold. Then (x; t)→ u(x; t) is locally H�older continuous in the interior of T .


That is; for every compact subset K of T , there exists a constant C =C(‖u‖∞; K)and �= �(‖u‖∞; K) in (0; 1) such that

|u(x1; t1)− u(x2; t2)| ≤C(|x1 − x2|� + |t1 − t2|�=2)

for every pair of points (x1; t1); (x2; t2)∈K .

3.1. Scaled cylinders and auxiliary logarithmic functions

We brie y sketch the argument to show that H�older regularity follows from a di-chotomy for the degenerate equation (1.1). The main idea is to construct a nestedsequence of suitable scaled cylinders and show that, in each such a cylinder, if thesolution is away from zero then the equation is nondegenerate and then the oscillationof the solution will decay in a controllable way. Otherwise, if the solution fails to beaway from zero then it will decay in H�older fashion.We will demonstrate that the proof for the decay estimates is actually the same as

that for the nondegenerate case if we choose the right form of logarithmic functionswhich will be de�ned on the cylinders that are scaled as in [2, 3, 18] to re ect thedegeneracy of the equation.Again, we would like to remind the reader that we are considering only nonnegative

solutions in order to simplify the exposition of this note. The case of signed solutioncan be treated similarly modulo some technical modi�cations. We shall also adapt thenotations of [18] (where signed solutions were investigated) for the sole purpose ofcomparison.Let �∈ (0; 1) be given. Fix a point (x0; t0)∈T and construct the cylinder

Q(R2−�; 3R) := (x0; t0) + Bx0 (3R)× [t0 − R2−�; t0];

where R∈ (0; 1) is so small such that the cylinder is contained in T . We set

�+ = supQ(R2−� ;3R)

u; �−= infQ(R2−� ;3R)

u; != �+ − �−

and (assuming u≥ 0) M = �+ =max{�+; �−; !}. We consider the cylinder

Q(dR2; R) := (x0; t0) + Bx0 (2R)× [t0 − 4dR2; t0]; where d := 1=�(M):

We will assume that

�(M)¿4R�: (3.3)

Otherwise, there would be nothing to prove. Note also that this implies the inclusionQ(dR2; R)⊂Q(R2−�; 3R).


Let �∈ (0; 1) be small enough such that �≤max{�=m; (2 − �)=2} and thereforeR2−2�; R2−�¡R� for all R∈ (0; 1). We consider the following logarithmic functions:

w1(x; t) := log(!+ R�

N1(u)

); N1(u)= 2(�+ − u) + R�; (3.4)

w2(x; t) := log(!+ R�

N2(u)

); N2(u)= 2(u− �−) + R�: (3.5)

Note that wi ≥−log 2. Again, in the lemmas below we will show that, under as-sumption (3:3), one of the above w1; w2 will be bounded from above by some universalconstant in some subcylinder of Q(dR2; R):

w1(x; t)≤C or w2(x; t)≤C; ∀(x; t)∈Bx0 (R)× [t0 − �dR2; t0]; (3.6)

where �∈ (0; 1) to be determined independently of R. It is easy to see that either ofthe above inequalities implies the following decay estimate for the oscillation:

oscQ(�d(R=2)2 ; R=2)u≤ �1!+ CR� (3.7)

for some �1 ∈ (0; 1) and C are positive constants independent of u; R. We now choose apostive constant A such that �= �1+C=A¡1 and Am�1¿4 where �1 is the constant inthe structure condition (3:2). If !¿AR� then M¿AR� and therefore �(M)¿�1AmRm�

¿4R� (because �¿m� and R¡1). In this case, it follows that (see Eq. (3.7))

oscQ(�d(R=2)2 ; R=2)u≤ �1!+ CR�¡�!:

We summarize the above in

Lemma 3.2. There exist universal positive constants �; �; �∈ (0; 1) and A such thateither !≤AR� or oscQ(�d(R=2)2 ; R=2)u≤ �!:

This lemma is similar to Lemma 9.1 in [18, p.176] from which the H�older con-tinuity of u follows in a standard manner (see [3, 13, 18]). Because the constants inLemma 3.2 are independent of u; R (if the supremum norm ‖u‖∞ is uniformly bounded)we have the uniform H�older continuity of u. Therefore, we will need only to showEq. (3.6).

3.2. The lemmas

As in Lemma 2.3, we �rst observe the following.

Lemma 3.3. For any nonnegative test function �, the functions w1; w2 satisfy aninequality of the form∫

@w@t

� dx +∫�a(x; t; Dw)D� dx≤

∫

�b(x; t; Dw)� dx: (3.8)


The functions �a; �b satisfy the following structure conditions:

| �a(x; t; Dw)| ≤�(u)(�1|Dw|+ � 1);

�a(x; t; Dw)Dw≥�(u)�0|Dw|2 − � 0; (3.9)

�b(x; t; Dw)≤ � 2;

with � 0 = 4 0=N2(u); � 1 = 2 1=N (u) and � 2 = 2(2 0=N

2(u) + 2=N (u)).

The proof is similar to that of Lemma 2.3 of Section 2 so that we shall not repeatit here. However, for later references, we note here that

�a(x; t; Dw) =2a(x; t; u(x; t); Du)

N (u(x; t));

�b(x; t; Dw) = 2(b(x; t; u(x; t); Du)

N (u(x; t))− 2a(: : :)�Du

N 2(u)

)

and

2a(: : :)DuN 2(u)

≥ 2�(u)�0|Du|2N 2(u)

− 2 0N 2(u)

; (3.10)

b(x; t; u(x; t); Du)N (u(x; t))

≤ �1(�+ + R�)�(u)|Du|2N 2(u)

+ 2

N (u): (3.11)

We now wish to estimate the supremum of w in terms of its L2 norm. SinceEq. (3.8) is degenerate, we need to show that on the subset where that function ispositive there will be certain comparison property for the solution u in order to handlethe degeneracy in the di�usion term. We must also show that one of the functionswi’s vanishes on a subset of large measure, a key factor in obtaining the L2 estimates.These facts are provided in the following lemma whose proof will be postponed to theend of this section to make our presentation smoother.

Lemma 3.4. We can construct a function w which is either w1 or w2 such that; forR=R or R=2, on the set of (x; t) where w+(x; t)¿0 the value of the solution u(x; t)can be compared with M. That is

w+(x; t)¿0⇒C1M ≤ u(x; t)≤M ⇒C1�(M)≤�(u)≤C2�(M): (3.12)

Here, C1; C1; C2 are positive universal constants. Moreover; let Q0w := {w+ =0}∩Q(dR

2; R) then we can �nd constants �∈ (0; 1) and ¿0 independent of R such that

meas(Q0w ∩Bx0 (R)× [t0 − 4dR2; t0 − 2�dR2])¿ meas(Q(dR2; R))∼dRn+2: (3.13)

From now on we will simply write R by R and let w; � be the function and theconstant stated in the above lemma. Given Eq. (3.12) of this lemma, we can estimate


the suppremum of w in terms of its L2 norm as follows (recall that i are boundedfunctions).

Lemma 3.5. There exists a positive constant C independent of R such that

supBx0 (R)×[t0−�dR2 ; t0]

w≤C

{ 1

�dRn+2

∫∫Bx0 (2R)×[t0−2�dR2 ; t0]

(w+)2 dx dt

}1=2+ 1

:

(3.14)

Proof. Using a change of variable in t we can assume that �=1. By Eq. (3.12),Eq. (3.8) is nondegenerate on the set where w¿0 so that the proof of Eq. (3.14) isstandard by using iteration technique. We will sketch only the main points here. Fora; b¿0, we consider the cylinder Q=Q(t; t′; R; a; b) :=Bx0 (R+ aR)× [t′ − bdR2; t] anda cut-o� function �(x; t) satisfying the following properties:

0≤ �(x; �)≤ 1 ∀(x; �);

�(x; �)≡ 0 if �≤ t′ − bdR2 or x 6∈Bx0 (R+ aR);

�(x; �)≡ 1 if �≥ t and x∈Bx0 (R);

|Dx�| ≤ 1aR

;∣∣∣∣@�@t∣∣∣∣ ≤ 1

bdR2:

(3.15)

For some �≥ 1, we test the inequality of w by = �2(w+)�. Integrating by part andusing Eq. (3.9) and the Young inequality, we easily get

maxt′−bdR2≤�≤t

∫�2(w+)1+�(x; �) dx +

�2�02

∫∫Q�(u)�2(w+)�−1|Dw|2 dx dt

≤C�∫∫

Q�(u)(w+)1+�(�2 + |D�|2) + �2

((w+)�−1

N+(w+)�

N 2

)

+ �(w+)1+�

∣∣∣∣@�@t∣∣∣∣ dx dt: (3.16)

Because �(u)∼�(M) on the set w+ 6=0; N (u)≥R�¿R, and since we are assumingthat �(M)≥R� ≥R2−2�, we easily deduce


∫�2(w+)1+�(x; �) dx +

�2�0�(M)2

∫∫Q�2(w+)�−1|Dw|2 dx dt

≤ C(�0; �1)��(M)max{a2; b}R2

∫∫Q((w+)1+� + 1) dx dt: (3.17)


For i=1; 2; : : : let us take a= b=1=2i and de�ne R0 = 2R; T0 = t0 − 2dR2 and

Ri=Ri−1 − 12i

R; Ti=Ti−1 +12i

dR2; Qi=Bx0 (Ri)× [Ti; t0]; �=2i :

Making a change of variables d �t=�(M)dt and using an iteration argument toEq. (3.17) it is standard to get Eq. (3.14).

In the same way as in Lemma 2.4 of the previous section, we have

Lemma 3.6. There exist a universal constant C and �∈ (0; 1) independent of R suchthat

1dRn+2

∫∫Bx0 (2R)×[t0−2�dR2 ; t0]

(w+)2 dx dt≤C:

Proof. Let �(x) be a cut-o� function for Bx0 (2R) and Bx0 (R). We now go back toEq. (3.8) and replace � by �2. Using Eqs. (3.10) and (3.11), we obtain

ddt

∫w�2 dx +

∫�2�(u)|Dw|2 dx≤C

∫

(�(u)|Dw|�|D�|+ �2

N 2(u)

)dx:

The Young inequality applies to the �rst integrand on the right-hand side and thefact that |Dx�| ≤ 1=R and N (u)≥R� and �(M)¿4R�¿4R2−� give

ddt

∫w�2 dx +

∫�2�(u)|Dw|2 dx≤ C�(M)

R2

∫�2 dx: (3.18)

Let V (t)=∫ w(x; t)�2 dx=

∫ �2 dx. Let w+; w− ≥ 0 be the positive and negative

parts of w, so that w=w+ − w−. We also denote V+(t)=∫ w+(x; t)�2 dx=

∫ �2 dx.

We now set I2 := [t0 − 4dR2; t0 − 2�dR2] and

0t = {x∈Bx0 (2R): w+(x; t)= 0}; and m(t)=meas(0t ); t ∈ I2:

Because on the set w¿0 we have that �(u)∼�(M). So,

C�(M)∫�2|Dw+|2 dx≤C�(M)

∫w≥0

�2|Dw|2 dx≤∫�2�(u)|Dw|2 dx (3.19)

for some universal constant C. Using the Poincar�e–Moser inequality, we have

C�(M)R2

∫�2(w+ − V+)2 dx≤�(M)

∫�2|Dw+|2 dx: (3.20)


By reducing the above integral to the smaller set 0t , we have from Eq. (3.18) andthe above estimates that∫

�2 dx× d

dtV (t) +

�(M)R2

(V+)2(t)m(t)≤C�(M)Rn−2:

From Eq. (3.13), we have∫I2m(t) dt=meas(Q0)≥ meas(Q2)∼dRn+2. Also, we

have V+(t)≥V (t). So, we can apply Lemma 2.5 here (with d=1=�(M) and V =V+)to see that there exists t1 ∈ I2 such that V (t1)≤A.Integrating Eq. (3.18) over [t1; t2] for t2 ∈ I1 := [t0−2�dR2; t0] (thus, |t2−t1| ≤ 4dR2),

we get

V (t2)∫�2 dx +

∫ t2

t1

∫�2�(u)|Dw|2 dx dt≤CRn + V (t1)

∫�2 dx: (3.21)

Note that V (t)≥−log 2, for all t ∈ I := [t0−4dR2; t0]. The fact that V (t1)≤A implies

−log 2≤V (t)≤C; ∀t ∈ I1 and∫I1

∫�2�(u)|Dw|2 dx dt≤CRn

for some universal constant C depends only on A. From −log 2≤w we also see thatw−, therefore, V− are bounded. So, V+ =V + V− is bounded from above. FromEqs. (3.19), (3.20) and the above estimates we deduce

0≤V+(t)≤C; ∀t ∈ I1 and�(M)R2

∫I1

∫�2(w+ − V+(t))2 dx dt≤CRn:

We easily see that this gives the estimate in the lemma.

Finally, to conclude the proof of the main theorem, we go back to the proof ofLemma 3.4 for which we need a lemma [18, Lemma 5.1] (see also its original versionin [2]). We want to give here a less involved proof for, however, a slightly generalversion of it.

Lemma 3.7. Assume Eq. (3.3). Given a; b∈ (0; 1) with a¿b there is a number �0depending on a; b but independent of R such that if

meas{(x; t)∈Q(dR2; R): u(x; t)¡�− + a!}≤ �meas(Q(dR2; R)) (3.22)

for some �∈ (0; �0) then

u(x; t)¿�− + b! for a:e: (x; t)∈Q

(d(R2

)2;(R2

)): (3.23)

Proof. Without loss of generality, we may assume that �−=0 so that M =! (in fact,this is the degenerate situation). Consider the following function on Q(dR2; R):

W =min{b!; (a!− u)+}:Let � be the cut-o� function as in the proof of Lemma 3.5 and �≥ 1. We test the

equation of u by �2W�. Since DW 6=0 i� b!¡u¡a! so that �(u)∼C(a; b)�(!)∼


�(M) where DW 6=0, using this fact and Eq. (3.3) we easily derive


∫�2W 1+�(x; �) dx +

�2�0�(M)2

∫∫Q�2W�−1|DW |2 dx dt

≤ C(�0; �1)��(M)max{a2; b}R2

∫∫Q(W 1+� + 1) dx dt: (3.24)

As before, from this and a standard iteration argument we derive easily

supQ(d(R=2)2 ; (R=2))

W ≤C(

1meas(Q(dR2; R))

∫∫Q(dR2 ; R)

W 2 dx dt)1=2

: (3.25)

Since W (x; t) 6=0 i� (x; t)∈∑ := {u(x; t)¡a!}. So, if meas(∑)≤ �meas(Q(dR2; R))

then we have from Eq. (3.25) that

supQ(d(R=2)2 ; (R=2))

W ≤C�1=2 supQ(dR2 ; R)

W =C�1=2! (!=M):

If � is su�ciently small such that C�1=2¡b=2 then the above and the de�nition ofW give

u(x; t)≥ (a− C�1=2)!; ∀(x; t)∈Q

(d(R2

)2;(R2

)):

Since a¿b, we see that if � is small enough then Eq. (3.23) is veri�ed.

Proof of Lemma 3.4. Choose a= 12 and b= 1

4 in the above lemma and let �0¿0 bethe constant given by that lemma (it can be smaller but �xed). Set

Q−u := {(x; t)∈Q(dR2; R): u(x; t)¡�− + !=2= (�+ + �−)=2}:

In fact, Q−u is the near degeneracy subset where u is closer to zero (recall also that

�− ≥ 0). We consider alternatively two cases.(a) If meas(Q−

u )¿�0 meas(Q(dR2; R)), the near degeneracy subset is large, we de�new(x; t)=w1(x; t), R=R and Q0w =Q−

u . It is clear that w+1 =0 on Q−

u and w+¿0 onQ(dR2; R)\Q−

u where �+≥ u≥ (�+ + �−)=2. Since �− ≥ 0, Eq. (3.12) is veri�ed.(b) Otherwise, if meas(Q−

u )≤ �0 meas(Q(dR2; R)), the equation is less degenerate butwe have to work a little more! We set w(x; t)=w2(x; t), R=R=2 and Qw =Q−

u ∩Q(d(R=2)2; R=2).

From the above lemma, if �0 is su�ciently small then u(x; t)≥ �−+!=4=3�−=4+�+=4≥ �+=4 (since �− ≥ 0) in Q(d(R=2)2; R=2). We see that Eq. (3.12) holds andthe equation is actually nondegenerate in Q(d(R=2)2; R=2). To check Eq. (3.13), weobserve that

meas(Qw)≤meas(Q−u )≤ �0 meas(Q(dR2; R))=C�0 meas

(Q

(d(R2

)2;R2

))


for some universal constant C. We can choose �0 smaller such that C�0¡1. With suchchoice of �0, the function w+ vanishes on the set Q0w :=Q(d(R=2)2; R=2)\Qw and

meas(Q0w)¿(1− C�0)meas

(Q

(d(R2

)2;R2

)): (3.26)

In both cases, we see that Eq. (3.12) holds and Eq. (3.13) is veri�ed for some ¿0but with �=0. To complete the proof we need to show Eq. (3.13) for some �¿0.But this is easy since, for any �¿0,

meas({w+ =0}⋂

Bx0 (R)× [t0 − 4dR2; t0 − 2�dR2])≥ ( − 2�)meas(Q(dR2; R)):

Hence, if we choose �= =4 then Eq. (3.13) follows.

4. Boundary regularity

In this section we show that the interior regularity results can be extended up tothe lateral parabolic boundary of T . The idea is exactly the same as that for theinterior case, i.e. we need to show that one of the logarithmic functions w1; w2 can bemajorized by a universal constant so that the decay property for the oscilation of thesolution u holds (see Lemma 3.2). Inspecting the proof for the interior case, we cansee that this can be achieved if the following two inequalities are veri�ed:a. the energy inequality (3.17) of Lemma 3.5,b. inequality (3.18) of the proof of Lemma 3.6.The functions � in these inequalities are now de�ned respectively in Q(t; t′; R; a; b)

(see Lemma 3.5) and Bx0 (2R) with x0 is now a point on @.The only technicalities need to be checked are the estimates for the boundary inte-

grals which occur in this situation.

4.1. The case of variational or Neumann boundary data

Consider the problem

@u@t=div(a(x; t; u; Du)) + b(x; t; u; Du) (x; t)∈T ;

a(x; t; u; Du) · nST = g(x; t; u) on ST ;

u(x; 0)= u0(x) x∈;

(4.1)

where nST denotes the outer unit normal to the lateral parabolic boundary ST of T .On the boundary data g(x; t; u) we assume that

(G) g is continuous over ST ×R and admits an extension �g(x; t; u(x; t)) over T suchthat

‖Du �g(x; t; u(x; t)); Dx �g(x; t; u(x; t))‖∞;T ≤C¡∞ (4.2)


and

g2(x; t; u)≤C�(u): (4.3)

Note that this condition can be veri�ed easily if apriori uniform estimate for the L∞

norm of u is already known.Taking into account the boundary condition, we have the integral form of the equa-

tion for u as follows:∫

@u@t

� dx +∫a(x; t; u; Du)D� dx=

∫@

g(x; t; u)� d� +∫b(x; t; u; Du)� dx (4.4)

for all �∈W 1;12 (T ).

First, let us show that w still satis�es an integral inequality similar to Eq. (3.8) ofLemma 3.3. By taking � in Eq. (4.4) to be �=N (u) and following the lines of theproof of that lemma we can see that only the boundary integral term, which appearsin the integration by part, will need a careful investigation here. Let �∈C2(; Rn) beany vector �eld satisfying � · n=1 on @. Then we estimate the resulting boundaryintegral as follows:∫

@g

�N (u)

d�=∫div(

g��N (u)

)dx

=∫

(�Dxg+ div(�))N (u)

�+g�

N (u)D� dx

+∫Dug

��DuN (u)

+ 2g��DuN 2(u)

dx:

We treat the last integrals as follows. In the �rst integral, because of Eq. (4.2), theterm (Dxg + div(�))=N (u) can be included in the de�nition of �b. Similarly, g=N (u)can go with �a. Finally, because we will later replace the function � by �2(w+)� in theproof of Lemma 3.5, we estimate the last two term by using the following inequalities(note that 2Du=N (u)Dw and N (u)≥R�):

g(w+)�Du

N 2(u)∼ g(w+)�

DwN (u)

≤ �0g2(w+)�−1|Dw|2 + C(�0)(w+)�+1R2−2�

R2;

Dug(w+)�DuN (u)

∼ (w+)�Dw≤ �0�(u)(w+)�−1|Dw|2 + C(�0)(w+)�+1

�(u):

For the �rst estimate we use the facts that g2≤�(u), �(M)≥R2−2�. For the second,we recall that on the set w+¿0 one has �(u) ∼ �(M) and the latter is assumed tobe greater than R� so that 1=�(M)≤C�(M)=R2� ≤C�(M)=R2 (since R; �¡1). We cansee that, by choosing �0 small enough, Eq. (3.17) is veri�ed.Finally, we check inequality (3.18) of the proof of Lemma 3.6. We need to esti-

mate the boundary integral of g�2=N (u) which would appear on the right-hand side ofEq. (3.18). Because @ is smooth, the (n − 1) dimensional measure of the boundary


portion @∩ supp(�) is comparable to Rn−1. This and the facts that N (u)≥R� and�(M)≥R1−� yield∫

@g

�2

N (u)d�≤C1Rn−1−� ≤C2

�(M)R2

Rn ≤C�(M)R2

∫�2 dx;

where C1; C2; C are some universal constants. This show that Eq. (3.18) is again sat-is�ed.We conclude that the H�older continuity for u can be extended up to the boundary.

For the nondegenerate case, one can just ignore the factors �(u);�(M) in the abovearguments.

4.2. The case of Dirichlet boundary data

Let u be a weak solution of Eq. (4.1) with the boundary condition now is

u(x; t)=�(x; t) (x; t)∈ ST = @× [0; T ]; (4.5)

in the sense of the traces over ST . On the Dirichlet data � we assume that(D) � admits an extension �� over T and ��∈C�()∩W 1;2() for some �∈ (0; 1).In the nondegenerate case, by substracting �� from u and making a change of vari-

able we can, and will, assume that � is identically zero on ST , i.e. u assumes thehomogeneous Dirichlet boundary data. However, for the degenerate case, where all ofour previous argument has been carried out for the case when the degeneracy occurs atu=0, such a change of variable will make the new equation degenerate when u= −�so that our previous calculation cannot be carried over directly. 1 Similar argument tothat of [3, Section III.12] can be used to cover this situation. On the other hand, onecan still follow the same idea to rework the proof with modi�ed logarithmic functionsas in [4] for elliptic equations to obtain regularity result near the boundary. We willnot pursuit this problem further in this note but restrict ourselve to the case of homo-geneous Dirichlet boundary condition � ≡ 0 (in fact, this is the degenerate situation)in the following explanation.Consider a point (x0; t0)∈ ST . Again, to prove the H�older continuity of u up to ST

we need only to check the validity of the two integral inequalities mentioned in thebeginning of this section.First, we observe that u|ST =0 implies that �−=0 and therefore w|ST =0 (see

the de�nitions (2.2) of Section 2, Eqs. (3.4) and (3.5) of Section 3). Using this fact,we see that the test function �2w� (�≥ 1), which was used in the proof of the en-ergy estimate (3.17), vanishes on the boundary so that there is no boundary integralappearing in the argument. We obtain easily the energy estimate in this case.On the other hand, to obtain estimate (3.18) we must slightly modify the test function

�2=N (u) as follows. Suppose w=w1. We notice that the function

2uN (u)

=2�+ + R�

N (u)− 1= N (0)

N (u)− 1

1 The author thanks the referee for pointing out this point to him.


vanishes on ST and its partial derivatives are just multiples of those of 1=N (u) by theconstant factor N (0). Therefore, by testing the equation of u by 2�2u=N (u) (insteadof �2=N (u) as in the proof of Lemma 3.6.) and dividing through by N (0) we see thatEq. (3.18) now becomes

ddt

∫w�2 dx +

∫�2�(u)|Dw|2 dx≤

∫�(u)|Dw|�|D�|+ �2

N 2(u)+

ut�2

N (0)dx:

Following the lines of the proof of the lemma we see that the next steps are tointegrate the above inequality over the time interval I1; I2. We need only to take careof the new term ut�2=N (0). Using the Fubini theorem (formally) and the fact thatN (0)≥M ≥ u we have for i=1; 2 that

1N (0)

∫Ii

∫ut�2 dx≤C

Rn sup uM

≤CRn:

Then we can see that the proof of the lemma goes through and gives the uniformestimate for the mean L2 norm of w.The same argument can be applied to the nondegenerate cases to show the H�older

continuity near the boundary.We then conclude that (see [3, 18]):

Theorem 4.1. Assume (G), (D), respectively, for the boundary conditions. The H�olderestimates of Theorems 2.1 and 3.1 can be extended up to the lateral boundary.

From this and the interior regularity results we conclude that

Corollary 4.2. Assume the conditions of Theorems 2.1 (respectively, 3.1) and 4.1. Letu be a bounded weak solution of Eq. (1.1). Then (x; t)→ u(x; t) is globally H�oldercontinuous in the cylinders T ′ ; T = [T ′; T ]× where T¿T ′¿0. That is, there existsa constant C =C(‖u‖∞;T′ ; T ) and �= �(‖u‖∞;T′ ; T ) in (0; 1) such that

|u(x1; t1)− u(x2; t2)| ≤C(|x1 − x2|� + |t1 − t2|�=2)for every pair of points (x1; t1); (x2; t2)∈T ′ ;T .

5. Convergence to global attractors in stronger norms

The H�older regularity results allow us to revisit the systems considered in [6, 9] andstrengthen the results on the global attractors. The systems considered in those papersare of the form

@ui

@t=Aiui + fi(x; u; Dui); t¿0; x∈; i=1; : : : ; m;

Biui= v0i on @; t¿0;

ui(0; x)= u0i (x) in ;

(5.1)


where Ai’s are nondegenerate or degenerate elliptic operators of divergence form andBi’s are nonlinear boundary operators. Speci�cally, Aiu=div(ai(x; u; Du)) with ai sat-is�es either Eq. (2.1) as in [6] or Eq. (3.1) as in [9].We showed that if the dynamical systems {S(t)}, which are generated by the above

PDE systems, have absorbing sets in Lp for some �nite p large enough then they alsohave such sets in L∞. As we mentioned before, in these papers we were only able toprove the compactness of the trajectories in the weaker metric of L2 so that we couldonly conclude that there exists a global attractor, which is a subset in L∞, for {S(t)}and the solutions converge to that global attractor in the L2 norm.Now we can consider the dynamical systems {S(t)} in the Banach space X =∏mi=1 C() of bounded continuous functions with the usual uniform supremum norm.

Given that the L∞ norms of the solutions are eventually bounded uniformly with re-spect to the initial data, because the system is coupled only in its reaction terms, wecan think of the (bounded) components ui’s in the fi’s as parameters and therefore re-duce the regularity problem to that of a scalar equation (1.1). Corollary 4.2 establishesthe compactness of the set {S(t)B: t≥ t0} for any t0¿0 and for any bounded set Bin X .We can conclude that

Corollary 5.1. Assume that system (5.1) has an absorbing set in X =∏m

i=1 C().The associated dynamical system {S(t)}t ≥ 0 posseses a global attractor A in X .Moreover, the solutions converge to A in the uniform supremum norm and A is asubset of

∏mi=1 L

∞()∩C�() for some �¿0.

We refer to [6, 9] for assumptions and proof of the existence of an absorbing set inX for system (5.1).

References

[1] D.G. Aronson, J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. RationalMech. Anal. 25 (1967) 81–122.

[2] E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math.J. (1983) 83–118.

[3] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer, Berlin, 1993.[4] L. Dung, A summary on the methods of regularity theory for quasilinear elliptic equations and some

contributions to the singular cases, ICTP Thesis, Trieste, Italy, 1993.[5] L. Dung, Boundary C1;� regularity of singular semilinear elliptic equations, Nonlinear Anal. TMA 26

(1) (1996) 67–80.[6] L. Dung, Dissipativity and global attractor for a class of quasilinear parabolic systems, Comm. Partial

Di�erential Equations 22 (3&4) (1997) 413–433.[7] L. Dung, On a class of singular quasilinear elliptic equations with general structure and distribution

data, Nonlinear Anal. TMA 28 (11) (1997) 1879–1902.[8] L. Dung, H�older regularity for a class of strongly coupled parabolic systems, J. Di�erential Equations

(1998), in press.[9] L. Dung, Global attractors for a class of nonlinear degenerate parabolic systems, J. Int. Di�erential

Equations (1998), to appear.[10] D. Gilbarg, N.S. Trudinger, Elliptic Partial Di�erential Equations of Second Order, Springer, Berlin,

1983.


[11] J. Hale, Asymptotic Behavior of Dissipative Systems. American Math. Soc. Math. Surveys andMonographs, vol. 25, AMS, Providence 1988.

[12] A.V. Ivanov, On the regularity of generalized solutions of quasilinear degenerate parabolic systems ofsecond order, Proc. Steklov Inst. Math. 3 (1991) 25–63.

[13] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’tseva, Linear and quasilinear Equations of ParabolicType, AMS Transl. Monographs, vol. 23, AMS, Providence 1968.

[14] G.M. Lieberman, Boundary and initial regularity for solutions of degenerate parabolic equations,Nonlinear Anal. TMA 20 (1993) 551–570.

[15] G.M. Liebermann, Maximum estimates for solutions of degenerate parabolic equations in divergenceform, J. Di�erential Equations 113 (1994) 543–571.

[16] J. Moser, A harnack inequality for parabolic equations, Comm. Pure Appl. Math. 17 (1964) 101–134;Correction 20 (1967) 231–236.

[17] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958) 931–954.[18] M.M. Porzio, V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate

parabolic equations, J. Di�erential Equations 103 (1993) 146–178.[19] N.S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21

(1968) 205–226.[20] N.S. Trudinger, On the regularity of generalized solutions of linear non-uniformly elliptic equations,

Arch. Rational Mech. Anal. 42 (1971) 50–62.[21] V. Vespri, Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations,

J. Math. Anal. Appl. 181 (1994) 104–131.

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Remarks on Hölder continuity for parabolic equations and convergence to global attractors