HÖLDER CONTINUITY FOR DEGENERATE PARABOLIC
EQUATIONS
APPROVED BY SUPERVISING COMMITTEE:
Dung Le, Ph.D., Chair
Gelu Popescu, Ph.D.
Fengxin Chen, Ph.D.
Weiming Cao, Ph.D.
ACCEPTED:
Dean, Graduate School
HÖLDER CONTINUITY FOR DEGENERATE
PARABOLIC EQUATIONS
by
MINH TUAN KHA, B.S.
THESIS
Presented to the Graduate Faculty of
The University of Texas at San Antonio
in Partial Fulfillment
of the Requirements
for the Degree of
MASTER OF SCIENCE IN MATHEMATICS
THE UNIVERSITY OF TEXAS AT SAN ANTONIO
College of Sciences
Department of Mathematics
August 2011
UMI Number: 1498612
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ACKNOWLEDGEMENTS
I would like to deeply thank my advisor Dr. Dung Le for his constant guidance
and inspiration through the years. From the bottom of my heart, I also want to thank
Mrs. Thu for her warm kindness during the time I study and live in San Antonio.
For serving in my thesis committee along with their valuable recommendations,
I would like to express my gratitude to Dr. Cao, Dr. Chen, Dr. Popescu. I am also
very grateful to all faculty and staff of Department of Mathematics, the University of
Texas at San Antonio, in particular Dr. Dimitri Gokhman, Dr. Sandy Norman, Mrs.
Wanda Crotty and Mrs. Rosemarie, for their numerous helps related to my teaching
duties and paper works in my study program. Moreover, I would like to thank Dr.
Popescu and Dr. Richardson for their concern and advice about my study.
My special thanks go to my previous professors, especially Dr. Duong Minh Duc,
and all those who greatly guided me in the first steps into Mathematics. It is due to
their excellent, dedicated lectures and instructions that I developed more and more
my passion in Mathematical areas.
Next, I am deeply grateful to my family for their love, confidence and encouragement.
My final thanks are to my friend Nguyen Doan Ha Quyen for all the encouragement, my colleagues Luu Minh Duc and Le Nguyen Minh Khoa for all interesting and
useful Math discussions together with their help in daily life. Also, my UTSA friends
who helped me get used to and enjoyed life here.
August 2011
ii
HOLDER CONTINUITY FOR
DEGENERATE PARABOLIC EQUATIONS
Minh Tuan Kha, M.S.
The University of Texas at San Antonio, 2011
Supervising Professor: Dung Le, Ph. D.
In this thesis, we will prove the Hölder continuity for bounded weak solutions of a
large class of degenerate parabolic equations, which are called the porous medium
equations. By using auxiliary logarithmic functions, we introduce a new approach
that combines both De Giorgi and Möser iteration methods. We consider both the
interior and Neumann boundary cases. Then at the end of this thesis, we present
some applications to some particular degenerate triangular systems.
iii
TABLE OF CONTENTS
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Chapter 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Chapter 2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1
General notations and definitions . . . . . . . . . . . . . . . . . .
6
2.2
Auxiliary results
Chapter 3
. . . . . . . . . . . . . . . . . . . . . . . . . . .
10
Interior Hölder continuity for the degenerate scalar
case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.1
Scaled cylinders and auxiliary logarithmic functions . . . . . . . .
14
3.2
The lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.3
Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . .
33
Chapter 4
The case of Neumann boundary data . . . . . . . . . .
40
Chapter 5
Applications to degenerate parabolic systems . . . . .
45
5.1
Food Chain Models and Triangular Systems . . . . . . . . . . . .
45
5.2
Hölder continuity for systems of model I . . . . . . . . . . . . . .
49
5.3
Hölder continuity for systems of model II . . . . . . . . . . . . . .
51
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
Vita
iv
Chapter 1
Introduction
Many relevant phenomena, not only in the natural sciences but also in engineering
and economics, are modeled by (systems of) partial differential equations (PDEs)
that display some sort of degeneracy. Examples include the motion of multi-phase
fluids in porous media, the melting of crushed ice (in general, phase transition), the
behavior of composite materials or the pricing of assets in financial markets. Due
to its significance in applications, the class of degenerate parabolic equations is an
important branch in the analysis of partial differential equations. The reader can
view more applications of degenerate and singular PDEs in those above topics in the
book [15].
In this thesis, we investigate the Hölder continuity problem for bounded solutions
of degenerate parabolic equations of the form
∂u
= div(a(x, t, u, Du)) + b(x, t, u, Du),
∂t
(x, t) ∈ QT
(1.1)
where QT = [0, T ] × Ω with Ω is a bounded open set in RN . The diffusion term
a(x, t, u, Du) is degenerate. In details, we will mainly focus on the Hölder continuity
problem for porous media equations, whose the simplest model has the following form:
∂u
= div(|u|m Du) where m > 0.
∂t
(1.2)
When m > 0, the above equation is degenerate since its modulus of ellipticity |u|m
vanishes at points where u = 0. In general, the nature and origin of the degeneracy
maybe quite different but it produces a common effect in the equation, namely weakening of its structure and the prospect that some of the properties of its solutions be
1
lost. From the point of view of regularity theory, the difficulty is to understand to
what extent this weaking of the structure, at the points where the partial differential
equation becomes degenerate, compromises the regularizing effect that is typical of
parabolic equations.
First of all, the problem and related results are classical and well known. There is
already a considerable mathematical litereature in the study of regularity of solutions
to this problem. At first, the regularity result for nondegenerate linear parabolic
equation was proven by Nash in 1957 [12], and DeGiorgi in 1958 for elliptic equations. Later, Möser extended his brilliant technique for elliptic equations to prove
the Harnack inequality for bounded solutions of linear parabolic equations and thus
obtain the regularity results (see [10] and [11]). His method had been then generalized to quasilinear cases by Aronson and Serrin [1], and Trudinger [14] (for scalar
nondegenerate case). This method based on the Harnack principle is itself very important theoretically. However, the derivation is complicated and difficult to cover
the degenerate cases.
On the other hand, the method of level sets or truncation technique of DeGiorgi
had been generalized by Ladyzhensakaya et al to obtain Hölder estimates for weak
solutions to quasilinear equations (nondegenerate). Other authors also extended the
results for degenerate elliptic equations from these techniques. The continuity of
a solution at a point follows from measuring its oscillation in a sequence of nested
and shrinking cylinders, with vertex at that point, and showing that the oscillation
converges to zero as the cylinders shrink to the point. In other words, the method
try to derive certain controllable decay estimates for the oscillation of the solution
in a sequence of nested cylinders which implies the Hölder continuity. This decay
estimate is a byproduct of the Harnack inequality in Möser’s method or the result of
an analysis of two alternatives of the level sets of the solution in De Giorgi’s method.
However, both the methods of De Giorgi and Möser could not be extended in
2
degenerate parabolic case. The issue about Hölder continuity of weak solutions of
degenerate parabolic equations remained open until the mid 1980’s when Di Benedetto
showed that the solutions of general quasilinear equations of the type of (p-Laplacian)
∂u
= div(|Du|p−2 Du) in QT
∂t
(1.3)
are locally Hölder continuous for p > 2. This is also another central example of degenerate parabolic equations. Note that if p > 2, the equation is degenerate in the space
part since its modulus of ellipticty |Du|p−2 vanishes at points where |Du| = 0. Suprisingly, the same techniques could be suitably modified to establish the local Hölder
continuity of any local solution of quasilinear porous medium-type equations. There
results follow, one way or another, from the single unifying the idea of intrinsic scaling: the diffusion processes in the equations evolve in a time scale determined instant
by instant by the solution itself, so that, roughly speaking, they can be regarded as
the heat equation in their own intrinsic time-configuration.
In fact, the technique combining intrinsic scaling and De Giorgi’s method had
been applied sucessfully to degenerate case. The idea behind the method of intrinsic
scaling is to exhibit this iterative process in cylinders that reflect the structure of the
equation. However, the proof will involve with dealing measure-theoretic tools and
an analysis of two alternates of level sets.
In this thesis, we will introduce a method which uses auxiliary logarithmic functions
to derive decay estimates of oscillations. The method had been used in for nondegenerate elliptic equations and in for degenerate equations. The main ingredients can be
summarized as follows:
1. Construct suitable logarithmic functions, denoted by ω, whose upper bound
implies the decay estimates.
2. Estimate the local suppremum of ω in terms of its L2 norm. This can be done
3
by observing that ω is a subsolution to certain equation.
3. Estimate the L2 norm of ω and complete the proof, thanks to step 1.
Actually, the idea of using logarithmic functions is also very old. For example, the
reader can view Chapter VII in [6] for using auxiliary logarithmic functions in order
to show the Hölder continuity of weak solutions of linear elliptic PDEs. However,
different forms of logarithmic functions had been used by other authors for different
purposes. Mainly, they were devised to obtain only certain auxiliary estimates as in
Moser’s work to prove Harnack inequalities. However, their uses did not play a direct
role in obtaining decay estimates as in the method presented here.
The method seems to be more direct and less technical than the above ones. In
particular, we combine both Möser’s and De Giorgi’s methods to prove step 2 and 3.
This reduces the complexity of the analysis of two alternates which are proved entirely
by De Giorgi’s method. By our method, we do not need to separate two alternates,
and just focus on the above three procedures which is more straightforward.
The thesis is organized as follows.
In Chapter 2, we recall some standard notations, definitions of functional spaces,
and some well known imbedding results, some auxiliary results that will be used
throughout the thesis.
In Chapter 3, we will prove the Hölder continuity of bounded weak solutions of
degenerate parabolic equations which cover the type of (1.2). In section 3.1, we
will introduce the auxiliary logarithmic functions ωκ which are constructed from the
weak solution u, and we explain that the problem is a consequence of proving the
boundedness of these logarithmic functions ωκ . In the next section 3.2, we gather
all lemmas that are main ingredients of the proof. In details, Lemma 3.5 is the
result of step 2 which is proved by using Möser iteration method and Lemma 3.6 is
exactly the work of step 3. Before that, we also use De Giorgi’s method to prove
Lemma 3.4 and this is also one of the crucial steps in the first alternate of Vespri
4
and Porzio’s method in [13]. Finally, in section 3.2, we exhibit a complete proof of
proving the main Theorem 3.1 from the Proposition 3.2 in the previous section. Note
that the procedure to prove Hölder continuity from a propostion like Proposition 3.2
is classical in nondegenerate case (see Ladyzhenskaya et al [7]). Because we work in
intrinsic cylinders, it is a little bit more complicated than the nondegenerate case.
However, the method still has the same spirit as Ladyzhenskaya’s method.
In Chapter 4, we consider the equations (1.2) with Neumann boundary data
∂a(x, t, u, Du)
+ g(x, t, u) = 0.
∂ν
(1.4)
We still have the Hölder continuity for this problem. Because almost details of the
proof are involved with integral estimates, we just repeat the same procedure showed
in Chapter 3 with some modifications on all estimates related to the boundary data
g in the proof.
In Chapter 5, we introduce some small applications to degenerate parabolic systems
(Triangular Systems). More clearly, we focus on two kinds of triangular models. The
first part of this chapter presents the connection of these triangular systems and
Food Chain models. The remaining part of this chapter shows the proof of Hölder
continuity of bounded weak solutions of these models.
Finally, the method we present in this thesis from chapter 3 to chapter 4 is obtained
in the paper [5] by my advisor- Dr. Dung Le. In Chapter 5, we use a result from the
Master Thesis [9].
5
Chapter 2
Preliminaries
2.1
General notations and definitions
The main purpose of this section is to introduce some general notations and definitions
that will be used throughout the thesis. Most of them will be vector notations and
some function spaces.
For x ∈ Rn , we write x = (x1 , x2 , . . . , xn ). We denote the point z = (x, t) ∈ Rn+1 ,
in which x = (x1 , . . . , xn ) ∈ Rn and t > 0.
Let x = (x1 , . . . , xn ) ∈ Rn and a = (aij ) is any m × n matrix. Their norms are
given by
n
X
1
|x| = (
|xi |2 ) 2
(2.1)
i=1
and
n X
m
X
1
|a| = (
|aij |2 ) 2
(2.2)
i=1 j=1
respectively.
The parabolic norm of a point z = (x, t) ∈ Rn+1 is defined by
1
|z| = (|x|2 + t) 2
(2.3)
Furthermore, we often use Ω as a bounded domain of Rn , n ≥ 1 with its boundary
∂Ω. For a fixed point x ∈ Ω̄, let BR (x) be the ball centered at x in Rn with radius
R, and B̂R (x) = Ω ∩ BR (x). If no ambiguity can arise, we can write BR , B̂R instead
of BR (x), B̂R (x).
For our parabolic problems, let QT denote the cylindrical domain Ω × [0, T ]. Also
6
let,
ST = ∂Ω × [0, T ], ΓT = ST ∪ (Ω × {0})
(2.4)
denote the lateral boundary and the parabolic boundary of ΩT respectively.
∂u
∂u
, and ut for
, Dij u for
For a real-valued function u, we denote Di u or uxi for
∂xi
∂t
∂ 2u
. Here the derivatives can be classical or weak ones. We also use ∇u or Du
∂xi ∂xj
for (ux1 , . . . , uxn ).
In the case of a real-valued vector function u = (u1 , u2 , . . . , um ), where m > 1, ∇u
P
mean (∇u1 , . . . , ∇um ). Also, when m = n, we use div(u) for ni=1 Di ui .
A function η(x) (or η(x, t)) is said to be a cutoff function for the domain Ω (or
QT ) if it is continuous in Ω̄ (or Q¯T ), has first-order piecewise continuous derivatives,
vanishes on the boundary of this domain ∂Ω (or ΓT ), and has its values contained
between zero and one.
In particular, if the domain Ω is a ball BR (x) (or cylinder QR (x, t)), we understand
that the cutoff function η in this case receives value 1 on a smaller ball Br (x) (or
cylinder Qr (x, t)) where r is usually comparable to R. We can always assume that
the integral of η over the domain Ω is equal to 1.
We now recall some definitions of some well-known functional spaces. We shall
denote by C k (Ω), k = 0, 1, . . . the space of functions which have continuous derivatives
up to the order k, and by C ∞ (Ω) the space of infinitely differentiable functions in Ω,
and by C k (Ω̄) the space of functions in C k (Ω) such that their derivatives up to the
order k are continuous in Ω̄, and by C0k (Ω) the space of functions in C k (Ω̄) such that
their supports are contained in Ω.
The spaces C k (Ω̄) are Banach spaces with the norm
kukC k =
X
|α|≤k
sup |Dα u(x)|
x∈Ω
7
(2.5)
For any 0 ≤ α ≤ 1, we denote by C α (Ω) the space of α-Hölder continuous functions
in Ω, which are continuous functions u such that the following quantity is finite
|u(x) − u(y)|
|x − y|α
x6=y∈Ω
[u]α := sup
(2.6)
The spaces C α (Ω) are Banach spaces with the norm
kukC α = kukC 0 + [u]α
(2.7)
We shall denote by Lp (Ω), p ≥ 1 the spaces of all measurable functions such that
the norm
p1
|u(x)| dx
Z
p
kukp,Ω =
(2.8)
Ω
kuk∞,Ω = sup |u(x)|
(2.9)
x∈Ω
is finite.
Lq,r (QT ) is the Banach space of all measurable functions in QT = Ω × [0, T ] with
a finite norm
Z
T
Z
kukq,r,QT =
0
|u(x, t)|q dx
rq
! r1
dt
(2.10)
Ω
in which r, q ≥ 1. In other words, Lq,r (QT ) = Lr (0, T ; Lq (Ω)).
If q = r, then Lq,q (QT ) and k.kq,q,QT will be denoted by Lq (QT ) and k.kq,QT respectively.
W k,p (Ω), the Sobolev space, is the Banach space of the elements of Lp (Ω) that
have generalized derivatives up to order k are in Lp (Ω) as well. The norm in W k,p is
defined by
kukk,p,Ω =
X
|β|≤k
8
kDβ (u)kp,Ω
(2.11)
W01,p (Ω) is the completion of C0∞ (Ω) under the norm
kvkW 1,p (Ω) = kDvkp,Ω
0
(2.12)
We introduce spaces where typically solutions of parabolic equations in divergence
form are found.
Let m, p ≥ 1 and consider the Banach spaces
V m,p (QT ) = L∞ (0, T ; Lm (Ω)) ∩ Lp (0, T ; W 1,p (Ω))
(2.13)
V0m,p (QT ) = L∞ (0, T ; Lm (Ω)) ∩ Lp (0, T ; W01,p (Ω))
(2.14)
and
both equipped with the norm, v ∈ V m,p (QT ),
kvkV m,p (QT ) = sup kv(., t)km,Ω + kDvkp,QT
(2.15)
0≤t≤T
When m = p, we set V p,p (QT ) = V p (QT ) and V0p,p (QT ) = V0p (QT ).
For any 0 ≤ α ≤ 1, we denote by C α,α/2 (QT ) the space of α-Hölder continuous
functions in QT with a finite norm
kukC α,α/2 = kuk∞,QT +
sup
z1 6=z2 ∈QT
|u(z1 ) − u(z2 )|
d(z1 , z2 )α
(2.16)
where d(z1 , z2 ) = |x1 − x2 | + |t1 − t2 |1/2 is the parabolic distance between two points
zi = (xi , ti ), i = 1, 2.
We recall the definitions of the Lorentz space L(R, ν), where R, ν > 0, which is the
collection of nonnegative measurable functions on Q(R, R2 ) = BR (x0 ) × [t0 , t0 + R2 ]
satisfy the following: if f ∈ L(R, ν) there exists a constant kf kL(R,ν) such that for
any measurable set A ⊂ Q(R, R2 ) we have
9
Z
f (x, t)dxdt ≤ kf kL(R,ν) |A|ν
(2.17)
A
where |A| denotes the Lebesgue measure of the set A.
2.2
Auxiliary results
In this section, we collect some auxiliary results to be used throughout this thesis.
First, we recall the Young inequality, which is used commonly in proving many estimates of solutions of PDEs.
Lemma 2.1 Let a, b be positive and p, q > 1 such that
1
1
+ = 1. Then for any
p
q
> 0, there exists a constant C() > 0 such that
ab ≤ ap + C()bq
(2.18)
On the right hand side of the Young’s inequality, the coefficient of ap could be made
small arbitrarily, thus can be absorbed by a larger term containing ap on the left hand
side later. This technique of using Young’s inequality will be used throughout this
thesis.
We state and prove a lemma concerning the geometric convergence of sequences of
numbers. This lemma is proved in Lemma 4.1, Chapter I of [3].
Lemma 2.2 Let {Yn }, n=0,1,2. . . be a sequence of positive numbers, satisfying the
recursive inequalities
Yn+1 ≤ Cbn Yn1+α
(2.19)
where C, b > 1 and α > 0 are given numbers. If
2
Y0 ≤ C −1/α b−1/α
10
(2.20)
then {Yn } converges to zero as n goes to ∞.
Consider the cylinder QT and let N be the dimension of the domain Ω. We need the
Sobolev embedding for parabolic space V0m,p (QT ). This space is embedded in Lq (QT )
for some q > p. In a precise way, from Proposition 3.1 , Chapter 1 of [3] we have
Proposition 2.3 There exists a constant γ depending only upon n, p, m such that
for every v ∈ V0m,p (QT )
ZZ
q
|v(x, t)| dxdt ≤ γ
q
p/N
Z
m
|Dv(x, t)| dxdt
sup
|v(x, t)| dx
ZZ
QT
p
0<t<T
QT
Ω
(2.21)
where q =
p(N + m)
. Moreover, kvkq,QT ≤ γkvkV m,p (QT )
N
A useful consequence of the Sobolev embedding is the following inequality
Proposition 2.4 Let p > 1. There exists a constant γ depending only upon N, p
such that for every v ∈ V0p (QT ),
p
kvkpp,QT ≤ γ||v| > 0| N +p kvkpV p (QT )
Proof: By applying the Sobolev embedding with m = p, there exists a number γ
such that
kvk p(N +p) ,QT ≤ γkvkV p (QT )
N
and a simple application of Hölder inequality implies
kvkp,QT ≤ kvk p(N +p) ,QT kχ{|v|>0} k(N +p),QT
N
Combining these two estimates, we get Proposition (2.4)
Another inequality we will use widely is the Poincaré-Moser inequality. The proof
of this inequality can be found in Lemma 6.12, Chapter V of [8]
11
Lemma 2.5 Let Ω be a bounded convex set in IRN and let η ∈ C(Ω) satisfy 0 ≤ η ≤ 1
in Ω and the set {η ≥ k} is convex for all k ∈ (0, 1). Let v ∈ W 1,p (Ω). Define
Z
vη dx
R = diamΩ
and
vΩ = ZΩ
.
η dx
Ω
There exists a constant C depending only on N, p such that
Z
p
|v(x) − vΩ | η dx ≤ CR
p
Z
Ω
|Dv|p η dx.
(2.22)
|Dv|p η dx.
(2.23)
Ω
Moreover, for Σ = {x|v(x) = 0}, we have
Z
p
v η dx ≤ C
Ω
RN p
|Σ|(N −1)p/N
Z
Ω
Finally, we state here an elementary lemma which has been used in the later proof.
We would like to present it more general than needed for a later use in the next
section.
Lemma 2.6 Let R, d > 0 be given constants and V (t), V̄ (t), m(t) be functions on an
interval I := [a, b] such that V̄ (t) ≥ V (t) on I. Assume that
Z
Ω
η 2 dx ×
d
1 2
V (t) +
V̄ (t)m(t) ≤ G(t),
dt
dR2
∀t ∈ I,
and
Z
G(t) dt ≤ CRn .
I
Here η is a cutoff function such that its support is contained in a ball whose radius
R
is comparable to R. If I m(t)dt ≥ C1 Rn+2 for some universal constant C1 and
|b − a| ∼ dR2 then there exists t1 ∈ I and a positive constant A depending only on
12
C, C1 such that V (t1 ) ≤ A.
Here and in the sequel, we use the notation A ∼ B to mean that the quantities
(usually nonnegative) A and B are comparable. That is, there exist positive constants
C1 , C2 independent of the quantities in question such that C1 B ≤ A ≤ C2 B.
Proof: Assume that V (t) ≥ A > 0 for all t in I := [a, b] so that V̄ 2 (t) ≥ V 2 (t).
We have
Z
Ω
V 0 (t)
1
G(t)
η dx 2 +
m(t) ≤ 2 .
2
V (t) dR
A
2
Integrating over I and making use of the assumption on G(t), m(t) and the fact
that |b − a| ∼ dR2 , we get
1
C1 R ≤ 2
R
n
Z
1
1
2CRn
1
2C
m(t)dt ≤ (
−
)
η 2 dx +
≤ Rn ( + 2 ).
2
V (a) V (b) Ω
A
A A
I
Z
By choosing A large enough (independent of R, d), we see that the above inequality
gives a contradiction. Hence, there must exist t1 ∈ I such that V (t1 ) ≤ A.
13
Chapter 3
Interior Hölder continuity for the degenerate
scalar case
3.1
Scaled cylinders and auxiliary logarithmic functions
In this section, we consider the following scalar parabolic equation
∂u
− div(a(x, t, u, Du)) = b(x, t, u, Du)
∂t
(3.1)
We impose the following structure condition
a(x, t, u, Du)Du ≥ ν0 Φ(u)|Du|2 − ψ0 (x, t),
p
|a(x, t, u, Du)|
≤ ν1 Φ(u)|Du| +
|b(x, t, u, Du)|
≤ ν2 Φ(u)|Du| + ψ2 (x, t),
Φ(u)ψ1 (x, t),
(3.2)
and the following condition on Φ(u)
Φ1 |u|m
≤
Φ(u)
≤
Φ2 |u|m ,
(3.3)
where ν0 , ν1 , Φ1 , Φ2 , m > 0. In fact, one can see that the proof still applies to more
general structure for Φ. In particular, one may also consider the singular case where
m ∈ (−1, 0).
For simplicity, we assume that the functions ψi ’s on the right hand side of (3.2)
are bounded nonnegative functions and, in addition, we will consider only nonnegative
solutions to (3.1) although the proof for the case of signed solutions is similar modulo
some minor modifications.
14
Let fix a point (t0 , x0 ) ∈ QT and let R > 0. We denote the cylinder Q(R, r) :=
Q(x0 , t0 , R, r) := BR (x0 ) × [t0 − r, t0 ].
Theorem 3.1 Let u be a locally bounded weak solution of (3.1). Assume (3.2). Then
(x, t) → u(x, t) is locally Hölder continuous in QT . That is, for every compact subset
K of QT , there exists a constant C = C(kuk∞,K , dist(K, ST )) and α = α(kuk∞,K ) in
(0, 1) such that
|u(x1 , t1 ) − u(x2 , t2 )| ≤ C(|x1 − x2 |α + |t1 − t2 |α/2 ),
(3.4)
for every pairs of points (x1 , t1 ), (x2 , t2 ) ∈ K.
For (xi , ti ) in the interior of QT the above result for regular diffusion equations is
well known (e.g. [7, Theorem 7.1]). For degenerate case, interior Hölder regularity
for (3.1) was also considered in [2] and [3].
We briefly sketch the argument to show that the Hölder regularity follows from a
dichotomy for the degenerate equation (3.1). The main idea is to construct a nested
sequence of suitable scaled cylinders, and show that in each such a cylinder, if the
solution is away from zero then the equation is nondegenerate and 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 a Hölder fashion. This is the idea of the method
of level sets or truncation techniques of De Giorgi that had been generalized by
Ladyzhenskaya et al and later by other mathematicians (DiBenedetto, Porzio and
Vespri, Ivanov,. . . ) to obtain the Hölder estimates for weak solutions to quasilinear
equations (nondegenerate or degenerate).
Meanwhile, there is a less-well-known technique using auxiliary logarithmic functions to derive decay estimates of oscillations. This method has been used in [6] for
nondegenerate elliptic equations. The method seems to be more direct and less technical than the above one since we do not need to have two alternatives here. The
15
main ingredients can be summarized as follows:
1. Construct suitable logarithmic functions, denoted by w, whose upper bound
implies the decay estimates of oscillations.
2. Estimate the supermum norm of w in terms of its L2 norm and the parameters
of the equation (3.1).
3. Estimate the L2 norm of w and complete the proof due to step 1.
We set up the geometry of the cylinders we will work with.
Let ε ∈ (0, 1) be given. Fix a point (x0 , t0 ) ∈ QT and construct the cylinder
Q(R2−ε , 3R)
:= Bx0 (3R) × [t0 − R2−ε , t0 ],
where R ∈ (0, 1) is so small such that the cylinder is contained in QT . We set
µ+ =
sup
Q(R2−ε ,3R)
u,
µ− =
inf
Q(R2−ε ,3R)
u,
ω = µ+ − µ− ,
and (assuming u ≥ 0) M = µ+ = max{µ+ , µ− , ω}. We consider the cylinder
Q(dR2 , R) := Bx0 (2R) × [t0 − 4dR2 , t0 ],
where d := 1/Φ(M ).
We will assume that
Φ(M ) > 4Rε .
(3.5)
Otherwise, there would be nothing to prove since the oscillation is then comparable
to the radius. Note also that this implies the inclusion Q(dR2 , R) ⊂ Q(R2−ε , 3R).
Let α ∈ (0, 1) be small enough such that α ≤ min{ε/m, (1 − ε)/2} and therefore
R2−2α , R1−α , R2−α < Rε for all R ∈ (0, 1).
For κ ∈ {−1, 1}, we consider the following logarithmic functions in Q(R2−ε , 3R).
16
wκ (x, t) = log
ω + Rα
Nκ (u)
,
(3.6)
where
N1 (u) = 2(µ+ − u) + Rα ,
and N−1 (u) = 2(u − µ− ) + Rα .
Note that wκ ≥ − log 2. Again, in the lemmas below we will show that, under the
assumption (3.5), one of the above wκ will be bounded from above by some universal
constant in some subcylinder of Q(dR2 , R).
w1 (x, t) ≤ C
or w−1 (x, t) ≤ C,
∀(x, t) ∈ Bx0 (R) × [t0 − σdR2 , t0 ],
(3.7)
where σ ∈ (0, 1) to be determined independently of R.
It is not hard to see that either of the above inequalities implies the following decay
estimate for the oscillation
oscQ(σd(R/2)2 ,R/2) u ≤ η1 ω + CRα ,
(3.8)
for some η1 ∈ (0, 1) and C are positive constants independent of u, R.
In fact, without of loss generality, we can assume that w1 ≤ C in Q(σd(R/2)2 , R/2).
Thus
CRα + (2C − 1)ω ≥ 2C(u − µ− ) ≥ 2CoscQ(σd(R/2)2 ,R/2) u.
2C − 1
< 1.
2C
We now choose a postive constant A such that η = η1 + C/A < 1 and Am Φ1 > 4
Hence, we derive (3.8) with η1 =
where Φ1 is the constant in the structure condition (3.3). If ω > ARα then M > ARα
and therefore Φ(M ) > Φ1 Am Rmα > 4Rε (because ε > mα and R < 1). In this case,
17
it follows that (see (3.8))
oscQ(σd(R/2)2 ,R/2) u ≤ η1 ω + CRα < ηω.
We summarize the above in
Proposition 3.2 There exist universal positive constants α, σ, η ∈ (0, 1) and A such
that
ω ≤ ARα
either
or
oscQ(σd(R/2)2 ,R/2) u ≤ ηω.
To prove this, we will need only to show (3.7) in the next section. This proposition
is similar to Lemma 9.1 in [13, p.176] (nondegenerate version) from which the Hölder
continuity of u follows in a standard manner (see [3, 7, 13]). However, we will delay
a complete proof for the main Theorem 3.1 as a consequence of Proposition 3.2 until
the final section.
Before going further, we show that wκ are subsolutions of some parabolic equations.
We observe that
Dx wκ = −
Dx Nκ
2κ
=
Dx u,
Nκ
Nκ (u)
∂wκ
2κ ∂u
=
.
∂t
Nκ (u) ∂t
(3.9)
For any nonnegative test function η, we multiply the equation of u by φ = κη/Nκ (u)
and integrate over Ω to get
Z
Ω
∂u
φ dx +
∂t
Z
Z
aDφ dx =
Ω
bφ dx
Ω
Using (3.9) and the fact that
Dφ =
κDη κηDNκ
κDη κηDwκ
−
=
+
,
2
Nκ
Nκ
Nκ
Nκ
18
(3.10)
we obtain
1
2
Z
Ω
∂wκ
η dx +
∂t
Z
Z
(Aκ Dη + Aκ Dwκ η) dx =
Ω
Ω
κb
η dx,
Nκ
(3.11)
where Aκ = κa/Nκ .
By the ellipticity condition we have
Aκ Dwκ = 2κ2
aDu
2ψ0
aDu
|Du|2 2ψ0
1
=
2
≥
2ν
Φ
− 2 = ν0 Φ|Dwκ |2 − 2 .
0
2
2
2
Nκ
Nκ
Nκ
Nκ
2
Nκ
(3.12)
Moreover,
|Du|
ψ2
ψ2
b
≤ ν2 Φ
+
≤ ν2 Φ|Dwκ | +
Nκ
Nκ
Nκ
Nκ
Hence, from (3.11), wκ satisfies
1
2
Z
Ω
∂wκ
η dx+
∂t
Z
Ω
1
(Ak Dη + ν0 Φ|Dwκ |2 η) dx ≤
2
Z
(ν2 Φ|Dwκ | + Ψ)η) dx, (3.13)
Ω
where
Ψ :=
ψ2
2ψ0
+ 2.
Nκ
Nκ
Note that since Nκ ≥ Rα ≥ 1
kΨk∞ ≤ C(kψ0 k∞ + kψ2 k∞ ).
19
(3.14)
3.2
The lemmas
In this section, we will focus on the proof of our main Proposition 3.2. We now wish
to estimate the supremum of w in terms of its L2 norm. Since (3.11) is degenerate,
we need to show that on the subset where that function is positive there will be
certain comparison property for the solution u in order to handle the degeneracy in
the diffusion term. We must also show that one of the functions wi ’s vanishes on a
subset of large measure, a key factor in obtaining the L2 estimates. In details, we
have the following result
Lemma 3.3 We can construct a function w which is either w1 or w−1 such that, for
R̄ = 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 =⇒ C1 M ≤ u(x, t) ≤ M =⇒ C̄1 Φ(M ) ≤ Φ(u) ≤ C̄2 Φ(M ).
(3.15)
Here, C1 , C̄1 , C̄2 are positive universal constants. Moreover, let Q0w := {w+ =
0} ∩ Q(dR̄2 , R̄) then we can find constants σ ∈ (0, 1) and γ > 0 independent of R
such that
meas(Q0w ∩ Bx0 (R̄) × [t0 − 4dR̄2 , t0 − 2σdR̄2 ]) > γmeas(Q(dR̄2 , R̄)) ∼ dRn+2 . (3.16)
Before presenting the proof of Lemma 3.3, we need a lemma [13, lemma 5.1] (see
also its original version in [2]). We want to give here a less involved proof for, however,
a slightly general version of it.
Lemma 3.4 Assume (3.5). Given 0 < b < a < 1 there is a number λ0 depending on
20
a, b but independent of R such that if
meas{(x, t) ∈ Q(dR2 , R) : u(x, t) < µ− + aω} ≤ τ meas(Q(dR2 , R)),
(3.17)
for some τ ∈ (0, λ0 ) then
u(x, t) > µ− + bω
for a.e.
2 !
R
R
(x, t) ∈ Q d
.
,
2
2
(3.18)
Proof: Without loss of generality, we could assume µ− = 0 so that M = ω (in
fact, this is the degenerate situation).
R
R
Denote Rn = + n+1 , Qn = Q(dRn2 , Rn ) and an = b +
2
2
Consider the following functions on Q(dR2 , R).
1
(a
2n
− b).
Wn = (an ω − u)+ .
and we set An = {(x, t) ∈ Qn /Wn (x, t) > 0}. Note that we have
DWn = −Du,
∂u
∂Wn
=
∂t
∂t
if Wn > 0.
For each t0 ∈ (t0 − 4dRn2 , t0 ), we set Qt0 = Bx0 (2Rn ) × (t0 − 4dRn2 , t0 ) for simplifying
notation.
For each n > 0, we construct a cutoff function η in the cylinder Qn := Bx0 (2Rn ) ×
[t0 − 4dRn2 , t0 ] satisfying the following properties.
21
0 ≤ η(x, τ ) ≤ 1 ∀(x, τ ),
if τ ≤ t0 − 4dRn2 or x 6∈ Bx0 (2Rn ),
η(x, τ ) ≡ 0
(3.19)
η(x, τ ) ≡ 1
|Dx η| ≤
if (x, τ ) ∈ Qn+1 ,
2n
,
R
|
∂η
4n
|≤
.
∂t
dR2
By testing the equation (3.10) of u by φ = −η 2 Wn , then integrating over (t0 −
4dRn2 , t0 ), and applying integration by parts, we easily get:
Z
η
2
Wn2 (x, t0 )
ZZ
2ηηt Wn2 dxdt
dx −
Qt0
Ω
ZZ
2
(3.20)
2
(2bη Wn − 2a(η DWn + 2ηDηWn )) dxdt
=
Qt0
Otherwise, from the structure conditions
ZZ
ZZ
2
Qn
Qt0
Qt 0
ZZ
ZZ
ν1 Φ(u)η|Dη||DWn |Wn dxdt+
aηDηWn dxdt ≤
Qt0
Qn
ZZ
ψ0 χAn η 2 dxdt
ν0 Φ(u)|DWn | η dxdt −
aDWn η dxdt ≥
ZZ
−
ZZ
2 2
Φ(u)ψ1 η|Dη|Wn dxdt
Qn
ZZ
2
p
(ν2 Φ(u)η 2 Wn |DWn | + ψ2 η 2 Wn ) dxdt
bη Wn dxdt ≤
Qn
Qt0
Combining these estimates and recalling (3.20) and that t0 ∈ (t0 − 4dRn2 , t0 ) is arbitrary, we obtain
Z
supt0 −4dRn2 <t0 <t0
ZZ
2
ηηt Wn2 dxdt +
Qn
ZZ
+4
η
2
Wn2 (x, t0 )
ZZ
Ω
Qn
ZZ
ZZ
ψ0 χAn η 2 dxdt + 4ν1
Qn
Φ(u)η|Dη|Wn |DWn | dxdt
Qn
ZZ
p
Φ(u)ψ1 η|Dη|Wn dxdt + 2ν2
Qn
Φ(u)η 2 |DWn |2 dxdt ≤
dx + 2ν0
2
ZZ
ψ2 η 2 Wn dxdt
Φ(u)η Wn |DWn | dxdt + 2
Qn
Qn
(3.21)
22
By Young’s inequality (2.1), we derive:
Z
supt0 −4dRn2 <t0 <t0
ZZ
η
2
Wn2 (x, t0 )
dx +
Ω
(ψ0 + ψ12 + ψ22 )χAn dxdt + C
≤C
ν0
2
ZZ
Qn
ZZ
Φ(u)η 2 |DWn |2 dxdt
Qn
(Φ(u)η 2 + Φ(u)|Dη|2 + ηt + 1)Wn2 dxdt
Qn
4n+1 C(ν0 , ν1 )Φ(M )
≤
R2
ZZ
Wn2
ZZ
(ψ0 + ψ12 + ψ22 )χAn dxdt
dxdt + C
Qn
Qn
(3.22)
Note that
ZZ
(ψ0 +
ψ12
+
ψ22 )χAn
ZZ
dxdt ≤ C(kψ0 k∞ , kψ1 k∞ , kψ2 k∞ )
Qn
χAn dxdt.
Qn
Let
Vn = (an ω − max(u, an+1 ω))+ .
Then we have u > an+1 ω ≥ bω and thus Φ(u) ≥ C(b)Φ(M ) on the set {(x, t) :
DVn (x, t) 6= 0}. Since Vn ≤ Wn , we have:
Z
supt0 −4dRn2 <t0 <t0
η
2
Vn2 (x, t0 )
dx +
Ω
Z
≤ supt0 −4dRn2 <t0 <t0
η
2
Wn2 (x, t0 )
C(b) ν20
dx +
Ω
ν0
2
ZZ
Φ(M )η 2 |DVn |2 dxdt
Qn
ZZ
(3.23)
2
2
Φ(u)η |DWn | dxdt
Qn
At this point, we could make a change of time variable τ = Φ(M )t and note that we
will implicitly understand that all functions in our below estimates are rewritten in the
new time variable τ . Hence, if we denote τ0 = Φ(M )t0 , Q˜n = B2Rn (x0 ) × [τ0 − 4Rn2 , τ0 ]
and A˜n = {(x, τ ) ∈ Q˜n /Wn (x, τ ) > 0}, from (3.22) we have
23
Z
η
supτ0 −4Rn2 <τ <τ0
n+1
4
C
2
Vn2 (x, τ )
ZZ
dx +
Q˜n
Ω
ZZ
R2
Q˜n
Wn2
ZZ
dxdτ + Cd
Q˜n
|D(ηVn )|2 dxdτ ≤
(3.24)
χA˜n dxdτ
An application of (2.4) to the left hand side of the above estimate gives us
2
kηVn k22,Q˜n ≤ γ|A˜n | N +2 kηVn k2V 2 (Q˜n )
On the other hand, Vn (x, τ ) = (an − an+1 )M when (x, τ ) ∈ Ãn+1 . Hence
kηVn k22,Q˜n
≥
kVn k22,Q̃n+1
2
≥ (an −an+1 ) M
2
ZZ
Q̃n+1
χÃn+1 dxdτ = (an −an+1 )2 M 2 |Ãn+1 |
Moreover, due to Wn ≤ an M
ZZ
Q˜n
Wn2
dxdτ ≤
a2n M 2
ZZ
Q˜n
χÃn dxdτ = a2n M 2 |Ãn |
Thus from (3.24)
(an − an+1 )2 M 2 |Ãn+1 | ≤
2
4n+1 C
|Ãn |1+ N +2 + Cd|Ãn |
2
R
Divide by |Q̃n+1 | and introduce the quantity Xn =
(3.25)
|An |
|A˜n |
=
.
|Qn |
|Q˜n |
1+ N2+2
M 2 (an − an+1 )2 Xn+1 ≤ C4n M 2 a2n Xn
1+ N2+2
+ CdR2 Xn
(3.26)
2
Since CM m ≥ Φ(M ) > 4R , we have M 2 > CR m and thus if we choose such that
(1 +
2
)
m
≤ 2, then dR2 < CM 2 for some constant C, which implies
1+ N2+2
Xn+1 ≤ C4n Xn
24
(3.27)
By applying the fast geometric convergence (2.2), we can deduce Xn tends to 0 when
n approaches to infinity provided that X0 is small enough. But this is equivalent to
the condition
meas{(x, t) ∈ Q(dR2 , R) : u(x, t) < µ− + aω} ≤ τ meas(Q(dR2 , R))
(3.28)
for a small parameter τ , which is our assumption.
Because |Qn | is bounded by |Q1 | for all n, |An | tends to 0 when n goes to infinity.
R
Since limn→∞ an = b and lim Rn = , the conclusion of our lemma holds.
n→∞
2
We return to the proof of our comparison lemma
Proof of Lemma 3.3: Choose a = 1/2 and b = 1/4 in the above lemma and let
λ0 > 0 be the constant given by that lemma (it can be smaller but fixed). Set
2
−
+
−
Q−
u := {(x, t) ∈ Q(dR , R) : u(x, t) < µ + ω/2 = (µ + µ )/2}.
In fact, Q−
u is the near degeneracy subset where u is closer to zero (recall also that
µ− ≥ 0). We consider alternatively two cases.
Case I
2
If meas(Q−
u ) > λ0 meas(Q(dR , R)), the near degeneracy subset is large, we define
+
−
+
w(x, t) = w1 (x, t), R̄ = R and Q0w = Q−
u . It is clear that w1 = 0 on Qu and w > 0
+
+
−
−
on Q(dR2 , R)\Q−
u where µ ≥ u ≥ (µ + µ )/2. Since µ ≥ 0, (3.15) is verified.
Case II
2
Otherwise, if meas(Q−
u ) ≤ λ0 meas(Q(dR , R)), the equation is less degenerate. We
R 2 R
, 2 ).
set w(x, t) = w2 (x, t), R̄ = R/2 and Qw = Q−
u ∩ Q(d 2
From the above lemma (3.4), if λ0 is sufficiently small then u(x, t) ≥ µ− + ω/4 =
3µ− /4 + µ+ /4 ≥ µ+ /4 (since µ− ≥ 0) in Q(d(R/2)2 , R/2). We see that (3.15) holds
and the equation is actually nondegenerate in Q(d(R/2)2 , R/2). To check (3.16), we
25
observe that
meas(Qw ) ≤
meas(Q−
u)
2
R
R
≤ λ0 meas(Q(dR , R)) = Cλ0 meas(Q(d
, )),
2
2
2
for some universal constant C. We can choose λ0 smaller such that Cλ0 < 1. With
2
such choice of λ0 , the function w+ vanishes on the set Q0w := Q(d R2 , R2 )\Qw and
meas(Q0w )
2
R
R
> (1 − Cλ0 )meas(Q(d
, )).
2
2
(3.29)
In both cases, we see that (3.15) holds and (3.16) is verified for some γ > 0 but
with σ = 0. To complete the proof we need to show (3.16) for some σ > 0. But this
is easy since, for any σ > 0,
meas({w+ = 0} ∩ Bx0 (R̄) × [t0 − 4dR̄2 , t0 − 2σdR̄2 ]) ≥ (γ − 2σ)meas(Q(dR̄2 , R̄)).
Hence, if we choose σ = γ/4 then (3.16) follows.
From now on we will simply write R̄ by R and let w, σ be the function and the
constant stated in the above lemma. Given (3.15) of this lemma, we can estimate
the suppremum of w in terms of its L2 norm as follows (recall that ψi are bounded
functions).
Lemma 3.5 There exists a positive constant C independent of R such that
sup
Bx0 (R)×[t0 −σdR2 ,t0 ]
w≤C
1
{
σdRn+2
!
ZZ
(w+ )2 dxdt}1/2 + 1 .
Bx0 (2R)×[t0 −2σdR2 ,t0 ]
(3.30)
26
Proof: Using a change of variable in t we can assume that σ = 1. By (3.15),
(??) is nondegenerate on the set where w > 0 so that the proof of (3.30) is standard
by using iteration technique of Möser. For a, b > 0, we consider the cylinder Q =
Q(t0 , R, a, b) := Bx0 (R + aR) × [t0 − 2bdR2 , t0 ] and a cut-off function η(x, t) satisfying
the following properties.
0 ≤ η(x, τ ) ≤ 1 ∀(x, τ ),
if τ ≤ t0 − 2bdR2 or x 6∈ Bx0 (R + aR),
η(x, τ ) ≡ 0
|Dx η| ≤
1
,
aR
|
(3.31)
∂η
C
|≤
.
∂t
bdR2
For any λ ≥ 1 and t0 ∈ (t − 2bdR2 , t), we test the inequality (3.13) of w by
∂w
∂wκ
=
ψ = η 2 (wk )λ , where wk = w+ +k with k = kψ0 k∞ +kψ1 k∞ +kψ2 k∞ . Thus
∂t
∂t
and Dwκ = Dw.
1
2
Z
Ω
∂wκ
ψ dx +
∂t
Z
Ω
1
(Ak Dψ + ν0 Φ|Dwκ |2 ψ) dx ≤
2
Z
(ν2 Φ|Dwκ | + Ψ)ψ) dx,
Ω
Then by integrating over (t − 2bdR2 , t0 ) and applying integration by parts, we obtain:
! ZZ
∂η
2η wκλ+1 dxdt +
Ak (λη 2 wκλ−1 Dwκ + 2ηDηwκλ ) dxdt
η 2 wκλ+1 dx −
∂t
Ω
Qt0
Qt0
ZZ
ZZ
ZZ
1
2 λ
2
2 λ
+ ν0
Φη wκ |Dwκ | dxdt ≤
ν2 Φη wκ |Dwκ | dxdt +
Ψη 2 wκλ dxdt
2
Qt0
Qt0
Qt0
(3.32)
1
2(λ + 1)
Z
ZZ
where Qt0 = Bx0 (R + aR) × [t0 − 2bdR2 , t0 ].
27
Again, recall that Aκ =
κa
and by using the structure condition (3.2) with Young
Nκ
inequality (2.1)
ZZ
−
Ak ηDηwκλ
ZZ
Qt0
Qt0
ZZ
wκλ ηDη(ν1 Φ(u)|Dwκ | +
dxdt ≤
(Φ(u)η 2 wκλ−1 |Dwκ |2 + C()Φ(u)|Dη|2 wκλ+1 +
≤
Qt0
ZZ
Ak η
2
wκλ−1 Dwκ
ZZ
η 2 wκλ−1 (
dxdt ≥
Qt0
Qt0
p
ψ1
Φ(u) ) dxdt
Nκ
p
wλ+1
Φ(u)η|Dη| κ ) dxdt
Nκ
(3.33)
ν0
2ψ0
Φ(u)|Dwκ |2 −
dxdt)
2
Nκ
ν0
2η 2 wκλ+1
( Φ(u)η 2 wκλ−1 |Dwκ |2 −
) dxdt
2
Nκ
ZZ
≥
Qt0
ZZ
(3.34)
(ν2 Φη 2 wκλ |Dwκ | + Ψη 2 wκλ ) dxdt
Qt0
ZZ
(Φ(u)η 2 wκλ−1 |Dwκ |2 + C()Φ(u)η 2 wκλ+1 + η 2 wκλ+1 (
≤C
Qt0
1
1
+ 2 )) dxdt
Nk Nk
(3.35)
Combining these estimates (3.33),(3.34),(3.35) and recalling (3.32), we easily get
Z
sup
η
t0 −2bdR2 <t0 <t0
2
wk1+λ (x, t0 )
Ω
ZZ
≤ Cλ
λ2 ν0
dx +
2
ZZ
Φ(u)η 2 wkλ−1 |Dwk |2 dxdt
Qt0
wκλ+1 (Φ(u)(η 2 + |Dη|2 ) + η|
Qt0
∂η
λ
1
| + η2(
+ 2 )) dxdt.(3.36)
∂t
Nk Nk
Thanks to the key lemma (3.3), we have Φ(u)|Dwκ |2 ∼ Φ(M )|Dwκ |2 and thus:
ZZ
Φ(u)η
2
wkλ−1 |Dwk |2
ZZ
Φ(M )η 2 wkλ−1 |Dwk |2 dxdt
dxdt ≥ C
Qt 0
Qt0
28
Moreover, due to the choice of α that makes R2−α , R2−2α < R or
1 1
Φ(M )
, 2 ≤
,
Nκ Nκ
R2
from (3.36) we have:
Z
sup
t0 −2bdR2 <t0 <t0
ZZ
λ2 ν0 Φ(M )
η 2 (wk+ )λ−1 |Dw|2 dxdt
η
dx +
2
Qt0
ZZ
2
C(ν0 , ν1 )λ Φ(M )
≤
(wk+ )1+λ dxdt.
max{a2 , b}R2
Qt0
(3.37)
2
Ω
(wk+ )1+λ (x, t0 )
Making a change of variables dτ = Φ(M )dt, τ0 = Φ(M )t0 to rewrite the above
estimate (3.37):
Z
sup
η
τ0 −2bR2 <τ <τ0
2
λ 2 ν0
dx +
2
2 ZZ
(wk+ )1+λ (x, τ )
Ω
≤
C(ν0 , ν1 )λ
max{a2 , b}R2
ZZ
Bx0 (R+aR)×[τ0 −2bR2 ,τ0 ]
Bx0 (R+aR)×[τ0 −2bR2 ,τ0 ]
η 2 (wk+ )λ−1 |Dw|2 dxdτ
(wk+ )1+λ dxdτ .
(3.38)
(Note that from now, all functions in integrations are rewritten in the new time
variable τ )
N +2
, then a simple application of the Sobolev embedding
N
leads to:
Let us denote χ =
λ+1
(2.3) to ηwκ 2
!1/χ
ZZ
Bx0 (R+aR)×[τ0
−2bR2 ,τ
η 2k wκχ(λ+1) dxdτ
0]
(3.39)
ZZ
C(ν0 , ν1 )λ2
≤
(wk+ )1+λ dxdτ .
a2 + b
2
Bx0 (R+aR)×[τ0 −2bR ,τ0 ]
For i = 0, 1, 2, . . . let us take ai =
1
1
1
,
b
=
+
. Define R0 = 2R,
i
2i+1 + 1
2
2i+1
T0 = τ0 − 2R2 and
1
Ri = Ri−1 − i R = (1+2−i )R;
2
1
Ti = Ti−1 + i R2 ;
2
29
Qi = Bx0 (Ri )×[Ti , τ0 ];
λ+1 = χi .
Hence, Ri = Ri+1 + ai Ri+1 and Ti = τ0 − 2bi Ri2 . For each n, we can choose the
cutoff function η such that η ≡ 1 if (x, τ ) ∈ Qn+1 . Using this cutoff η in (3.39) when
R = Rn+1 , a = an , λ + 1 = 2χn :
ZZ
1/χ
n+1
wκ2χ
ZZ
n
(wk+ )2χ dxdτ .
n 2n
≤ C(ν0 , ν1 )4 χ
dxdτ
Qn+1
(3.40)
Qn
ZZ
Denote In =
Qn
n
wκ2χ dxdτ
Pn
1
induction on n: In ≤ C
Note that these series
i=0 2χi
P∞
1/2χn
n
Pn
(4χ2 )
1
i=0 2χn ,
n
, then we have In+1 ≤ C 1/2χ (4χ2 )n/2χ In . By
i
i=0 2χi
P∞
n
i=0 2χn
I0 .
are convergent and thus this leads to
(3.41)
lim supn→∞ In ≤ CI0
!1/2
ZZ
Since limn→∞ In = sup(x,t)∈Bx0 (R)×[t0 −dR2 ,t0 ] wκ and I0 =
Bx0 (2R)×[t0 −2dR2 ,t0 ]
wκ2 dxdt
we have the conclusion.
Before going futher, we have a quick note about the local L2 norm of
p
Φ(u)|Du|.
Let R > 0. We write BR = BR (x0 ), IR = [t0 − R2 , t0 ] and QR = BR × IR . For a
measurable function u on Q(R) we denote its mean value over Q(R) by uR . That is,
R
1
uR := |Q(R)|
udxdt.
Q(R)
Let U = esu , where s > 0 to be determined later. By testing (3.10) with esu η with
η a cut-off function in Q = Q(2R, 4R2 ) we get
Z
su
e η dx
Ω
|tt00 −4R2
Z
su
Z
se aDuη ≤
+
Q
su
Z Z
e (|aDη| + bη + |ηt |) +
Q
t
gesu η dσ,
∂Ω
(3.42)
30
,
Note that we can choose η such that η = 1 on Q(R, R2 ) and η(t0 − 4R2 ) = 0. Since
aDu ≥ ν0 Φ(u)|Du|2 − ψ0 , the above gives the following estimate
Z
su
Z
2
se (ν0 Φ(u)|Du| − ψ0 )η ≤
Q
Cesu (Φ(u)|Du|(|Dη| + η) +
p
Φ(u)ψ1 |Dη| + ψ2 η + |ηt |)
Q
(3.43)
One easily see that, if s is sufficiently large, by Young’s inequality we obtain
ZZ
Q(R,R2 )
Φ(u)|Du|2 dxdt ≤ C(kuk∞,Q¯T )Φ(M )Rn .
(3.44)
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 w−1 , by a constant
independent of R. The following lemma gives an estimate for the right hand side of
(3.5) and therefore concludes the proof of the theorem.
Lemma 3.6 There exists a positive constant C depending on kuk∞,Q¯T but independent of R such that either w1 or w−1 satisfies the following
1
dRn+2
Proof:
ZZ
w2 dxdt ≤ C.
Bx0 (2R)×[t0
−2R2 ,t
0]
Let η(x) be a cut-off function for Bx0 (2R), i.e. η(x) ≡ 1 in Bx0 (2R),
η(x) ≡ 0 outside Bx0 (4R) and |Dx η| ≤
1
.
2R
We go back to (3.13) and replace η by η 2 .
By Young’s inequality, we have
|Dw|η|Dη| ≤ εη 2 |Dw|2 + C(ε)|Dη|2 ,
ν2 |Dw|η 2 ≤ εη 2 |Dw|2 + C(ε)η 2 .
For ε sufficiently small, we obtain in a standard way that
d
dt
Z
2
Z
wη dx +
Ω
η 2 Φ(u)|Dw|2 dx ≤ H(t),
Ω
31
(3.45)
where
Z
H(t) := C(
(Φ(u)|Dη|2 + Φ(u) +
Ω
ψ2
2ψ0
+ 2 )η 2 dx.
Nκ
Nκ
(3.46)
By our assumption on the structure of the equation ,we observe that
Z
H(t) dt ≤ CdΦ(M )Rn = CRn ,
for any interval I with |I| ≤ CdR2 .
(3.47)
I
Z
Z
η 2 dx. Let w+ , w− ≥ 0 be the positive and negative
Ω
Ω
Z
Z
+
2
+
−
+
parts of w, so that w = w −w . We also denote V (t) =
w (x, t)η dx/
η 2 dx.
Let V (t) =
2
w(x, t)η dx/
Ω
Ω
We now set I2 := [t0 − 4dR2 , t0 − 2σdR2 ] and
Ω0t = {x ∈ Bx0 (2R) : w+ (x, t) = 0},
and m(t) = meas(Ω0t ),
t ∈ I2 .
Because on the set w > 0 we have that Φ(u) ∼ Φ(M ). So,
Z
CΦ(M )
2
Z
+ 2
2
η |Dw | dx ≤ CΦ(M )
Ω
Z
2
η |Dw| dx ≤
w≥0
η 2 Φ(u)|Dw|2 dx, (3.48)
Ω
for some universal constant C. Using the Poincaré-Moser inequality (2.5), we have
CΦ(M )
R2
Z
Z
η (w − V ) dx ≤ Φ(M )
2
+
+ 2
Ω
η 2 |Dw+ |2 dx.
(3.49)
Ω
By reducing the above integral to the smaller set Ω0t , we have from (3.45) and the
above estimates that
Z
Ω
η 2 dx ×
d
Φ(M ) + 2
V (t) +
(V ) (t)m(t) ≤ H(t) (Φ(M )Rn−2 ).
dt
R2
32
Z
m(t)dt = meas(Q0 ) ≥ γmeas(Q2 ) ∼ dRn+2 . Also, we
From (3.16), we have
I2
+
have V (t) ≥ V (t). So, we can apply Lemma 2.6 here (with d = 1/Φ(M ) and
V̄ = V + ) to see that there exists t1 ∈ I2 such that V (t1 ) ≤ A.
Integrating (3.45) over [t1 , t2 ] for t2 ∈ I1 := [t0 − 2σdR2 , t0 ] (thus, |t2 − t1 | ≤ 4dR2 ),
we get
Z
V (t2 )
2
Z
t2
Z
2
n
Z
η Φ(u)|Dw| dxdt ≤ CR + V (t1 )
η dx +
Ω
2
t1
Ω
η 2 dx.
(3.50)
Ω
Note that V (t) ≥ − log 2, for all t ∈ I := [t0 − 4dR2 , t0 ]. The fact that V (t1 ) ≤ A
implies
Z Z
− log 2 ≤ V (t) ≤ C,
∀t ∈ I1
and
I1
η 2 Φ(u)|Dw|2 dxdt ≤ CRn ,
Ω
for some universal constant C depends only on A. From − log 2 ≤ w we also see that
w− , therefore, V − are bounded. So, V + = V + V − is bounded from above. From
(3.48), (3.49) and the above estimates we deduce
+
0 ≤ V (t) ≤ C,
∀t ∈ I1
and
Φ(M )
R2
Z Z
I1
η 2 (w+ − V + (t))2 dxdt ≤ CRn .
Ω
We easily see that this gives the estimate in the lemma.
3.3
Proof of Theorem 3.1
We finally prove the Hölder continuity of bounded weak solutions through an iterative scheme designed from all previous results. An immediate consequence of
33
Proposition 3.2 is the following:
Proposition 3.7 There exists constants A > 1, η < 1 and α > 0 that can be determined a priori upon only the data and supQ(R2−ε ,3R) u such that if we construct the
following sequences ω0 = ω, R0 = R, M0 = M and
Rn = C −n R0 ,
ωn+1 = max{ηωn , ARnα }
(3.51)
Mn = max{ωn , supQn−1 u}
where
Qn = Q(an Rn2 , Rn ),
1
= Φ(Mn )
an
(3.52)
Then for all n = 0, 1, 2, . . .
Qn+1 ⊂ Qn , oscQn u ≤ ωn
Proof:
(3.53)
We sketch the idea of the proof first. Recall the inclusion Q(dR2 , R) ⊂
Q(R2−ε , 3R) implies the relation
oscQ(dR2 ,R) u ≤ ω
(3.54)
The proof will show that it would have been suficient to work with a number ω and a
cylinder Q(a0 R2 , R) linked by (3.54). This relation is in general not verifiable a priori
for a given cylinder, since its dimension would have to be intrinsically defined in terms
of the essential oscillation of u within it. The role of having introduced the cylinder
Q(R2−ε , 3R) and having assumed (3.5) is that (3.54) holds true for the constructed
box Q(a0 R2 , R). It is part of the proof to show that, at each step, the cylinders Qn
and the essential oscillation of u within them satisfy the intrisic geometry dictated
by (3.54), and thus we can repeat the Proposition 3.2 for these cylinders.
34
When n = 0, Q0 = Q(a0 R2 , R) satisfy oscQ0 ≤ ω0 . We rewrite
R2
σd(R/2) = σ
= a1
4Φ(M )
2
σΦ(M1 )
4Φ(M )
R2
Next we will show that there exists a constant γ > 0 such that Φ(M ) ≤ γΦ(M1 ). Let
η be the same constant in Proposition 3.2.
If µ+ ≥ 2µ− then since M = µ+ ≤ 2ω
Φ(M ) ≤ Φ(2ω) ≤ Φ(2η −1 ω1 ) ∼ CΦ(ω1 ) = CΦ(M1 )
(3.55)
If µ+ ≤ 2µ− then since supQ(a0 R2 ,R) u ≥ µ−
Φ(M ) ≤ Φ(2µ− ) ≤ Φ(2M1 ) ∼ CΦ(M1 )
We can choose C such that C 2 =
(3.56)
σγ
+ 4 because this leads to
4
σd(R/2)2 ≥ a1 R12
and R/2 ≥ R1
(3.57)
and hence
Q1 = Q(a1 R12 , R1 ) ⊂ Q(σd(R/2)2 , R/2) ⊂ Q0
We separate two cases in order to show oscQ1 u ≤ ω1
Case 1:
If ω ≤ ARα then from (3.58)
oscQ1 u ≤ oscQ0 u ≤ ω0 ≤ AR0α ≤ ω1
Case 2:
35
(3.58)
If ω > ARα , then by Proposition 3.2 we must have
oscQ1 u ≤ oscQ(σd(R/2)2 ,R/2) u ≤ ηω ≤ ω1
In both cases, the relation (3.54) is still preserved at the step n = 1. The entire
process can now be repeated inductively starting from Q1 .
A consequence of this Proposition is
Lemma 3.8 There exists constants γ > 1 and α0 ∈ (0, 1) that can be determined a
priori only interms of the data, such that for all the cylinders
0 < ρ ≤ R,
oscQ(a
0
Q(a0 ρ2 , ρ),
1
= Φ(M )
a0
u ≤ γ(ω + R2α0 )
ρ2 ,ρ)
ρ α0
R
(3.59)
(3.60)
Proof: From the iterative construction of ωn it follows that
ωn+1 ≤ ηωn + ARnα
and by iteration
ωn ≤ η n ω + A
n−1
X
!
η i A−α(n−i) Rα
i=0
We may assume without loss of generality that α is so small that η ≤ A−α . Then
n
ωn ≤ η ω + An
R
An
α
Let now 0 < ρ ≤ R be fixed. There exists a non-negative integer n such that
C −(n+1) R ≤ ρ ≤ C −n R
36
This implies the inequalities
(n + 1) ≥ log
ρ −1
R log C
or η n ≤ η −1
and also since we can modify A such that A2 ≥ C 2 =
An
R
An
Let α0 = min(α/2;
α
≤ A1+α log
ρ −1/ log C
R
ρ | log η|/ log C
R
σγ
+4
4
ρα ≤ C(α)Rα/2 ρα/2
| log η|
) then
log C
ωn ≤ γ(ω + R
2α0
)
ρ α0
R
We observe that since Mn ≤ M , the cylinder Q(a0 ρ2 , ρ) is included in Qn =
Q(an Rn2 , Rn ). Thus
oscQ(a0 ρ2 ,ρ) u ≤ ωn
by applying the above Proposition 3.7 to conclude the proof.
Now, we prove the interior regularity of u. Assume M̄ = supQ̄T u. Let S be the
parabolic boundary of QT and K be any compact subset of QT .
Define the intrinsic parabolic distance from K to S by
n
o
1
1
2
2
d(K, S) = inf |x − y| + Φ(M̄ ) |t − s|
(3.61)
where the infimum is taken on the set {(x, t) ∈ K, (y, s) ∈ S}.
For convenience, we rephrase more precisely our main Theorem 3.1 as the following
theorem
Theorem 3.9 Let u be a bounded weak solution of (3.1). Then there exists constants
37
α0 > 0, C = C(M̄ , K, S, α) > 1 such that for any compact subset K ∈ ΩT and for any
two points (xi , ti ) ∈ K, i = 1, 2,
|u(x1 , t1 ) − u(x2 , t2 )| ≤ C
|x1 − x2 | + (Φ(M̄ )|t1 − t2 |)1/2
d(K, S)
α0
(3.62)
Proof: Fix (xi , ti ) ∈ K, i=1,2, such that t2 > t1 and construct the cylinder
Q = (x2 , t2 ) + Q(a0 R2 , R)
(3.63)
If we choose 2R = d(K, S) then this cylinder is contained in QT since for any (x1 , t1 ) ∈
Q, we have:
q
q
1/2
|x1 − x2 | + Φ(M̄ )|t1 − t2 | ≤ R + Φ(M̄ )a0 R = 2R = d(K, S)
(3.64)
To prove the Hölder continuity in the time t-variable, we can assume first that
t2 − t1 < a0 R2
Then, there exists ρ ∈ (0, R) such that t2 − t1 = a0 ρ2 , i.e,
ρ=
q
Φ(M̄ )|t2 − t1 |1/2
The oscillation decay estimate of Lemma 3.8 is applied in the cylinder
(x2 , t2 ) + Q(a0 ρ2 , ρ)
38
(3.65)
implies
|u(x2 , t2 ) − u(x2 , t1 )| ≤ γ
M̄ + d(K, S)2α0
d(K, S)α0
α /2
Φ(M̄ )|t2 − t1 | 0
(since ω ≤ M̄ )
(3.66)
If t2 − t1 ≥ a0 R2 then
|u(x2 , t2 ) − u(x1 , t1 )| ≤ 2M̄ ≤ 4M̄
(Φ(M̄ )|t2 − t1 |)1/2
d(K, S)
The Hölder continuity in space variables is proved analogously.
39
(3.67)
Chapter 4
The case of Neumann boundary data
Assume that ∂Ω is piecewise smooth. Consider the equation
∂u
− div(a(x, t, u, Du)) = b(x, t, u, Du)
∂t
(4.1)
in QT satisfies the structure (3.2) together with the boundary condition
∂a(x, t, u, Du)
+ g(x, t, u) = 0.
∂ν
(4.2)
On the boundary data g, we assume that the function g(., t, u(., t)), for a.e t ∈ (0, T ),
admits an extension into Ω which we denote with G(., t, u(., t)) such that there exists
a constant C > 0
(|G| + |Du G|)2 ≤ CΦ(u), |Dx G| ≤ C
(4.3)
Note that from now, we always assume that u is a bounded weak solution.
Taking into account the boundary condition, we have the integral form of u as the
following
1
2
Z
Ω
∂u
φ dx +
∂t
Z
Z
aDφ dx =
Ω
Z
bφ dx +
Ω
∂a
φ dσ.
∂Ω ∂ν
First, let us show that wk are still subsolutions of some parabolic equations. By
η
taking φ to be
in the above equation, we have
N (u)
1
2
Z
Ω
∂wk
η dx +
∂t
Z
Z
Ak Dη dx =
Ω
Ω
Z
kb
η
(
− Ak Dwk )η dx +
g
dσ.
Nk (u)
∂Ω Nk (u)
40
By (3.12),
1
2
Z
Ω
∂wk
η dx +
∂t
Z
Z
ā(x, t, Dwk )Dη dx ≤
Ω
Z
b̄(x, t, Dwk )η dx +
Ω
g
∂Ω
η
dσ
Nk (u)
(4.4)
where
ā = Ak
b̄ =
2ψ0
kb
+
Nk (u) Nk 2 (u)
(4.5)
We need a careful investigation the boundary integral term here. Let ζ ∈ C 2 (Ω, RN )
be any vector field such that ζ.n = 1 on ∂Ω. Then we write the boundary term as
the following way
Z
η
Gηζ
g
dσ =
div
dx
Nk (u)
∂Ω Nk (u)
Ω
Z
Z
Z
Z
Gζ
ζηDu
ζηDu
ζDx G + Gdiv(ζ)
η dx +
Dη dx +
Du G
dx +
2G 2
dx
=
Nk (u)
Nk (u)
Nk (u)
Ω Nk (u)
Ω
Ω
Ω
(4.6)
ζDx G + Gdiv(ζ)
We treat the last integrals as follow. In the first integral, the term
Nk (u)
Gζ
can be included in the definition of b. Similarly, the term
can go with a. Then
Nk (u)
using the condition (4.3)
Z
p
2aDu
Du
−
2C
Φ(u) 2
2
Nk (u)
Nk (u)
1
1
2ψ0
≥ Φ(u)ν0 |Dwk |2 − εΦ(u)|Dwk |2 − c(ε) 2
−
2
Nk (u) Nk 2 (u)
a.Dwk ≥
We can choose ε such that the new a, b satisfy the following structure:
41
(4.7)
|a| ≤ Φ(u)ν1 |Dwk | +
p
Φ(u)ψˆ1
1
a.Dwk ≥ Φ(u)ν0 |Dwk |2 − ψˆ0
4
(4.8)
|b| ≤ ν2 Φ(u)|Dwk | + ψˆ2
where
p
2
(|ζ|
+
|D
ζ|)(1
+
Φ(u))
ψ
+
C
sup
2ψ
C(ψ
+
1)
|ζ|
ψ
+
C
sup
2
0
0
1
Ω
Ω
, ψˆ2 =
.
ψˆ0 =
, ψˆ1 =
2
2 +
Nk
Nk
Nk
Nk
(4.9)
Then we only need to check the two last integrals in (4.6) because we will later replace
the function η by η(wk+ )λ in the proof of lemma (3.5). Again, a simple use of Young
inequality and conditions on G give us
ζ(wk+ )λ |Du|
ζ(wk+ )λ |Du|
1
+ λ−1
+ λ+1
2
|Du G|
+ |G|
≤ εΦ(u)(wk ) |Dwk | + C(ε)(wk )
1 + 2α
Nk (u)
R
Nk 2 (u)
(4.10)
which is also less or equal than
εΦ(u)(wk+ )λ−1 |Dwk |2 + C(ε)(wk+ )λ+1
1
R2
(4.11)
Hence, we still have (3.37) in this case. Then other details in this lemma would follow
similarly.
Second, we check the validity of estimates (3.45),(3.46),(3.47) in the proof of Lemma 3.6.
In fact, our function H in (3.46) will become:
Z
H(t) := C(
Ω
Z
ψ2
2ψ0 2
κgη 2
(Φ(u)|Dη| + Φ(u) +
+ 2 )η dx −
dσ)
Nκ
Nκ
∂Ω Nκ
2
(4.12)
Since ∂Ω is smooth, the (n − 1) dimensional surface measure of ∂Ω ∩ supp(η) is
κgη 2
can be estimated
comparable to Rn−1 . Thanks to this, the boundary integral of
Nκ
42
in this way:
κgη 2
dσ ≤ CdRn+1−α ≤ CRn
∂Ω Nκ
Z Z
I
(4.13)
Thus, the property (3.47) is still available.
Finally, we estimate the boundary integral that appears when we test the equation
with η 2 Wn in Lemma 3.4. It is only necessary to consider the case that the center
(x0 , t0 ) of the cylinder Q(dR2 , R) lies on the boundary ∂Ω. We choose a local system
of coordinates as a portion of the hyperplane xN = 0 and B2R ∩ Ω ⊂ {xN > 0}.
Set B̂2R = B2R ∩ ∂Ω, which is the (N − 1) dimensional cube, i.e B̂2R = {x̃ =
(x1 , . . . , xN −1 ) ∈ RN −1 / max1≤i≤(N −1) |xi | < 2R}.
Then
Z
2
Z
Z
2R
gη Wn dx̃ =
B̂2R
B̂2R
0
∂
Gη 2 Wn dxN dx̃
∂xN
(4.14)
By using (4.3) and Young inequality,
∂
∂xN
(Gη 2 Wn ) ≤ |GxN |Wn η 2 + |G||DWn |η 2 + |G|Wn η|Dη| + |Gu ||DWn |Wn η 2
p
p
p
2
2
2
≤ C Wn η + Φ(u)|DWn |η + Φ(u)Wn |Dη|η + Φ(u)|DWn |Wn η
≤ εΦ(u)η 2 |DWn |2 + C(ε) (Wn2 (η 2 + Φ(u)|Dη|2 ) + η 2 χAn )
2
W
n
2
≤ εΦ(u)η 2 |DWn |2 + C(ε)Φ(M )
+ η χAn
R2
(4.15)
where in the last estimate we have to use the property 1 ≤
Φ(M )
.
R2
Consequently, we obtain inequality (3.23) again.
By repeating the proof of the interior case, we also get Proposition 3.2 and furthermore, the constants in Proposition 3.2 are independent of u, R (if the supremum
norm kuk∞ is uniformly bounded). We have the uniform Hölder continuity of u. We
summarize the boundary regularity in this theorem.
Theorem 4.1 Assume the Neumann boundary condition (4.3). Let u be a bounded
43
weak solution of equation (3.1). Then u is globally Hölder continuous in the cylinders
Ω[T0 ,T1 ] = [T0 , T1 ] × Ω where T1 > T0 > 0. This means that there exists constants
C = C(kuk∞,Ω[T0 ,T1 ] ) and α = α(kuk∞,Ω[T0 ,T1 ] ) ∈ (0, 1) such that
|u(x1 , t1 ) − u(x2 , t2 )| ≤ C(|x1 − x2 |α + |t1 − t2 |α/2 ),
for every pairs of points (x1 , t1 ), (x2 , t2 ) ∈ Ω[T0 ,T1 ] .
44
(4.16)
Chapter 5
Applications to degenerate parabolic systems
In this chapter, we will introduce some simple models of degenerate parabolic systems.
Most of these results could be regarded as the consquences of the previous work on
one single degenerate parabolic equation.
5.1
Food Chain Models and Triangular Systems
A food chain is usually defined as a series of organisms showing feeding relationships.
The bottom of the chain is called producers which produce food and provide the
system with source energy, and each level of consumption is called a trophic level. A
typical example of food chain can be presented as: grass-grasshopper-mouse-snakehawk. In general, a food chain in the natural environment can be represented in an
order as
1. Autotrophs (Producers),
2. Herbivores (Primary Consumers),
3. Carnivores (Secondary Consumers),
In the above example of food chain, a species only feeds on the next lower one in the
chain. To model the food chain in mathematical language, we consider a system of
partial differential equations as follow
∂u
= div(A(u)Du) + f (u)
∂t
(5.1)
Here u is a vector-valued functions from a domain QT to Rm . Hence we can write
u = (u1 , u2 , . . . , um ). We use the function ui (x, t) to describe the density of population
of the ith species at a give position x and time t.
45
The source term f, in ecological context, can represent the birth-death process or the
competition within the species. For example, in a logistic model, f = u(a − bu),
where a is the intrinsic growth rate and b is the competition coefficient of the species.
The diffusion part div(A(u)Du) describes how species interacts with others. Note
that the structure of the diffusion matrix A(u) depends on the corresponding model.
To illustrate how the structure of the diffusion matrix is connected with the model,
we can take some simple examples. First, if we assume that each spieces does not
interact with others (no prey and predator concept in this model), then our matrix
A is a diagonal matrix since the change in population of any species only depends
on its self diffusion rate. Second, if we can suppose that any species moves toward
the region of high density concentration of its prey and does not pay attention to
its predator. Then the change in population of any species only depends on its self
diffusion rate and the cross diffusion rate of i.e its prey, not on its predator. Thus
for 1 ≤ i ≤ m − 1, the diffusion part of the ith species only has two terms aii (u),
ai(i+1) (u) whereas the diffusion of the mth species only has the term amm (u). In short,
the matrix A will be of the 2-diagonal form






A=






a11 (u) a12 (u)


...
...



... ...




..
. am−1,m (u)

amm (u)
In general, we can assume that all the lower ones of a given species are its preys. Then
the resulting model is the so-called pyramid model and the corresponding matrix A
46
will be upper triangular

a11 (u) . . . . . .

...


A=
...




a1m (u) 
.. 
. 

.. 
. 


amm (u)
In this system, the higher order term div(aii (u)Du) represents random diffusive flux
of the ith species and the cross diffusion terms div(aij (u)Du); j > i describe the
dependence of the ith species on its prey. The terms fi (u) represent reactions including
reproduction or death rates and so on.
Many results of Hölder continuity of triangular parabolic systems were established
when the diffusion matrix A(u) = (aij (u)) (λ, Λ > 0) satisfies the following ellipticity
condition
λ|ξ|2 ≤ aij (u)ξi ξj ≤ Λ|ξ|2
∀u ∈ IRm .
(5.2)
or all self-diffusion rates and cross diffusion terms aij (u) are functions that are bounded
below and above by two positive numbers. In these models, the systems are nondegenerate (or regular). To see these results, the reader can view in [4] (or in [9] for
even near triangular systems).
A natural question arises about the Hölder continuity problem of the reaction diffusion systems of triangular types when the diffusion matrix A(u) does not have the
property (5.2). In fact, each species can behave in many very different manners. The
self-diffusion rates and cross diffusion terms could be of porous medium types or pLaplacian types, not always just regular types. This is also the main motivation of
our work in the next section.
47
In details, we consider the following models
Model I: p-Laplacian and porous media types

a11 (u) . . . . . .

...


A=
...




a1m (u) 
.. 
. 

.. 
. 


amm (u)
Here we consider the case that only the 1st species u1 behaves like porous medium
type while other species u2 , . . . , um behave like p-Laplacian types (p ≥ 2). For
simplicity, we can understand that a11 (u) ∼ Φ(u1 ) ∼ |u1 |α , α > 0 and aii (u) ∼
|(Du2 , . . . , Dum )|p−2 , where i > 1.
Model II: porous media and regular types

a11 (u) . . . . . .

..

.

A=
...




a1m (u) 
.. 
. 

.. 
. 


amm (u)
In this model, we consider the case that only the mth species um behaves like porous
medium type while other species u1 , . . . , um−1 behave like regular types (Laplacian).
For simplicity, we can understand that amm (u) ∼ Φ(um ) ∼ |um |β , β > 0 and aii (u) ∼
1, where i < m.
For simplifying notation, we just consider the case of 2 spieces in proving the below
results, but still keep the spirit of the method when using for the general case with
m spieces. We sketch the idea quickly here. To deal with triangular systems, we use
upward method. This means that we will start with the equation at the bottom of the
matrix, i.e the equation of the mth species. By using the knowledge on one equation,
48
we can know good information about um without difficulties. Then we will go up to
the next equation of the (m − 1)th spieces. The information on um will support more
properties for the equation of um−1 , and hence we still get the Hölder continuity of
um−1 . And we repeat this process until we reach the first equation.
5.2
Hölder continuity for systems of model I
Given a positive number p ≥ 2.
We consider a triangular system which has the following form:
∂u1
= div(a(x, t, u1 , u2 , Du1 )) + div(b(x, t, u1 , u2 , Du2 )) + f1 (x, t, u1 , u2 , Du1 , Du2 )
∂t
(5.3)
∂u2
= div(|Du2 |p−2 Du2 ) + f2 (x, t, u1 , u2 )
(5.4)
∂t
with the following structure condition is imposed:
a(x, t, u1 , u2 , Du1 )Du1
≥ ν0 Φ(u1 )|Du1 |2 − ψ0 (x, t),
|a(x, t, u1 , u2 , Du1 )|
≤ ν1 Φ(u1 )|Du1 | +
p
Φ(u1 )ψ1 (x, t),
(5.5)
|b(x, t, u1 , u2 , Du2 )|
≤ ν2
p
α
Φ(u1 )|Du2 | , α ≥ 0
|f1 (x, t, u1 , u2 , Du1 , Du2 )| ≤ ν3 Φ(u1 )|Du1 | + ψ2 (x, t).
Here u2 is a vector-valued function and f2 is bounded on any compact set of their
arguments (x, t, u1 , u2 ). If f2 does not depend on u1 , the boundedness assumption is
replaced by the condition
|f2 (x, t, u2 , Du2 )| ≤ ν4 |Du2 |p−1 + ψ3 (x, t)
where ψ3 ∈ Lq with q >
N +2
.
2
49
(5.6)
Theorem 5.1 Assume that the flux vectors a, b and f1 satisfy the structure condition
(5.5) .If u1 , u2 are bounded weak solutions to (5.3),(5.4) then u1 , u2 are interior Hölder
continuous.
Before giving a proof, we need the following result from Lemma 4.2, Chapter VIII in
[3]
Lemma 5.2 Let u be a weak solution to the system
∂u
= div(|Du|p−2 Du) + f (u)
∂t
, where f (u) satisifes the condition (5.6). Then |Du| ∈ L∞
loc (QT ) and thus, u is locally
Hölder continuous.
Proof: We begin with the equation for u2 . Since ui are bounded, f2 (x, t, u1 , u2 )
α
is bounded so that the above Lemma 5.2 implies u2 ∈ Cloc
for some positive α and
moreover, Du2 is locally bounded. Next, we rewrite the equation for u1 in the form
∂u1
= div(A(x, t, u1 , Du1 )) + B(x, t, u1 , Du1 )
∂t
(5.7)
A(x, t, u1 , Du1 ) = a1 (x, t, u1 , u2 , Du1 ) + b(x, t, u1 , u2 , Du2 )
(5.8)
B(x, t, u1 , Du1 ) = f1 (x, t, u1 , Du1 , u2 , Du2 )
(5.9)
with
. By using structure condition (5.5) and Young inequality, we have:
A(x, t, u1 , Du1 )Du1 ≥
|A(x, t, u1 , Du1 )|
where γ0 (x, t) = ψ0 (x, t) +
1
ν Φ(u1 )|Du1 |2
2 0
≤ ν1 Φ(u1 )|Du1 | +
2ν2
|Du2 |2α
ν0
− γ0 (x, t),
(5.10)
p
Φ(u1 )γ1 (x, t),
and γ1 (x, t) = ψ1 (x, t) + ν2 |Du2 |α
50
The structure conditions (3.2) are satisfies for A and B since |Du2 | is a bounded
function. Thus, Theorem 3.1 implies the interior Hölder continuity of u1 .
5.3
Hölder continuity for systems of model II
Consider the triangular system of two equations
∂u
= div(A(u, v)) + div(B(u, v)))
∂t
(5.11)
∂v
= div(a(x, t, v, Dv)) + b(x, t, u, v, Dv)
∂t
(5.12)
Here u and v are scalar functions. We impose the structure condition (3.2) on the
coefficients a, b of the equation (5.12). With the equation (5.11) of u, we assume that
A(u, v)Du ≥ λ1 |Du|2 ,
|A(u, v)| ≤ λ2 |Du|,
|B(u, v)| ≤ λ3
(5.13)
p
Φ(v)|Dv|.
where λi , i = 1, 2, 3 are positive constants.
We still have a similar conclusion for this model
Theorem 5.3 Assume that u, v are bounded weak solutions to (5.11),(5.12) then u, v
are locally Hölder continuous.
By our theorem (3.1), we just only need to prove the Hölder continuity of u. The main
key of this work is to apply the decay estimates of nondegenerate parabolic equations.
We state here the result about the decay estimate which was summarized in Section
2.3.2 of [9]. Note that in [9], the result was proved even for Dirichlet problem, but
we just need the interior case here.
51
Lemma 5.4 Let u be a weak solution to the equation on a domain QT = Ω × [T0 , T1 ]
∂u
= div(A(x, t)Du)) + div(F )
∂t
(5.14)
where A satisfies the ellipticity condition (5.13).
Then there exists positive constants α, β, C such that the following decay estimate is
hold for any R > ρ > 0 and QR ⊂ QT :
ZZ
ρ
|Du| dxdt ≤ C( )n+α
R
2
Qρ
ZZ
ZZ
2
|F |2 dxdt + CRn+β (5.15)
|Du| dxdt + C
QR
QR
ZZ
|F |2 dxdt ≤
Moreover, the solution u is locally Hölder continuous on QT if
QR
CR
n+µ
for some µ > 0 and R > 0 such that Q2R ⊂ QT .
From the equation (5.11), we see that our F is B(u, v), and also note that |B(u, v)|2
is bounded by Φ(v)|Dv|2 . We recall the Theorem 3.1 to derive a local L2 norm of
Φ(v)|Dv|2
Lemma 5.5 There exist positive constants µ = µ(kvk∞ ) and C = C(kvk∞ ) such
that
ZZ
Φ(v)|Dv|2 dxdt ≤ CRn+µ
(5.16)
Q(R,R2 )
, if Q2R ⊂ QT .
Proof: Let R > 0 and η be a cutoff function for QR := Q(R, R2 ), that is η = 1 in
QR and η vanishes outside Q2R and |Dη| ≤
1
R
and 0 ≤ ηt ≤
1
.
R2
Testing the equation
(1.1) with (v − vR )η 2 and use structure condition and Young inequality to get
52
Z
ZZ
2 2
ν0 Φ(v)|Dv|2 η 2 dxdt
(v − vR ) η dx +
supt
B2R
Q2R
ZZ
ZZ
≤
ν2 Φ(v)|Dv||v − vR |η 2 dxdt
2ν1 Φ(v)|Dv||v − vR |ηDη dxdt +
Q2R
Q2R
ZZ
+
(ψ0 + ψ12 + ψ22 )η 2 dxdt + C
ZZ
Q2R
|v − vR |2 (|Dη|2 + ηηt ) dxdt
Q2R
(5.17)
.
Using the boundedness of cutoff η and the fact that v is Holder continuous, there
are positive constants C, α such that |v(x, t) − vR | ≤ CRα for (x, t) ∈ Q2R . We have
then:
ZZ
2
2
ZZ
|v − vR |2 ηηt dxdt ≤ CR2α
|v − vR | |Dη| dxdt +
Q2R
Q2R
1
|Q2R | ≤ CRn+2α
2
R
(5.18)
By the boundedness of ψi , the integral of (ψ0 + ψ12 + ψ22 )η 2 can be majorized by
CRn+2 . Moreover, using Young inequality and the above estimate we get
ZZ
Φ(v)|Dv||v − vR |(η|Dη| + η 2 ) dxdt
Q2R
ZZ
(5.19)
ZZ
Φ(v)|Dv| η dxdt + C(ε)
2 2
≤ε
Q2R
2
2
2
Φ(v)|v − vR | (|Dη| + η ) dxdt
Q2R
The last term is bounded by CRn+2α . By choosing ε small enough and combining
these estimates we obtain the desired estimate with µ = 2α and complete our proof.
Proof of Theorem 5.3: By combining the lemma (5.4) and the lemma (5.5), theorem (5.3) is verified.
Remark: By the same way, our result is still hold for the triangular system which u
53
is a vector-valued function and v is a scalar function.
54
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56
VITA
Minh Kha was born in Ho Chi Minh City, Viet Nam on Sep 23, 1987. He studied in Department of Mathematics in University of Sciences, Ho Chi Minh city, Viet
Nam and got a Bachelor degree over there. In his senior year there, he learnt some
fundamental stuff of the regularity theory in PDEs, which plays an important role in
classical and contemporary analysis.
In Fall 2009 he came to the U.S. to pursue his Master in Mathematics at the University of Texas at San Antonio. From 2009-2011, he has worked on some problems
relating to reaction diffusion systems including nondegenerate and degenerate equations.
He is now single and going to study his Ph.D. in Mathematics at Texas A and M
University.