On the $\omega$-limit set of a nonlocal differential equation: application of rearrangement theory

0.16 MB·

by  Thanh Nam Nguyen

On the ω-limit set of a nonlocal differential
equation: application of rearrangement theory
Thanh Nam Nguyen∗
January 26, 2016
6
1
0
2

n Abstract. We study the ω-limit set of solutions of a nonlocal ordinary
a
J differential equation, where the nonlocal term is such that the space integral

5 of the solution is conserved in time. Using the monotone rearrangement
2
theory, we show that the rearranged equation in one space dimension is the

] same as the original equation in higher space dimensions. In many cases, this
P
property allows us to characterize the ω-limit set for the nonlocal differential
A
equation. More precisely, we prove that the ω-limit set only contains one
.
h
element.
t
a
m
[

1 1 Introduction
v
1
9 The aim of the present paper is to study the ω-limit set of solutions of the
4
6 initial value problem
0
.
1
g(u)p(u)
0
6 ut = g(u)p(u)−g(u) ZΩ x ∈ Ω, t ≥ 0,
1 (P)
:  g(u)
v  ZΩ
i
X u(x,0) = u (x) x ∈ Ω.
0

r 
a Here Ω ⊂ IRN(N ≥ 1) is an open bounded set, g,p : IR → IR are continuously
differentiable and u is a bounded function. More precise conditions on g,p
0
and u will be given later. A typical example is given by the functions g(u) =
0
u(1−u) and p(u) = u. In this case, the equation becomes
u2(1−u)
u = u2(1−u)−u(1−u) ZΩ .
t
u(1−u)
ZΩ
∗Laboratoire de Mathe´matique, Analyse Nume´rique et EDP, Universite´ de
Paris-Sud, F-91405 Orsay Cedex, France
1
The corresponding parabolic equation
u2(1−u)
1
u = ∆u+ u2(1−u)−u(1−u) ZΩ .
t
ε2
u(1−u)
 
 ZΩ 
 
has been used by Brassel and Bretin[1, Formula (9)] to approximate mean
curvature flow with volume conservation. It has been also proposed by Na-
gayama [6]todescribe abubblemotionwithachemical reaction. Hesupposes
furthermore that the volume of the bubble is preserved in time. Mathemati-
cally, it is expressed in the form of the mass conservation property
u(x,t)dx = u (x)dx for all t ≥ 0. (1)
0
ZΩ ZΩ
We refer to Proposition 2.2 for a rigorous proof of this equality.
We will consider Problem (P) under some different hypotheses on the
initial function u . Problem (P) possesses a Lyapunov functional whose form
0
depends on the hypothesis satisfied by u (see section 4 for more details).
0
Inthispaper, wealwaysconsider thefollowinghypotheses onthefunctions
g and p:
p ∈ C1(IR) is strictly increasing on IR,
g ∈ C1(IR),g(0) = g(1) = 0,g > 0 on (0,1) and g < 0 on (−∞,0)∪(1,∞).
(
We suppose that the initial function satisfies one of the following hypotheses:
(H1) u0 ∈ L∞(Ω), u0(x) ≥ 1 for a.e. x ∈ Ω, and u0 6≡ 1.
(H2) u0 ∈ L∞(Ω), 0 ≤ u0(x) ≤ 1 for a.e. x ∈ Ω, and Ωg(u0(x))dx 6= 0.
(H3) u0 ∈ L∞(Ω), u0(x) ≤ 0 for a.e. x ∈ Ω, and u0 6≡R0.
Note that Hypothesis (H1) (and also (H3)) implies that Ωg(u0) 6= 0.
Before defining a solution of Problem (P), we introduRce the notation
g(u)p(u)
F(u) := g(u)p(u)−g(u)ZΩ . (2)
g(u)
ZΩ
Definition 1.1. Let 0 < T ≤ ∞. The function u ∈ C1([0,T);L∞(Ω)) is
called a solution of Problem (P) on [0,T) if the three following properties
hold
(i) u(0) = u ,
0
(ii) g(u(t)) 6= 0 for all t ∈ [0,T),
ZΩ
du
(iii) = F(u) in the whole interval [0,T).
dt
2
The ω-limit sets are important and interesting objects in the theory of
dynamical systems. Understanding their structure allows us to apprehend
the long time behavior of solutions of dynamical systems. In this paper, we
characterize the ω-limit set of solutions of Problem (P), which is defined as
follows:
Definition 1.2. We define the ω-limit set of u by
0
ω(u ) := {ϕ ∈ L1(Ω) : ∃t → ∞,u(t ) → ϕ in L1(Ω) as n → ∞}.
0 n n
In the above definition, we do not use the L∞-topology to define ω(u )
0
because the solution often develops sharp transition layers which cannot be
captured by the L∞-topology. Note also that as we will see in Theorem 2.5,
solutionsof(P)areuniformlyboundedsothatthetopologyofL1 isequivalent
to that of Lp with p ∈ [1,∞). For convenience, we refer to the books [7, 8]
for studies about dynamical systems as well as the structure of ω-limit sets.
An essential step to study ω(u ) is to show the relative compactness of the
0
solutionorbitsinL1(Ω). Inlocalproblems, thestandardcomparisonprinciple
can be applied to obtain the uniform boundedness of solutions. Furthermore,
in local problems with a diffusion term, such as local parabolic problems,
the uniform boundedness of solutions implies the relative compactness of
solution orbits in some suitable spaces by using Sobolev imbedding theorems.
However, the above scheme cannot be applied to Problem (P), due to the
presence of the nonlocal term as well as to the lack of a diffusion term.
By careful observation of the dynamics of pathwise trajectories (i.e. the
sets {u(x,t) : t ≥ 0} for x ∈ Ω), we show the existence of invariant sets
and hence the uniform boundedness of solutions. The difficulties connected
with the lack of diffusion term will be overcome by using ideas presented
in [3]. More precisely, applying the rearrangement theory, we introduce the
equi-measurable rearrangement u♯ and show that it is the solution of a one-
dimensional problem (P♯) (see section 3). Since the orbit {u♯(t) : t ≥ 0}
is bounded in BV(Ω♯), where Ω♯ := (0,|Ω|) ⊂ IR, it is relatively compact
in L1(Ω♯). We then deduce the relative compactness of solution orbits of
Problem (P), by using the fact that
ku(t)−u(τ)kL1(Ω) = ku♯(t)−u♯(τ)kL1(Ω♯). (3)
Note that the inequality ku(t) − u(τ)kL1(Ω) ≥ ku♯(t) − u♯(τ)kL1(Ω♯) follows
from a general property of the rearrangement theory. The important point
is that (3) involves an equality.
An other advantage of considering Problem (P♯) is that the differential
equations in (P♯) and (P) have the same form. Therefore we will study
the ω-limit set for Problem (P♯) rather than for Problem (P). Although
(P♯) possesses many stationary solutions, the one-dimensional structure of
Problem (P♯) allows us to characterize its ω-limit set, andthen deduce results
for that of (P).
3
The organization of this article is as follows: In section 2, we prove the
global existence and uniqueness of the solution as well as its uniform bound-
edness. Next in section 3, we recall and apply results from the arrangement
theory presented in [3] to obtain the relative compactness of the solution in
L1(Ω). In section 4, we prove that Problem (P) possesses Lyapunov function-
als and use them together with the relative compactness of the solution to
show that ω(u ) is nonempty and consists of stationary solutions. Moreover,
0
these stationary solutions are step functions. More precise properties of these
functions are given in Theorems 4.4 and 4.5. In section 5, we suppose that
one of the hypotheses (H1) or (H3) holds and prove that ω(u0) only contains
one element.
In the case that Hypothesis (H2) is satisfied, the structure of the ω-limit
set becomes more complicated than in the other cases since the solution can
develop many transition layers. More precisely, as we will see in Theorem 4.4,
elements in the ω-limit set may contain step functions taking three values
{0,1,ν} instead of the two values {1,µ} in the case (H1) and {0,ξ} in the
case (H3). As a consequence, it is more difficult to prove that the ω-limit set
contains a single element. We refer to our forthcoming paper [2] for a study
in more details of the case (H2) .
2 Existence and uniqueness of solutions of (P)
2.1 Local existence
First we prove the local Lipschitz property of the nonlocal nonlinear term F,
given by (2), in the space L∞(Ω).
Lemma 2.1 (Local Lipschitz continuity of F). Let v ∈ L∞(Ω) be such that
g(v(x))dx 6= 0. Then there exist a L∞(Ω)-neigbourhood V of v and a
Ω
constant L > 0 such that F(v) is well-defined for all v ∈ V and that
R
kF(v1)−Fe(v2)kL∞(Ω) ≤ Lkv1 −v2keL∞(Ω),
for all v ,v ∈ V.
1 2
Proof. Since g is continuous, the map v 7→ g(v) is continuous from L∞(Ω)
Ω
to L∞(Ω). It follows that there exist a constant α > 0 and a neighbourhood
R
V of v such that
g(v) ≥ α for all v ∈ V. (4)
(cid:12)ZΩ (cid:12)
(cid:12) (cid:12)
Without loss of generalit(cid:12)y, weem(cid:12)ay choose e
(cid:12) (cid:12)
V := {v ∈ L∞(Ω) : kv−vkL∞(Ω) ≤ ε},
for a constant ε > 0 small enough. We set
e e
c¯:= kvkL∞(Ω) +ε, f(s) := g(s)p(s),
4
and
K := max sup |f(s)|, sup |g(s)|, sup |f′(s)|, sup |g′(s)| .
[−c¯,c¯] [−c¯,c¯] [−c¯,c¯] [−c¯,c¯]
(cid:8) (cid:9)
Then the following properties hold and will be used later: for all v ,v ∈ V,
1 2
kf(v1)−f(v2)kL∞(Ω) ≤ Kkv1 −v2kL∞(Ω), (5)
and
kg(v1)−g(v2)kL∞(Ω) ≤ Kkv1 −v2kL∞(Ω).
We have
f(v ) f(v )
1 2
F(v )−F(v ) = [f(v )−f(v )]−g(v )ZΩ −g(v )ZΩ 
1 2 1 2 1 2
g(v ) g(v )
 1 2 
 ZΩ ZΩ 
 
g(v ) f(v ) g(v )−g(v ) f(v ) g(v )
1 1 2 2 2 1
= [f(v )−f(v )]− ZΩ ZΩ ZΩ ZΩ
1 2
g(v ) g(v )
1 2
ZΩ ZΩ
A
2
=: A − ,
1
A
3
where
A := f(v )−f(v ),
1 1 2
A := g(v ) f(v ) g(v )−g(v ) f(v ) g(v ),
2 1 1 2 2 2 1
ZΩ ZΩ ZΩ ZΩ
and
A := g(v ) g(v ).
3 1 2
ZΩ ZΩ
In the sequel, we estimate A , A and A . First the inequality (5) yields
1 2 3
kA1kL∞(Ω) ≤ Kkv1 −v2kL∞(Ω). (6)
Next we write A as
2
A = g(v ) f(v ) g(v )−g(v ) f(v ) g(v )
2 1 1 2 2 1 2
ZΩ ZΩ ZΩ ZΩ
+g(v ) f(v ) g(v )−g(v ) f(v ) g(v )
2 1 2 2 2 2
ZΩ ZΩ ZΩ ZΩ
+g(v ) f(v ) g(v )−g(v ) f(v ) g(v ),
2 2 2 2 2 1
ZΩ ZΩ ZΩ ZΩ
or equivalently,
A = [g(v )−g(v )] f(v ) g(v )
2 1 2 1 2
ZΩ ZΩ
+g(v ) [f(v )−f(v )] g(v )
2 1 2 2
ZΩ ZΩ
+g(v ) f(v ) [g(v )−g(v )],
2 2 2 1
ZΩ ZΩ
5
which in turn implies that
kA2kL∞(Ω) ≤ 3K3|Ω|2kv1 −v2kL∞(Ω). (7)
As for the term A , we apply (4) to obtain
3
|A | ≥ α2 > 0. (8)
3
Combining (6), (7) and (8), we deduce that
3K3|Ω|2
kF(v1)−F(v2)kL∞(Ω) ≤ K + α2 kv1 −v2kL∞(Ω).
(cid:16) (cid:17)
This completes the proof of Lemma 2.1.
Proposition 2.2. Let u ∈ L∞(Ω) satisfy g(u ) 6= 0. Then Problem (P)
0 Ω 0
has a unique local-in-time solution. Moreover, we have
R
u(x,t)dx = u (x)dx for all t ∈ [0,T (u )), (9)
0 max 0
ZΩ ZΩ
where T (u ) denotes the maximal time interval of the existence of solution.
max 0
Proof. Since F is locally Lipschitz continuous in L∞(Ω), the local existence
follows from the standard theory of ordinary differential equations. We now
prove (9). Integrating the differential equation in Problem (P) from 0 to t,
we obtain
t t
u(t)−u = u (s)ds = F(u(s))ds.
0 t
Z0 Z0
It follows that
t
u(x,t)dx− u (x)dx = F(u)dxds = 0,
0
ZΩ ZΩ Z0 ZΩ
where the last identity holds since
F(u)dx = 0.
ZΩ
This completes the proof of the proposition.
Lemma 2.3. If Tmax(u0) < ∞ and limsupt↑Tmax(u0)kF(u(t))kL∞(Ω) < ∞,
then u(T (u )−) := lim u(t) exists in L∞(Ω) and
max 0 t↑Tmax(u0)
g(u(T (u )−)) = 0.
max 0
ZΩ
Proof. For simplicity we write T instead of T (u ). Set
max max 0
M := limsupkF(u(t))kL∞(Ω) < ∞.
t↑Tmax
6
Then there exists 0 < T < T such that
max
kF(u(t))kL∞(Ω) ≤ 2M for all t ∈ [T,Tmax).
Consequently, for any t,t′ ∈ [T,T ), with t < t′, we have
max
t′
ku(t)−u(t′)kL∞(Ω) ≤ kF(u(s))kL∞(Ω)ds ≤ 2M|t−t′|.
Zt
Thus {u(t)} is a Cauchy sequence so that thelimit u(T −) := lim u(t)
max t↑Tmax
exists in L∞(Ω). If g(u(T −)) 6= 0, then, by Lemma 2.2, we can extend
Ω max
the solution on [T ,T + δ), with some δ > 0, which contradicts the
max max
R
definition to T . This completes the proof of the lemma.
max
2.2 Global solution
In this subsection, we fix u ∈ L∞(Ω) satisfying g(u ) 6= 0 and denote by
0 Ω 0
[0,T ) the maximal time interval of the existence of solution. Set
max
R
g(u)p(u)
λ(t) = ZΩ for all t ∈ [0,T ), (10)
max
g(u)
ZΩ
and study solutions Y(t;s) of the following auxiliary problem:
Y˙ = g(Y)p(Y)−g(Y)λ(t), t > 0,
(ODE) (11)

Y(0) = s,

where Y˙ := dY/dt. Weremark that the function u satisfies
u(x,t) = Y(t;u (x)) for a.e. x ∈ Ω and all t ∈ [0,T ). (12)
0 max
Lemma 2.4. Let s < s and let 0 < T < T . Assume that Problem (ODE)
max
possesses the solutions Y(t;s),Y(t;s) ∈ C1([0,T]), respectively. Then
e
Y(t;s) < Y(t;s) for all t ∈ [0,T]. (13)
e
Proof. Since Y(0;s) = s < s = Y(0;s), the assertion follows immediately
e
from the backward uniqueness of solution of (ODE).
e e
Theorem 2.5. Assume that one of the hypotheses (H1),(H2),(H3) holds.
Then Problem (P) possesses a global solution u ∈ C1([0,∞);L∞(Ω)). More-
over:
(i) If (H1) holds, then for all t ≥ 0,
1 ≤ u(x,t) ≤ ess sup u for a.e. x ∈ Ω. (14)
Ω 0
7
(ii) If (H2) holds, then for all t ≥ 0,
0 ≤ u(x,t) ≤ 1 for a.e. x ∈ Ω. (15)
(iii) If (H3) holds, then for all t ≥ 0,
ess inf u ≤ u(x,t) ≤ 0 for a.e. x ∈ Ω.
Ω 0
Proof. For simplicity, we set
a := ess inf u , b := ess sup u .
Ω 0 Ω 0
We only prove (i) and (ii). The proof of (iii) is similar to that of (i).
(i) First, we show that (14) holds as long as the solution u exists and then
deduce the global existence from Lemma 2.3. Let Y(t;s) be the solution of
(ODE). We remark that b ≥ 1 and that Y(t,1) ≡ 1 for all t ∈ [0,T ). The
max
monotonicity of Y(t;s) in s implies that as long as u,Y(t;b) both exist
1 ≡ Y(t,1) ≤ Y(t;u (x)) = u(x,t) ≤ Y(t;b) a.e. x ∈ Ω. (16)
0
The first inequality above implies the first inequality of (14) as long as the
solution u exists. It remains to prove the second inequality of (14). To that
purpose, it suffices to show that
Y(t;b) ≤ b (17)
as long as the solution Y(t;b) exists. In view of (16), we have 1 ≤ u(x,t) ≤
Y(t;b). Then the definition of g and the monotonicity of p imply that
g(Y(t;b)) ≤ 0, g(u(x,t)) ≤ 0, p(Y(t,b)) ≥ p(u(x,t)),
for a.e. x ∈ Ω. These properties, together with the definition of λ(t) in (10),
imply that
Y˙ (t;b) = g(Y(t,b))(p(Y(t;b))−λ(t))
g(u(x,t))p(u(x,t))dx
= g(Y(t,b))p(Y(t;b))− ZΩ 
g(u(x,t))dx
 
 ZΩ 
 
g(u(x,t))[p(Y(t;b)−p(u(x,t))]dx
= g(Y(t,b))ZΩ ≤ 0.
g(u(x,t))dx
ZΩ
Hence
Y(t,b) ≤ Y(0;b) = b,
whichcompletestheproofof (17). Thus(14)issatisfiedaslongasthesolution
u exists.
8
Next we show that the solution u exists globally. Suppose, by contradic-
tion, that T < ∞. We have for all t ∈ [0,T ),
max max
|g(u)p(u)| |g(u)||p(u)|
|λ(t)| ≤ Ω = Ω ≤ max{|p(1)|,|p(b)|}.
g(u) |g(u)|
R Ω R Ω
(cid:12)R (cid:12) R
It follows that the(cid:12)re exist(cid:12)s C > 0 such that kF(u(t))kL∞(Ω) ≤ C for all
t ∈ [0,T ). By Lemma 2.3, u(T −) := lim u(t) exists in L∞(Ω) and
max max t↑Tmax
g(u(T −)) = 0.
max
ZΩ
Since u(x,t) ≥ 1 for a.e x ∈ Ω,t ∈ [0,T ), u(x,T −) ≥ 1 for a.e.
max max
x ∈ Ω. Hence g(u(T −)) = 0 if and only if u(x,T −) ≡ 1. The
Ω max max
mass conservation property (cf. (9)) yields u = |Ω|. Hence u (x) = 1 for
R Ω 0 0
a.e x ∈ Ω. This contradicts Hypothesis (H1) so that Tmax = ∞.
R
(ii) Since Y(t,1) ≡ 1, Y(t,0) ≡ 0, we deduce that
0 ≡ Y(t,0) ≤ Y(t;u (x)) = u(x,t) ≤ Y(t,1) ≡ 1 a.e. x ∈ Ω.
0
This implies (15) as long as the solution u exists. We now prove that T =
max
∞. Indeed, suppose, by contradiction, that T < ∞. Since 0 ≤ u(x,t) ≤ 1
max
for a.e. x ∈ Ω, and all t ∈ [0,T ), g(u(x,t)) ≥ 0 for a.e. x ∈ Ω, and all
max
t ∈ [0,T ). Therefore
max
|g(u)p(u)| g(u)|p(u)|
|λ(t)| ≤ Ω = Ω ≤ max{|p(0)|,|p(1)|},
g(u) g(u)
R Ω R Ω
(cid:12)R (cid:12) R
forallt ∈ [0,Tmax).(cid:12)Itfollow(cid:12)sthatthereexistsC > 0suchthatkF(u(t))kL∞(Ω) ≤
C for all t ∈ [0,T ). By Lemma 2.3, u(T −) := lim u(t) exists in
max max t↑Tmax
L∞(Ω) and
g(u(T −)) = 0.
max
ZΩ
This implies that u(T −) only takes two values 0 and 1. Or equivalently,
max
Y(T −;u (x)) only takes two values 0 and 1. Thus the backward unique-
max 0
nessofthesolutionoftheinitialvalueproblem(ODE)impliesthatu (x)only
0
takes two values 0 and 1; hence g(u ) = 0. This contradicts Hypothesis
Ω 0
(H2) so that Tmax = ∞.
R
The result below follows from the proof of Theorem 2.5.
Corollary 2.6. Assume that one of the hypotheses (H1),(H2),(H3) holds
and let λ(t) be defined by (10). Then there exists C > 0 such that
|λ(t)| ≤ C
for all t ∈ [0,∞).
9
3 Boundedness of the solution and one-dimensional
associated problem (P♯)
All the results in this section are similar to those of [3, Section 3]. We recall
and state some important results. Let w be a function from Ω to IR and let
Ω♯ := (0,|Ω|) ⊂ IR. The distribution function of w is given by
µ (s) := |{x ∈ Ω : w(x) > s}|.
w
Definition 3.1. The (one-dimensional) decreasing rearrangement of w, de-
♯
noted by w♯, is defined on Ω = [0,|Ω|] by
w♯(0) := ess sup(w)
(18)
( w♯(y) = inf{s : µw(s) < y}, y > 0.
Remark 3.2. The function w♯ is nonincreasing on Ω♯ and we have µ (s) =
w
µ (s) for all s ∈ IR. Moreover, if a ≤ w(x) ≤ b a.e. x ∈ Ω, then
w♯
a ≤ w♯(y) ≤ b for all y ∈ Ω♯.
Theorem 3.3. Let one of the hypotheses (H1),(H2),(H3) hold. We define
u♯(y,t) := (u(t))♯(y) on Ω♯ ×[0,+∞). (19)
Then u♯ is the unique solution in C1([0,∞);L∞(Ω♯)) of Problem (P♯)
g(v)p(v)
dv
 = g(v)p(v)−g(v)ZΩ t > 0,
(P♯)  dt g(v)


 ZΩ
v(0) = u♯.
 0


Moreover, for allt ≥ 0,

u♯(y,t) = Y(t;u♯(y)) for a.e. y ∈ Ω♯, (20)
0
and the assertions (i), (ii), (iii) of Theorem 2.5 hold for the function u♯.
Lemma 3.4 ([3, Lemma 3.7]). Let u be the solution of (P) with u ∈ L∞(Ω)
0
and let u♯ be as in (19). Then
ku♯(t)−u♯(τ)kL1(Ω♯) = ku(t)−u(τ)kL1(Ω), (21)
for any t,τ ∈ [0,∞).
Corollary 3.5 ([3, Corollary3.9]). Let {t } be a sequence of positive numbers
n
such that t → ∞ as n → ∞. Then the following statements are equivalent
n
(a) u♯(t ) → ψ in L1(Ω♯) as n → ∞ for some ψ ∈ L1(Ω♯);
n
(b) u(t ) → ϕ in L1(Ω) as n → ∞ for some ϕ ∈ L1(Ω) with ϕ♯ = ψ.
n
The following proposition follows from similar results in [3, Lemma 3.5
and Proposition 3.10].
Proposition 3.6. Let one of the hypotheses (H1),(H2),(H3) hold. Then
{u(t) : t ≥ 0} is relatively compact in L1(Ω) and the set {u♯(t) : t ≥ 0} is
relatively compact in L1(Ω♯).
10