The Arabian Journal for Science and Engineering, Volume 35, Number 1D
May 2010, Pages 1–12
APPLICATION OF THE Ly -ENERGY METHOD TO THE NON LOCAL
THERMISTOR PROBLEM
MOULAY RCHID SIDI AMMI
Abstract. In this paper, we show the importance of the Ly -energy method in proving existence,
regularity, and uniqueness of solution to the well known non local thermistor problem and its
variants.
1. I n t r o d u c t i o n
The choice of the function spaces is essential in the study of the nonlinear partial di¤erential equations
(PDEs). For the energy method applied to nonlinear parabolic equations, even the Ly -space is not
frequently used because of its non-reflexivity and non-separability; the Ly -energy method could be a
suitable and powerful tool to prove existence, regularity, and uniqueness of solutions and could serve to
derive Cy for a class of non-linear PDEs, including some quasilinear or strongly nonlinear parabolic
equations. Moreover, it enables us to deal with the asymptotic behavior of solutions of PDEs using the
dynamical systems approach and proving the existence of absorbing sets which paves the way for the existence of the global attractor [10]. In this paper, we give the basic idea of the Ly -energy method by using
the so-called nonlocal thermistor problem.
Thermistor is a generic name for a device made from materials whose electric conductivity is highly
dependent on temperature. It can be a part of a conductor. We consider here the following nonlocal
form of the thermistor problem and its di¤erent general variants:
lf ðuÞ
ut su ¼ Ð
;
ð W f ðuÞ dxÞ 2
ð1Þ
lf ðuÞ
ut sp u ¼ Ð
;
ð W f ðuÞ dxÞ 2
ð2Þ
lf ðuÞ
ut sy u ¼ Ð
;
ð W f ðuÞ dxÞ 2
ð3Þ
associated to boundary and initial conditions of type:
uðx; tÞ ¼ 0; x A qW;
ðIBCÞ ¼
uðx; 0Þ ¼ u0 ðxÞ; x A W:
usual p-Laplacian is defined by sp u :¼ divðj‘uj p2 ‘uÞ and the so-called y-Laplacian sy u :¼
PThe
N
qu qu q 2 u
i; j¼1 qxi qxj qxi qxj , which is defined in a canonical way with the second derivatives in the local maximum
and minimum directions.
Problem (1) has a variety of applications. It represents the thermo-electric flow in a conductor [16]. In
this case, u is the temperature of the conductor, f ðuÞ is the temperature dependent electrical resistivity,
and l is a positive dimensionless parameter that can be identified with the square of the applied potential
di¤erence at the ends of a conductor. It also has been used to describe fuse wires, electric arcs, and fluorescent lights [10, 11, 14, 15]. It can model the phenomena associated with the occurrence of shear bands
(i) in metals being deformed under high strain rates [5, 6], (ii) in the theory of gravitational equilibrium of
Received February 23, 2008; Accepted December 4, 2009.
2010 Mathematics Subject Classification. 35K15, 35K55, 35K60, 35K65.
Key words and phrases. Ly -energy method, p-Laplacian, Ly -Laplacian, existence, uniqueness, regularity.
Research supported by the Centre for Research on Optimization and Control (CEOC) of the Portuguese Foundation
for Science and Technology (FCT), cofinanced by the European Community Fund FEDER/POCI 2010.
1
2
MOULAY RCHID SIDI AMMI
polytropic stars [13], (iii) in the investigation of the fully turbulent behavior of real flows, using invariant
measures for the Euler equation [7], (iv) in modelling aggregation of cells via interaction with a chemical
substance (chemotaxis) [21].
The infinity Laplace equation is nowadays one of the most trendy nonlinear partial di¤erential equations. This is due to the beautiful mathematical theory that has been put forward to understand it, starting from the pioneering work of Aronsson in the 1960’s, but also to the recent finding that it is related to
important applications in game theory, image processing, and mass transfer problems. See [4], [12], and
[3], which is an excellent survey on the subject, with plenty of clarifying examples. The ‘‘infinity Laplacian’’ operator is introduced in relationship with the Absolutely Minimal Lipschitz Extension (AMLE)
[1]. Its parabolic counterpart is much less popular but lately has started to attract more interest. However, hardly any attention has been given yet to the extension of the thermistor problem. One of the issues
touched in this paper concerns the existence and obtaining of an estimate, which is local in time, for the
space derivative of the solutions of ðPÞy . Let W be a bounded domain in RN with smooth boundary qW
and let g A W 1; y ðqWÞ, i.e., g is a Lipschitz continuous function defined on the boundary of W. Then
AMLE of g is given as follows:
8
1; y
ðWÞ; u=qW ¼ g;
>
<Find u A W
ðAMLEÞg ¼ EU H W bounded; Ev A W 1; y ðUÞ satisfying u=qU ¼ v=qU;
>
:j‘uj y a j‘vj y :
L ðUÞ
L ðUÞ
Aronsson [1, 2] showed that
ðAEEÞg sy u ¼ 0 in W;
u=qW ¼ g
is the Euler equation of the ðAMLEÞg problem and that any classical solution of ðAEEÞg is unique. Furthermore, Jensen [12] proved the existence of a unique viscosity solution for ðAEEÞg as well as a maximum principle for u. This motivates us to consider the sy extension to the basic problem (1) with the
simple Laplacian.
On the other hand, from the point of view of applications, it is known that large and high temperature
is an undesirable e¤ect and it may cause the thermistor to crack. This is another motivation to use the
Ly -energy method to establish a priori bound for u and to keep temperature from exceeding some extremal values. Recall that theoretical and numerical analysis of the thermistor problem with di¤erent
types of boundary and initial conditions has recently received a significant amount of attention [20, 22,
23]. For the Ly -energy method and its related applications, we refer the reader to [18, 19].
This work is organized as follows: in the next section, we use the Ly -energy method to prove the
existence and the uniqueness of (1)–(2). In Section 3, we are interested in a generalized form of the
quasi-linear parabolic thermistor problem associated with p-Laplacian equations. Finally, in Section 4,
we see how the Ly -energy method is applied to the more general strongly nonlinear parabolic thermistor
problem associated with the so-called infinity Laplacian sy .
2. A si m p l e f o r m f o r th e n o n lo c a l t h e r m i s t or p r o b l e m
Henceforth, we use the standard notation for Sobolev spaces. We set j:jp ¼ k:kL p for each p A ½1; þy.
W denotes a bounded domain of RN with smooth boundary. In the remainder of this paper, we denote by
c various constants that may depend on the data of the problem, and that are not necessarily the same at
each occurrence. Throughout, we make the following assumptions:
(H1) f : W ! R is a positive C 1 continuous function.
(H2) There exist positive constants c and a such that for all x A R we have
c a f ðxÞ a cjxj aþ1 þ c:
We recall now the following lemma which is useful for our purposes and which plays an essential role
in the sequel.
Lemma 2.1. ([18]) Suppose that a non-negative integrable function yðtÞ satisfies
ðt
yðtÞ a y0 þ b yðsÞ 1þn ds for almost every t A ½0; T ðn; b > 0Þ:
0
APPLICATION OF THE Ly -ENERGY METHOD TO THE NON LOCAL THERMISTOR PROBLEM
3
Then there exists a number T0 A ð0; T depending only on y0 , n, b such that
yðtÞ a y0 þ 1;
for almost every t A ½0; T0 :
We claim that all calculations are formal. We can verify computations by using cut-o¤ functions, truncations method, see [18], and using some approximation procedures like the Faedo Galerkin method, see
[17].
In this section, the Ly -energy method is exemplified for nonlinear parabolic equations (1) and (2).
More precisely, we have the following results.
Theorem 2.2. Given u0 A Ly ðWÞ, there exists a positive T0 ¼ T0 ðju0 jy ; aÞ such that (1)–(2) associated to
ðIBCÞ admit a unique solution u verifying
u A Ly ð0; T0 ; L p ðWÞ X Ly ðWÞÞ X L p ð0; T0 ; W01; p ðWÞÞ X W 1; p ð0; T0 ; L p ðWÞÞ;
(respectively p ¼ 2 for (1)).
Proof. Multiplying equations (1) and (2) by juj k2 u for k > 2, integrating over W and using Holder’s
inequality we obtain
ð
1 d k
aþk
k1
a
jujk þ Ik a c ðjuj aþ1 þ 1Þjuj k1 dx a cjujaþk
þ cjujk1
a cjujy
jujkk þ cjujkk1 ;
k dt
W
where
(
Ð
ðk 1Þ W juj k2 j‘uj 2 dx for ð1Þ;
Ik ¼
Ð
ðk 1Þ W juj k2 j‘uj p dx for ð2Þ:
We remark that all terms of Ik are nonnegatives. Then
1 d k
a
juj a cjujy
jujkk þ cjujkk1 :
k dt k
Setting yk ¼ jujk and dividing both sides of the above inequality by ykk1 , we have
dyk
a
a cjujy
yk þ c:
dt
Integrating over ð0; tÞ we get
yk ðtÞ a c þ yk ð0Þ þ c
ðt
a
juðsÞjy
yk ðsÞ ds:
0
Letting k ! y, it yields
juðtÞjy a c þ ju0 jy þ c
ðt
aþ1
juðsÞjy
ds:
0
By applying Lemma 2.1, there exists T0 depending only on a and ju0 jy such that
juðtÞjy a ju0 jy þ c
Recall that if u A W
1; p
p
qu
qt
for almost every t A ð0; T0 :
p
ð4Þ
p
ð0; T0 ; L ðWÞÞ, then A L ð0; T0 ; L ðWÞÞ. It follows that
qv
A L p ð0; T0 ; L p ðWÞÞ
u A v A L 2 ð0; T0 ; W 1; p ðWÞÞ;
qt
which is compactly embedded in L p ð0; T0 ; L p ðWÞÞ. Thus, from Lion’s Lemma of compacity (see [17]), u
makes sense at 0 and is continuous except on a set of measure null.
Uniqueness of solutions. Suppose that u1 and u2 are two di¤erent solutions ðu1 0 u2 Þ of problems (1)–(2),
verifying ðIBCÞ. Subtracting the equations verified by u1 and u2 , we obtain
lf ðu2 Þ
lf ðu1 Þ
Ð
Ð
ð W f ðu2 Þ dxÞ 2
ð W f ðu1 Þ dxÞ 2
dw
I ¼
w;
dt
w
4
MOULAY RCHID SIDI AMMI
where w ¼ u2 u1 , and
I¼
su2 su1 ¼ sw;
sp u2 sp u1 :
Since ui A Ly , we have
ð
lf ðu2 Þ
Ð
W
f ðu2 Þ dxÞ
lf ðu1 Þ
Ð
2 ð
W
f ðu1 Þ dxÞ 2
w
A Ly ;
Multiplying the above equation by w and using the monotonicity of the Laplacian and p-Laplacian
operators, we get
djwðtÞj22
a cjwðtÞj22 ;
dt
r
and so the uniqueness follows by Gronwall’s inequality.
3. A g e n e r a l i z ed f o r m o f q u a s i - l i n e a r p a r a b o l i c t h e r mi s t o r p r o b l e m
We consider in this section the following problem
lf ðuÞ
;
ut ¼ divðjuj r j‘uj p2 ‘uÞ þ Ð
ð W f ðuÞ dxÞ 2
ðx; tÞ A W ð0; yÞ
uðx; tÞ ¼ 0;
ðx; tÞ A qW ð0; yÞ
uðx; 0Þ ¼ u0 ðxÞ;
x in W;
ð5Þ
where W is an open bounded domain in RN with smooth boundary qW, and 2 < r < þy, 1 < p < þy.
Equation (5) is a generalized form of quasi-linear parabolic thermistor problem associated with pLaplacian equations [18]. Existence and regularity results of weak solutions have already been considered
such an equation, in particular r ¼ 0 and p ¼ 2 (see [9]). However, to the best of our knowledge, there
have been no results on the estimate for the space derivative ‘u of u in the general case (5). Then, we have
the following theorem.
Theorem 3.1. In addition to hypotheses (H1)–(H2), we further suppose that u0 A W 1; y ðWÞ. Then, there
exists a positive real number T0 ¼ T0 ðju0 jW 1; y Þ such that (5) admits a unique solution u satisfying
u A Ly ð0; T0 ; W01; y ðWÞÞ X W 1; 2 ð0; T0 ; L 2 ðWÞÞ:
Proof. The existence is proved by the Faedo-Galerkin method [17]. We consider a sequence of linearly
independent elements w1 ; . . . wm ; . . . of H01 ðWÞ, which is complete in H01 ðWÞ. For each m, we define an
approximate solution um of (5) as follows:
um ðtÞ ¼
m
X
gjm ðtÞwj ;
j¼1
l
hum0 ; wj i ¼ hdivðjum j r j‘um j p2 ‘um Þ; wj i þ Ð
h f ðum Þ; wj i;
ð W f ðum Þ dxÞ 2
1 a j a m;
um ð0Þ ¼ u0m ;
ð6Þ
where u0m is, for example, the orthogonal projection in H01 ðWÞ of u0 on the space spanned by w1 ; . . . ; wm :
Equations (6) are equivalent to an initial-value problem for a linear finite m-dimensional ordinary di¤erential equation (ODE) for the gjm . The existence and uniqueness is obvious by classical ODE’s theory.
This solution is shown to exist on a maximal interval ½0; tm Þ. We obtain further a priori estimates on um
which guaranteed that tm ¼ T0 , and after, we pass to limits on the approximate problem by straightforward standard compactness and monotonicity arguments, which allow us to assert that u is a solution of
problem (5). Indeed, we establish in the following lemma the Ly a priori bound for u.
APPLICATION OF THE Ly -ENERGY METHOD TO THE NON LOCAL THERMISTOR PROBLEM
5
Lemma 3.2. Suppose that the hypotheses (H1)–(H2) are satisfied. Then there exist positive numbers T0 and
c such that
juðtÞjy a ju0 jy þ c;
Et A ð0; T0 :
Proof. Multiplying (5) by juj k2 u for k > 2 and integrating over W, we have exactly as we did in the
proof of Theorem 2.2:
1 d k
juj þ Ik a cjujay jujkk þ cjujk1
k ;
k dt k
Ð
where Ik ¼ ðk 1Þ W juj rþk2 j‘uj p dx b 0. Arguing exactly as above in Theorem 2.2, we obtain juðtÞjy a
ju0 jy þ c Et A ð0; T0 , for a certain T0 .
r
Now we prove an a priori estimate of ‘u in Ly .
Lemma 3.3. Under the hypotheses of Theorem 3.1, there exist positive numbers T0 and c such that
j‘uðtÞjy a j‘u0 jy þ c;
Et A ð0; T0 :
Proof. For simplicity, we restrict ourselves to the 1-dimensional case. We rewrite (5) as:
lf ðuÞ
ut ¼ ð p 1Þjuj r jux j p2 uxx þ rjuj r2 ujux j p þ Ð
:
ð W f ðuÞ dxÞ 2
ð7Þ
Multiplying (7) by sk u :¼ ðjux j k2 ux Þx ¼ ðk 1Þjux j k2 uxx , we have
ð
1 d
jux jkk þ ðk 1Þð p 1Þ juj r jux j kþp4 juxx j 2 dx
k dt
W
ð
ð
l
¼ rðk 1Þ ujuj r2 jux j kþp2 uxx dx þ Ð
f 0 ðuÞjux j k dx:
2
ð W f ðuÞ dxÞ W
W
We set
I1 ¼ ðk 1Þð p 1Þ
ð
juj r jux j kþp4 juxx j 2 dx;
W
and
I2 ¼ rðk 1Þ
ð
ujuj r2 jux j kþp2 uxx dx:
W
Using the hypotheses, the Ly estimate of u and the fact that I1 b 0, we get
ð
1 d
jux jkk a I2 þ c jux j k dx:
k dt
W
ð8Þ
We want now to estimate I2 . Indeed, observe that jux j kþp2 uxx ¼ kþ 1p1 ðjux j kþp2 ux Þx . Then it follows
by Lemma 3.2 that
ð
ð
rðk 1Þ
rðr 1Þðk 1Þ
I2 ¼
ujuj r2 ðjux j kþp2 ux Þx dx a
juj r2 jux j kþp dx
kþ p1 W
kþ p1
W
r2
p
p
a rðr 1Þðju0 jy
þ cÞjux jy
jux jkk a cjux jy
jux jkk :
Then from (8), we have
1 d
p
jux jkk a cjux jy
jux jkk þ cjux jkk :
k dt
Setting again yk ¼ jux jk and dividing by ykk1 we get
d
p
yk a ðcjux jy
þ cÞ yk :
dt
6
MOULAY RCHID SIDI AMMI
Hence, integrating over ð0; tÞ, we obtain
ðt
ðt
p
jux ðtÞjk a jðu0 Þx jk þ c jux ðsÞjy
jux ðsÞjk ds þ c jux ðsÞjk ds;
0
Et b 0:
0
Letting k ! y, we have
jux ðtÞjy a jðu0 Þx jy þ c
ðt
pþ1
jux ðsÞjy
ds þ c
0
ðt
jux ðsÞjy ds:
0
On the other hand, by Young’s inequality we have
ð pþ1Þ1=ð pþ1Þ
a
jux ðtÞjy ¼ jux ðtÞjy
Then
jux ðtÞjy a ðc þ jðu0 Þx jy Þ þ c
ðt
1
pþ1
jux ðtÞjy
þ c:
pþ1
pþ1
jux ðsÞjy
ds;
Et b 0:
0
Applying thus Lemma 2.1, there exists T0 ¼ T0 ðju0 jW 1; y Þ > 0 such that
jux ðtÞjy a jðu0 Þx jy þ c;
Et A ð0; T0 :
r
This concludes the proof of Lemma 3.3.
Remark 3.4. To show that ut A L 2 ð0; T0 ; L 2 ðWÞÞ, a priori estimate of ut arises by formal di¤erentiation
with respect to time of (5) in consideration and using the Ly estimates of u and ‘u, or we can see it directly from the equation itself. The second member is in Ly ðWÞ. Also, juj r j‘uj p2 ‘u A Ly ðWÞ H L 2 ðWÞ.
Then divðjuj r j‘uj p2 ‘uÞ A L 2 ðWÞ. It yields ut A L 2 ðWÞ. For more details, see [8] and [9].
Uniqueness of solutions. Suppose that we have two di¤erent solutions u and v of (5). If we set w ¼ u v
lf ðuÞ
and HðuÞ ¼ Ð
, then, subtracting the equations satisfied by v from the one satisfied by u, we
ð W f ðuÞ dxÞ 2
obtain
wt ¼ divðjuj r j‘uj p2 ‘uÞ divðjvj r j‘vj p2 ‘vÞ þ ðHðuÞ HðvÞÞ ¼ divððjuj r j jvj r jÞj‘uj p2 ‘uÞ
þ divðjvj r ðj‘uj p2 j‘vj p2 Þ‘uÞ þ divðjvj r j j‘vj p2 ‘wÞ þ ðHðuÞ HðvÞÞ:
If we multiply by w and integrate over W, we get
ð
ð
1 d
2
r
r
p2
jwj ¼ ðjvj j juj jÞj‘uj ‘u‘w dx þ jvj r ðj‘vj p2 j‘uj p2 Þ‘u‘w dx
2 dt 2
W
W
ð
ð
ð
ðHðuÞ
HðvÞÞ 2
w dx a ðjvj r j juj r jÞj‘uj p2 ‘u‘w dx
jvj r j‘vj p2 j‘wj 2 dx þ
w
W
W
W
ð
ð
ðHðuÞ HðvÞÞ 2
jwj dx;
þ jvj r ðj‘vj p2 j‘uj p2 Þ‘u‘w dx þ
w
W
W
Ð
since W jvj r j‘vj p2 j‘wj 2 dx a 0. Using then the Ly estimate of the solutions and their space derivatives, Holder’s and Poincaré’s inequalities, we easily see that
ð
ð
ðjvj r j juj r jÞj‘uj p2 ‘u‘w dx a c w‘w dx a cjwj j‘wj a cjwj2 ;
2
2
2
W
W
ð
ð
jvj r ðj‘vj p2 j‘uj p2 Þ‘u‘w dx a c j‘wj 2 dx a cjwj2 ;
2
W
W
ð
ðHðuÞ HðvÞÞ 2 jwj dx a cjwj22 :
w
W
It yields
1 d
jwj2 a cjwj22 :
2 dt 2
The uniqueness follows by Gronwall’s inequality.
r
APPLICATION OF THE Ly -ENERGY METHOD TO THE NON LOCAL THERMISTOR PROBLEM
7
4. M o r e g e n er a l s t r o n g l y n o n l i n e a r p a r a b o l i c t h e r m i s t o r p r o b l em
In this section, we are interested in applying the Ly -energy method to the more general strongly nonlinear parabolic equations associated with the so-called y-Laplacian sy , namely:
8
lf ðuÞ
qu
>
Ð
in W ð0; þyÞ;
>
qt sy u ¼ ð
>
f ðuÞ dxÞ 2
>
W
>
<
u=qW ¼ 0
ðPÞy ¼
>
uð:; 0Þ ¼ u0 in W H RN ;
>
>
>
>
:s u :¼ P N qu qu q 2 u :
y
i; j¼1 qxi qxj qxi qxj
Let us now give the main result of this section.
Theorem 4.1. Let hypotheses (H1)–(H2) be satisfied. Then, there exists a unique solution u A W 1; y ðWÞ of
ðPÞy for any u0 A W 1; y ðWÞ.
2
qu
u
Proof. For simplicity, we denote qx
and qxqi qx
by ui and uij , respectively, and we use the summation coni
j
vention. First, we derive a priori estimate of u in Ly .
Lemma 4.2. There exists a T0 such that
juðtÞjy a ju0 jy þ c;
Et A ð0; T0 :
Proof. Multiplying ðPÞy by juj k2 u and using hypotheses (H1)–(H2), we have
ð
ð
1 d
l
k
k2
juðtÞjk ¼
ui uj uij juj u dx þ Ð
f ðuÞjuj k2 u dx
k dt
ð W f ðuÞ dxÞ 2 W
W
ð
ð
l
k2
¼ I3 þ Ð
f
ðuÞjuj
u
dx
a
I
þ
c
juj aþk dx;
3
ð W f ðuÞ dxÞ 2 W
W
where
ð
ð
ð
ð
1
ðk 1Þ
1
ui uj uij juj k2 u dx ¼
ui ðuj2 Þi juj k2 u dx ¼
ui uj2 juj k2 ui dx uii uj2 juj k2 u dx
I3 ¼
2 W
2
2 W
W
W
ð
ð
ð
ðk 1Þ
1
1 k
1
1 k
k2
2 2
juj
juj
ui uj juj
dx uii uj
dx a I4 ¼ uii uj
dx;
¼
2
2 W
k
2 W
k
W
j
j
ðk1Þ Ð
k2
2 2
since 2
dx a 0.
W ui uj juj
On the other hand, we have
ð 1
1
k
k
uii ujj juj þ uiij uj juj dx
I4 ¼
2k
W 2k
ð
ð
ð
ð
1
1
1
1
1
jsuj 2 juj k dx uij2 juj k dx ui uj uij juj k2 u dx a
jsuj 2 juj k dx I3 ;
¼
2k W
2k W
2 W
2k W
2
Ð 2 k
1
since 2k W uij juj dx a 0.
Gathering (10)–(11) we infer
ð
1
I3 a
jsuj 2 juj k dx:
3k W
Combining (9)–(12), we obtain
ð
ð
1 d
1
1
2
a
juðtÞjkk a
jsujy
jsuj 2 juj k dx þ c juj aþk dx a
jujkk þ cjujy
jujkk :
k dt
3k W
3k
W
Then, dividing both sides of (13) by jujkk1 and integrating over ð0; tÞ, we find
ð
ðt
1 t
2
a
juðtÞjk a ju0 jk þ
jsuðsÞjy
juðsÞjk ds þ c juðsÞjy
juðsÞjk ds:
3k 0
0
Letting k ! þy, we obtain
juðtÞjy a ju0 jy þ c
ðt
0
aþ1
juðsÞjy
ds:
ð9Þ
ð10Þ
ð11Þ
ð12Þ
ð13Þ
8
MOULAY RCHID SIDI AMMI
Applying again Lemma 2.1, there exists T0 > 0 such that
juðtÞjy a ju0 jy þ c:
for every t A ð0; T0 .
r
The crucial step for proof of Theorem 4.1 is the derivation of Ly —a priori estimate for ‘u.
Lemma 4.3.
j‘uðtÞjy a j‘u0 jy ;
Et A ð0; T0 :
Proof. Multiplying ðPÞy by sk u ¼ divðj‘uj k2 ‘uÞ, we get
ð
ð
1 d
l
k
k2
j‘uðtÞjk ¼ ui uj uij ðj‘uj ul Þl dx þ Ð
sk u: f ðuÞ dx
k dt
ð W f ðuÞ dxÞ 2 W
W
ð
l
¼ I5 þ Ð
sk u: f ðuÞ dx;
ð W f ðuÞ dxÞ 2 W
where
ð
1
ui ðuj2 Þi ðj‘uj k2 ul Þl dx
I5 ¼ ui uj uij ðj‘uj ul Þl dx ¼
2
W
W
ð
ð
1
1
uii ðuj2 Þðj‘uj k2 ul Þl dx þ
ui ðuj2 Þðj‘uj k2 ul Þli dx ¼ I6 þ I7 ;
¼
2 W
2 W
ð
k2
with
I6 ¼
1
2
and
I7 ¼
Notice that
1
I7 ¼
2
ð14Þ
ð
2
1
2
ð
W
ð
W
uil j‘uj ðj‘uj
W
ð15Þ
uii ðuj2 Þðj‘uj k2 ul Þl dx;
ui ðuj2 Þðj‘uj k2 ul Þli dx:
k2
ul Þi dx ð
W
ui uj ujl ðj‘uj k2 ul Þi dx
ð
1
uil j‘uj 2 ðj‘uj k2 uli þ ðk 2Þj‘uj k4 ul um umi Þ dx
¼
2 W
ð
ui uj ujl ðj‘uj k2 uli þ ðk 2Þj‘uj k4 ul um umi Þ dx a 0:
ð16Þ
W
Moreover, we have
ð
1
I6 ¼
uii j‘uj 2 ðj‘uj k2 ull þ ðk 2Þj‘uj k4 ul um uml Þ dx
2 W
ð
ð
ð
1
k2
1
2
k
k2
¼
jsuj j‘uj dx þ
uii j‘uj ul um uml dx ¼
jsuj 2 j‘uj k dx þ I8 ;
2 W
2
2 W
W
where
!
ð
ð
k2
k2
lf ðuÞ
k2
k2
I8 ¼
uii j‘uj ul um uml dx ¼
uii j‘uj
ut Ð
dx
2
2
ð W f ðuÞ dxÞ 2
W
W
ð
ð
k2
ðk 2Þ
l
k2
uii j‘uj ut dx uii f ðuÞj‘uj k2 dx
¼
Ð
2
2
ð W f ðuÞ dxÞ 2 W
W
ð
ð
ðk 2Þ 2
ðk 2Þ
k4
¼
ui j‘uj um umi ut dx ui j‘uj k2 uti dx
2
2
W
W
ð
ðk 2Þ
l
uii f ðuÞj‘uj k2 dx:
Ð
2
ð W f ðuÞ dxÞ 2 W
ð17Þ
ð18Þ
APPLICATION OF THE Ly -ENERGY METHOD TO THE NON LOCAL THERMISTOR PROBLEM
On the other hand, we have
ð
ð
ðk 2Þ
l
l
ðk 2Þ 2
k2
u
f
ðuÞj‘uj
dx
¼
ui j‘uj k4 um umi f ðuÞ dx
Ð
Ð
ii
2
2
ð W f ðuÞ dxÞ 2 W
ð W f ðuÞ dxÞ 2
W
ð
l
ðk 2Þ
ui2 j‘uj k2 f 0 ðuÞ dx:
Ð
2
2
ð W f ðuÞ dxÞ
W
9
ð19Þ
From (18)–(19), we have
!
ð
ðk 2Þ 2
l
k4
I8 ¼ ui j‘uj um umi ut Ð
f ðuÞ dx
2
ð W f ðuÞ dxÞ 2
W
ð
ð
ðk 2Þ
l
k2
k2 0
2
þ
u
j‘uj
f
ðuÞ
dx
ui j‘uj k2 uti dx
Ð
i
2
2
ð W f ðuÞ dxÞ 2 W
W
ð
ð
ð
ðk 2Þ
ðk 2Þ
l
k2
k
2
k2 0
2
j‘uj jsuj dx þ
ui j‘uj f ðuÞ dx ui j‘uj k2 uti dx
¼
Ð
2
2
2
2
ð W f ðuÞ dxÞ W
W
W
ð
ð
ðk 2Þ
l
k2
j‘uj k j f 0 ðuÞj dx ui j‘uj k2 uti dx
a
Ð
2
2
2
ð W f ðuÞ dxÞ W
W
ð
ðk 2Þ
ðk 2Þ d
ðk 2Þ
ðk 2Þ d
j‘uðtÞjkk a c
j‘uðtÞjkk j‘uðtÞjkk :
j‘uj k dx ð20Þ
ac
2
2k dt
2
2k dt
W
ðk2Þ Ð
Here, we use the fact that 2 W j‘uj k jsuj 2 dx a 0. Furthermore, we have
ð
ð
l
l
s
u:
f
ðuÞ
dx
¼
j‘uðtÞj k f 0 ðuÞ dx a cj‘uðtÞjkk :
ð21Þ
Ð
Ð
k
ð W f ðuÞ dxÞ 2 W
ð W f ðuÞ dxÞ 2 W
Gathering (14)–(21), we obtain
1 d
k2 d
k2
1
j‘uðtÞjkk a j‘uðtÞjkk þ c
j‘uðtÞjkk þ cj‘uðtÞjkk þ
k dt
2k dt
2
2
k2 d
k2
1
j‘uðtÞjkk þ c
j‘uðtÞjkk þ cj‘uðtÞjkk þ
a
2k dt
2
2
ð
jsuj 2 j‘uj k
W
ð
2
jsujy
j‘uðsÞj k ds:
W
Then
d
2
j‘uðtÞjkk a cðk 2Þj‘uðtÞjkk þ cj‘uðtÞjkk þ jsujy
j‘ujkk :
dt
Dividing by k b 1, we get
1 d
k2
c
1
2
j‘uðtÞjkk a c
j‘uðtÞjkk þ j‘uðtÞjkk þ jsujy
j‘ujkk :
k dt
k
k
k
ð22Þ
Dividing again both sides of (22) by j‘ujkk1 and integrating on ð0; tÞ, we get
ð
ð
ð
1 t
c t
cðk 2Þ t
2
jsuðsÞjy
j‘uðsÞjk ds þ
j‘uðsÞjk ds þ
j‘uðsÞjk ds:
j‘uðtÞjk a j‘u0 jk þ
k 0
k 0
k
0
Letting k ! þy, we obtain
j‘uðtÞjy a j‘u0 jy þ c
ðt
j‘uðsÞjy ds:
0
By Gronwall’s inequality, we conclude that there exists a T0 such that for all t A ð0; T0 Þ, we have
j‘uðtÞjy a cj‘u0 jy :
r
Remark 4.4. We can derive Cy -estimates of solutions by the Ly -energy method by di¤erentiating (1),
(2), and ðPÞy n-times with respect to space variables x and repeating the same procedure as in the previous sections.
10
MOULAY RCHID SIDI AMMI
Uniqueness of solutions to ðPÞy : Suppose that u and v are two weak solutions to ðPÞy corresponding to
the same initial and boundary data. As before, we make the di¤erence of the two equations verified by
the solutions u and v, we multiply it by w ¼ u v, and we have
ð
ð
ð
1 d
HðuÞ HðvÞ 2
jwj22 ¼
w dx;
ui uj uij w dx vi vj vij w dx þ
2 dt
w
W
W
W
or
ð
1
ui uj uij w dx ¼
2
W
ð
W
ui ðuj2 Þi w dx
1
¼
2
ð
W
uj2 uii w dx
1
2
ð
W
uj2 ui wi dx;
and similarly we have
ð
vi vj vij w dx ¼ W
1
2
ð
W
vj2 vii w dx 1
2
ð
W
vj2 vi wi dx:
Then,
ð
ð
ð
1
1
ðvj2 vii uj2 uii Þw dx þ
ðvj2 vi uj2 ui Þwi dx ¼ I þ II ;
2
2
W
W
W
W
ð
ð
ð
1
1
1
I¼
ðv 2 vii uj2 uii Þw dx ¼
ðv 2 uj2 Þvii w dx þ
u 2 ðvii uii Þw dx ¼ III þ IV ;
2 W j
2 W j
2 W j
ui uj uij w dx where
IV ¼
1
2
1
¼
2
ð
W
ð
W
ð
vi vj vij w dx ¼
uj2 ðvii uii Þw dx ¼
1
2
ð
W
uj2 wii w dx ¼
ðwi2 uj2 þ 2wwi uj uij Þ dx a c
ð
W
ð
W
wi ðuj2 wÞi dx ¼
1
2
ð
W
wi ðwi uj2 þ 2wuj uij Þ dx
wi2 þ cjwj2 j‘wj2 a cj‘wj22 þ cjwj2 j‘wj2 a cjwj22 :
In the same manner, we have
III ¼
1
2
ð
W
ðvj2 uj2 Þvii w dx a c
ð
W
jwj 2 jvj þ uj j dx a cjwj22 :
Coming back to II, we have
ð
ð
ð
1
1
1
2
2
2
2
ðv vi uj ui Þwi dx ¼
ðv uj Þvi wi dx þ
u 2 ðvi ui Þwi dx
2 W j
2 W j
2 W j
ð
ð
a c wj ðvj þ uj Þvi wi dx þ c uj2 wi2 dx a cj‘wj22 þ cj‘wj22 ¼ cj‘wj22 a cjwj22 :
W
W
Hence,
I þ II a cjwj22 :
On the other hand,
ð
HðuÞ HðvÞ 2
w dx a cjwj22 :
w
W
Collecting all the previous inequalities, we get
1 d
jwj2 a cjwj22 :
2 dt 2
The uniqueness follows by applying Gronwall’s inequality. In all the proof of uniqueness, we mainly
use the Ly -energy estimate of the solutions and their space derivatives, Holder’s and Poincaré’s inequalities.
r
Acknowledgement. The author would like to thank the referees for their helpful comments and whose
stimulating questions and remarks allowed us to improve the paper.
APPLICATION OF THE Ly -ENERGY METHOD TO THE NON LOCAL THERMISTOR PROBLEM
11
References
[1] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561.
[2] G. Aronsson, On the partial di¤erential equation ux2 uxx þ 2ux uy uxy þ uy2 uyy ¼ 0, Ark. Mat. 7 (1968), 395–425.
[3] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull.
Amer. Math. Soc. 41 (4) (2004), 439–505.
[4] T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as p ! y of sp up ¼ f and related extremal problems,
Rend. Sem. Mat. Univ. Pol. Torino, (1991), 15–68.
[5] J. W. Bebernes and A. A. Lacey, Global existence and finite-time blow-up for a class of non-local parabolic
problems, Adv. Di¤. Eqns. 2 (1997), 927–953.
[6] J. W. Bebernes, C. Li and P. Talaga, Single-point blow-up for non-local parabolic problems, Physica D 134
(1999), 48–60.
[7] E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional
Euler equations: a statistical mechanics description, Comm. Math. Phys. 143 (1992), 501–525.
[8] A. El Hachimi and F. de Thélin, Supersolutions and stabilisation of the solutions of the equation
ut sp u ¼ f ðx; uÞ, Part II, Publicacions Matematiques, 35 (1991), 347–362.
[9] A. El Hachimi and M. R. Sidi Ammi, Existence of global solution for a nonlocal parabolic problem, Electron.
J. Qual. Theory Di¤er. Equ. 1 (2005), 9 pages.
[10] A. El Hachimi and M. R. Sidi Ammi, Semidiscretization for a nonlocal parabolic problem, Int. J. Math. Math.
Sci. 2005 (10) (2005), 1655–1664.
[11] A. El Hachimi, M. R. Sidi Ammi and D. F. M. Torres, A dual mesh method for a non-local thermistor problem,
SIGMA Symmetry Integrability Geom. Methods Appl. 2 (2006), Paper 058, 10 pages (electronic). Available
online at http://www.emis.de/journals/SIGMA/2006/Paper058/index.html
[12] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Ration. Mech.
Anal. 123 (1) (1993), 51–74.
[13] A. Krzywicki and T. Nadzieja, Some results concerning the Poisson-Boltzmann equation, Zastos. Mat. 21 (2)
(1991), 265–272.
[14] A. A. Lacey, Thermal runaway in a non-local problem modelling ohmic heating. Part I: Model derivation and
some special cases, Euro. J. Appl. Math. 6 (1995), 127–144.
[15] A. A. Lacey, Thermal runaway in a non-local problem modelling ohmic heating. Part II: General proof of blow-up
and asymptotics of runaway, Euro. J. Appl. Math. 6 (1995), 201–224.
[16] A. A. Lacey, Di¤usion models with blow-up, J. Comp. Appl. Math. 97 (1998), 39–49.
[17] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.
[18] M. Ôtani, Ly -energy method and its applications. Nonlinear partial di¤erential equations and their applications,
GAKUTO Internat. Ser. Math. Sci. Appl. 20, Gakkōtosho, Tokyo, (2004), 505–516.
[19] M. Ôtani, Ly -energy method, basic tools and usage, Di¤erential Equations; Chaos and Variational Problems,
357–376, Progr. Nonlinear Di¤erential Equations Appl. 75, Birkhäuser Basel, 2008.
[20] M. R. Sidi Ammi and Delfim F. M. Torres, Numerical analysis of a nonlocal parabolic problem resulting from
thermistor problem, Math. Comput. Simulation 77 (2-3) (2008), 291–300.
[21] G. Wolansky, A Critical parabolic estimate and application to non-local equations arising in chemotaxis, Appl.
Anal. 66 (1997), 291–321.
[22] X. Xu, Existence and uniqueness for the nonstationary problem of the electrical heating of a conductor due to the
Joule-Thomson e¤ect, Int. J. Math. Math. Sci. 16 (1) (1993), 125–138.
[23] X. Y. Yue, Numerical analysis of nonstationary thermistor problem, J. Comput. Math. 12 (3) (2004), 213–223.
Department of Mathematics, Université Moulay Ismail, F. S. T. Errachidia, B. P. 509, Boutalamine,
Errachidia-Morocco
E-mail address: sidiammi@ua.pt, rachidisidiammi@yahoo.fr
12