Applied Mathematics E-Notes, 16(2016), 56-64 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Energy Decay Result In A Quasilinear Parabolic
System With Viscoelastic Term
Mohamed Ferhaty, Ali Hakemz
Received 27 June 2015
Abstract
In this paper we consider a quasilinear parabolic system of the form:
A(t)jut jm
2
ut
Lu +
Z
t
g(t
s)Lu(s)ds = 0
0
in a bounded domain , A(t) is a bounded and positive de…nite matrix,
Rn (n 1), initial data (u0 ; u1 ) are given functions belonging to suitable spaces
and g a continuously di¤erentiable decaying function. We use the lemma of Martinez to establish a general decay result. This improves the result obtained by
Messaoudi and Tellab [3].
1
Introduction
In this paper we consider
A(t)jut jm
2
ut
Lu +
Z
t
g(t
s)Lu(s)ds = 0;
m > 2;
(1)
0
subjected to the following boundary conditions
u(x; t) = 0; x 2 @ ; t
0;
(2)
and initial conditions
u(x; 0) = u0 (x); ut (x; 0) = u1 (x); x 2
where is a bounded open subset of Rn (n
matrix,
Lu =
div(M ru) =
;
(3)
1), A(t) is a bounded and positive de…nite
N
X
@
@x
i
i;j=1
ai;j (x)
@u
@xi
:
Mathematics Subject Classi…cations: 35L05, 35L15, 35L70, 93D15.
of Mathematics, USTO University, 31000 Oran, Algeria
z Laboratory ACEDP, Djillali Liabes University, 22000 Sidi Bel Abbes, Algeria
y Department
56
M. Ferhat and A. Hakem
57
The matrix M = (ai;j (x)), where ai;j 2 C 1 ( ), is symmetric and there exists a
constant a0 > 0 such that for all x 2
and = ( 1 ; 2 ; :::::; N ) 2 RN , we have
PN
2
a0 j j : Also,
i;j=1 ai;j (x) j i
N
X
@u(t) @ (t)
a(u(t); (t)) =
ai;j (x)
dx =
@xj @xi
i;j=1
and
N
X
a1 = max
i=1
kai;j k21
Z
!
M ru(t):r (t)dx;
:
The values of u are taken in Rn and A 2 C(R+ ) is a bounded square matrix satisfying
c0 j j2
(A(t) ; )
c1 j j2 ; 8t 2 R+ ;
2 Rn :
(4)
The initial data (u0 ; u1 ) are given functions belonging to suitable spaces and g a continuously di¤erentiable decaying function. To motivate our work, let us recall some
results regarding quasilinear parabolic system. This type of equation arises from a
variety of mathematical models in engineering and physical sciences. For example, in
the study of a heat conduction in materials with memory, the classical Fourier’s law is
replaced by the following form (cf. [7]):
Z t
q = dru
k(x; t)u(x; s)ds;
1
where u is the temperature, d is the di¤usion coe¢ cient and the integral term represents the memory e¤ect in the material. The study of this type of equations has
drawn a considerable attention and many results have been obtained, see ([2, 3, 6, 9]).
From a mathematical point of view one would expect that the integral term should be
dominated by the leading term in the equation. Messaoudi and Tellab [3] studied the
following system
Z t
A(t)jut jm 2 ut
u+
g(t s) u(s)ds = 0;
0
with the same conditions in (2)–(3) and obtained an energy decay result although the
memory term makes more complex situation. Berrimi and Messaoudi [6] showed that
if A satis…es
(A(t) ; ) c0 j j2 ; 8t 2 R+ ; 2 Rn
then the solutions with small initial energy decay exponentially for m = 2 and polynomially if m > 2. Very recently for a framework of blow-up in …nite time Liu and Chen
[2] studied the following system
Z t
m 2
A(t)jut j
ut
u+
g(t s) u(s)ds = jujp 2 u;
0
with the same conditions in (2)–(3) and proved a blow-up result for both positive and
negative initial energy under suitable conditions on g and p. Motivated by the previous
58
Energy Decay Result
works, in the present paper we investigate problem (1) in which we generalize the results
obtained in [3], supposing new conditions with which the stability is assured, by using
the lemma of Martinez.
Our work is organized as follows. In section 2, we present some preliminaries and
some lemmas. In section 3, the decay property is derived. Our result improves the one
in Messaoudi and Tellab [3].
2
Preliminary Results
In this section, we present some material needed for the proof of our main result. For
the relaxation function g we assume
(A0 ) g : R+ ! R+ is a bounded C 1 function satisfying
Z 1
g(0) > 0;
1
g(s)ds = l < 1;
0
and there exists a nonincreasing di¤erentiable function : R+ ! R+ satisfying
Z +1
g 0 (t)
(t)g(t); t 0;
(s)ds = +1:
0
(A1 ) We also assume that
2
2n
m
n
2
if n
3;
m
2;
if n = 1; 2:
REMARK 1. The same as in [3], there are many functions satisfying (A0 ). Examples of such functions are
g1 (t) = e
(t+1)
; 0<
1 and g2 (t) = (1 + t) ;
<
1:
We will also be using the embedding
H01 ( ) ,! Lp ( );
H01 ( ) ,! Lm ( ):
LEMMA 1 ([2]). Let E : R+ ! R+ be a nonincreasing function and : R+ ! R+
be a C 2 increasing function with (0) = 0 and limt!1 (t) = +1. Assume that there
exists c > 0 for which
Z T
E(t) 0 (t)dt cE(S); 8S 0:
S
Then
E(t)
E(0)e (
Rt
0
(s)ds)
; 8t
0;
M. Ferhat and A. Hakem
where
59
and ! are positive constants.
LEMMA 2 (Sobolev-Poincaré’s inequality). Let 2
ku km
cs kruk2
2n
n 2.
m
The inequality
for u 2 H01 ( )
holds with some positive constant cs .
LEMMA 3 ([3]). For u(:; t) 2 H01 ( ), we have
Z
Z
2
t
g(t
s)(u(x; t)
u(x; s))ds
dx
(1
l)
0
cs 2
(g u)(t);
a0
where cs is the Poincaré constant and l is given in (A1 ), and
(g u)(t) =
Z
0
3
t
g(t
s)
Z
a(u(x; t)
u(x; s); u(x; t)
u(x; s))dxds:
Asymptotic Behavior
In this section, we consider the energy decay of solutions associated to the system
(1)–(3). Similarly as in [7] we give a de…nition of a weak solution of the system (1)–(3):
DEFINITION 1. A weak solution of (1)–(3) is a function u(x; t) such that
u(x; t) 2 C [0; T ); [H01 ( )]n \ C 1 ((0; T ); [Lm ( )]n ) ;
which satis…es
Z tZ
Z tZ
A(s)jus (s)jm 2 us (x; s) (x; s)dsdx +
Lu(x; s) (x; s)dsdx
0
0
Z sZ tZ
+
g(t
) (x; s)Lu(x; )d dxds = 0;
0
0
for all t 2 [0; T ] and
2 C [0; T ); [H01 ( )]n :
REMARK 2. Similar to ([3, 9]), we assume the existence of a solution. For the
linear case (m = 2), one can easily establish the existence of a weak solution by the
Galerkin method. In the one-dimensional case (n = 1), the existence is established, in
a more general setting, by Yin [9].
Now we de…ne the "modi…ed" energy equation related with problem (1)–(3) by
E(t) =
1
1
(g u)(t) +
2
2
1
Z
0
t
g(s)ds a(u(t); u(t)):
60
Energy Decay Result
LEMMA 4. Let u(x; t) be the solution of (1)–(3). Then the energy equation satis…es
Z
1 0
1
E 0 (t)
(g u)(t)
g(t)a(u(t); u(t))
A(t)jut (t)jm dx:
2
2
PROOF. By multiplying (1) by ut (t), and integrating over
Z
1 d
a(u(t); u(t)) +
2 dt
A(t)jut (t)jm dx
Z Z
Note that
a(u(t); ut (t)) =
we get
t
g(t
0
s)M ru(s)rut (t)dsdx = 0:
(5)
1 d
a(u(t); u(t));
2 dt
following the ideas of [10], we can obtain
Z
=
g(t
0
N
X
i;j=1
=
s)
Z tZ
Z
@u(s) @ut (t)
dxds
@xj @xi
g(t
s)ai;j (x)
@u(t) @ut (t)
dxds
@xi @xi
g(t
g(t
0
1 d
2 dt
t
g(t
Z
@ut (s)
@xj
@ut (t)
dxds
@xi
d
a(u(t); u(t)) ds
dt
s)
0
Z
@u(t)
@xi
s)ai;j (x)
0
t
1 d
2 dt
=
s)ai;j (x)
N Z tZ
X
1
2
=
g(t
0
i;j=1
1
2
M ru(s)rut (t)dxds
0
N Z tZ
X
i;j=1
=
Z
t
d
a(u(t)
dt
s)
u(s); u(t)
u(s)) ds
t
g(t
s)a(u(t); u(t) ds
0
Z
t
g(t
s)a(u(t)
u(s); u(t)
u(s)) ds
0
Z
1
1 t 0
g(t)a(u(t); u(t)) +
g (t s)a(u(t) u(s); u(t) u(s))ds
2
2 0
Z t
1
1 d
1 d
(g u)(t) + (g 0 u)(t) +
a(u(t); u(t))
g(s)ds
2 dt
2
2 dt
0
1
g(t)a(u(t); u(t));
2
where
(g u)(t) =
Z
0
t
g(t
s)a(u(x; t)
u(x; s); u(x; t)
u(x; s))ds:
(6)
M. Ferhat and A. Hakem
61
By (5) and (6), we obtain
1 0
(g u)(t)
2
0
E (t)
Z
1
g(t)a(u(t); u(t))
2
A(t)jut (t)jm dx
0:
THEOREM 1. Let (u0 ; u1 ) 2 (H01 ( ))2 be given. Suppose that (A0 )–(A1 ) and (4)
hold. Then there exist two positive constants w and K, depending on the initial data
and c0 for which the solution of (1)–(3) satis…es
E(t)
w
Ke
Rt
(s)ds
0
:
PROOF. From now on, we denote by ci various positive constants which may be
di¤erent at di¤erent occurrences. We multiply the equation (1) by (t)u, integrate over
(S; T ), and use the boundary conditions to get
Z
T
S
+
Z
Z
(t)A(t)jut (t)jm
T
(t)
S
Z
M ru(t)
2
Z
Z
ut (t)u(t)dxdt
T
(t)a(u(t); u(t))dtdx
S
t
g(t
s)ru(s)dsdx = 0:
(7)
0
We then estimate
=
Z
Z
Z
t
(t):M ru(t)
g(t s)ru(s)dsdx
0
Z t
(t):M
g(t s)(ru(t) ru(s))ru(t)dsdx
0
Z
t
g(s)ds: (t)a(u(t); u(t));
(8)
0
by substituting (8) in (7) and adding the following term in (7)
1
2
Z
T
1
2
(t)(g u)(t)
S
Z
T
(t)(g u)(t):
(9)
S
So (7) becomes
Z
Z
T
(t)E(t)dt =
S
T
S
Z T
+
S
Z T
S
Z
Z
(t)A(t)jut (t)jm
(t)M:
Z
2
ut (t)u(t)dxdt
t
g(t
0
(t)(g u)(t)dt:
s)(ru(t)
ru(s)):ru(t)dsdxdt
(10)
62
Energy Decay Result
By Lemma 4, equation (4), the boundedness of A(t) and condition (A1 ), we see that,
8 > 0;
Z
Z
Z
m 2
m
A(t)jut (t)j
ut (t)u(t)dx
ju(t)j dx + c
jut (t)jm dx
Z
m
cm
kru(t)k
+
c
jut (t)jm dx
s
2E(0)
l
cm
s
m 2
2
c
c0
E(t)
E 0 (t):
Then
Z
T
S
Z
m 2
A(t)jut (t)j
cm
s
ut (t)u(t)dxdt
c
c0
m 2!Z
T
2
2E(0)
E(t) (t)dt
l
S
Z T
E 0 (t) (t)dt; 8 > 0: (11)
S
We also have
Z Z
t
M g(t
s)(ru(t)
ru(s))ru(t)dsdx
0
=
N
XZ
t
g(t
0
i;j=1
N Z
X
Z
N Z
1 X
a0
a0
0
t
aij (x)
Z
@u(t)
ds
@xj
t
g(t
@max
i;j=1
@
kaij k21 A E(t) +
N
X
i;j=1
@u(t)
@xi
kaij k21 A a(u(t); u(t)) +
1
dxds
dx
@u(s)
@xi
s)
@u(s)
@xi
2
1
N
X
@u(t)
@xi
@u(t)
@xj
0
i;j=1
0
aij (x)
0
i;j=1
+
s)
Z
N
(1
4a0
2
ds
N
(1
4a0
dx
l)(g u)(t)
l)(g u)(t);
(12)
and using the fact that
j (t)(g u)(t)j = (t)(g u)(t)
we obtain
Z
T
S
(t)(g u)(t)dt
Z
(g u)(t)
( E 0 (t));
T
S
( E 0 (t))dt
CE(S):
(13)
M. Ferhat and A. Hakem
63
By combining (10)–(13), we easily deduce the following
8
2
0
1
N
m 2
<
2
X
N
2E(0)
4( + 1) @ max
kaij k21 A +
1
cm
(1
s
:
l
4a0
i i N
i;j=1
Z T
(t)E(t)dt
39
=
l)5
;
S
c
(0) + e
c0
Finally, we get
E(S):
Z
T
S
(t)E(t)dt
CE(S);
8S
0;
by choosing , 2 , small enough and by the hypothesis l R< 1. By letting T go to
t
in…nity, one can easily see that (A0 ) is satis…ed with (t) = 0 (s)ds. This completes
the proof.
Acknowledgment. The author thanks the editor and the referees for their remarks
and suggestions to improve this paper.
References
[1] H. Levine, Some nonexistence and instability theorems for solutions of formally
parabolic equations, Archiv Rat. Mech. Anal., 51(1973), 371–386.
[2] G. Liu and H. Chen, Global and blow up of solutions for a quasilinear parabolic
system with viscoelastic and source terms, Mathematical Methods in Applied Sciences., 51(2013), 370–379.
[3] S. A. Messaoudi and B. Tellab, A general decay result in a quasilinear parabolic
system with viscoelastic term, Appl. Math. Lett., 25(2012), 443–447.
[4] S. A. Messaoudi, A note on blow up of solutions of a quasilinear heat equation
with vanishing initial energy, J. Math. Anal. Appl., 273(2002), 243–247.
[5] S. A. Messaoudi and N. E. Tatar, Uniform stabilization of solutions of a nonlinear
system of viscoelastic equations, Appl. Anal., 87(2008), 247–263.
[6] S. Berrimi and S. A. Messaoudi, A decay result for a quasilinear parabolic system,
Progr. Nonlinear Di¤erential Equations Appl., 53(2005), 43–50.
[7] J. A. Nohel, Nonlinear Volterra equations for heat ‡ow in materials with memory,
Integral and functional di¤erential equations (Proc. Conf., West Virginia Univ.,
Morgantown, W. Va., 1979), pp. 3–82, Lecture Notes in Pure and Appl. Math.,
67, Dekker, New York, 1981.
64
Energy Decay Result
[8] G. D. Prato and M. Iannelli, Existence and regularity for a class of integrodi¤erential equations of parabolic type, J. Math. Anal. Appl., 112(1985), 36–55.
[9] H. M. Yin, On parabolic Volterra equations in several space dimensions, SIAM J.
Math. Anal., 22(1991), 1723–1737.
[10] Y. Boukhatem and B. Benabderrahmane, Existence and decay of solutions for a
viscoelastic wave equation with acoustic boundary conditions, Nonlinear Analysis.,
97(2014), 191–209.