Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms
The global existence and nonexistence of solutions for a system of nonlinear wave equations with degenerate damping and source terms supplemented with initial and Dirichlet boundary conditions was shown by Rammaha and Sakuntasathien in a bounded domain Ω ⊂ Rⁿ, n = 1, 2, 3, in the case where the init...
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nasplib_isofts_kiev_ua-123456789-1653232025-02-10T01:46:45Z Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms Глобальне неіснування розв'язків системи нелінійних в'язкоеластичних хвильових рівнянь в виродженим затуханням та джерелами Ouchenane, D. Zennir, Kh. Bayoud, M. Статті The global existence and nonexistence of solutions for a system of nonlinear wave equations with degenerate damping and source terms supplemented with initial and Dirichlet boundary conditions was shown by Rammaha and Sakuntasathien in a bounded domain Ω ⊂ Rⁿ, n = 1, 2, 3, in the case where the initial energy is negative. A global nonexistence result on the solution with positive initial energy for a system of viscoelastic wave equations with nonlinear damping and source terms was obtained by Messaoudi and Said-Houari. Our result extends these previous results. We prove that the solutions of a system of wave equations with viscoelastic term, degenerate damping, and strong nonlinear sources acting in both equations at the same time are globally nonexisting provided that the initial data are sufficiently large in a bounded domain Ω of Rⁿ, n ≥ 1, the initial energy is positive, and the strongly nonlinear functions f₁ and f₂ satisfy the appropriate conditions. The main tool of the proof is based on the methods used by Vitillaro and developed by Said-Houari. Глобальне існування та неіснування розв'язків системи нєлінійних хвильових рівнянь із виродженим затуханням та джерелами, доповненої початковими умовами та граничними умовами Діріхле, було встановлено Rammaha та Sakuntasathien у обмеженій області Ω ⊂ Rⁿ , n = 1, 2, 3, при від'ємній початковій енергії. Результат про глобальне неіснування розв'язку системи нелінійних в'язкоеластичних хвильових рівнянь із нелінійним затуханням та джерелами при додатній початковій енергії було отримано у роботі Messaoudi та Said-Houari. Наш результат узагальнює ці попередні результати. Доведено, що розв'язки системи хвильових рівнянь із в'язкоеластичним членом, виродженим затуханням та сильно нелінійними джерелами, що діють одночасно в обох рівняннях, глобально не існують, якщо початкові дані є достатньо великими в обмеженій області Ω в Rⁿ , n ≥ 1, початкова енергія є додатною, а сильно нелінійні функції f₁ та f₂ задовольняють відповідні умови. Доведення базується на методах, що були використані у роботі Vitillaro та розвинуті у роботі Said-Houari. 2013 Article Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms / D. Ouchenane, Kh. Zennir, M. Bayoud // Український математичний журнал. — 2013. — Т. 65, № 5. — С. 654–669. — Бібліогр.: 29 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165323 517.98 en Український математичний журнал application/pdf Інститут математики НАН України |
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Статті Статті Ouchenane, D. Zennir, Kh. Bayoud, M. Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms Український математичний журнал |
| description |
The global existence and nonexistence of solutions for a system of nonlinear wave equations with degenerate damping and source terms supplemented with initial and Dirichlet boundary conditions was shown by Rammaha and Sakuntasathien in a bounded domain Ω ⊂ Rⁿ, n = 1, 2, 3, in the case where the initial energy is negative. A global nonexistence result on the solution with positive initial energy for a system of viscoelastic wave equations with nonlinear damping and source terms was obtained by Messaoudi and Said-Houari. Our result extends these previous results. We prove that the solutions of a system of wave equations with viscoelastic term, degenerate damping, and strong nonlinear sources acting in both equations at the same time are globally nonexisting provided that the initial data are sufficiently large in a bounded domain Ω of Rⁿ, n ≥ 1, the initial energy is positive, and the strongly nonlinear functions f₁ and f₂ satisfy the appropriate conditions. The main tool of the proof is based on the methods used by Vitillaro and developed by Said-Houari. |
| format |
Article |
| author |
Ouchenane, D. Zennir, Kh. Bayoud, M. |
| author_facet |
Ouchenane, D. Zennir, Kh. Bayoud, M. |
| author_sort |
Ouchenane, D. |
| title |
Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms |
| title_short |
Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms |
| title_full |
Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms |
| title_fullStr |
Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms |
| title_full_unstemmed |
Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms |
| title_sort |
global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms |
| publisher |
Інститут математики НАН України |
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2013 |
| topic_facet |
Статті |
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https://nasplib.isofts.kiev.ua/handle/123456789/165323 |
| citation_txt |
Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms / D. Ouchenane, Kh. Zennir, M. Bayoud // Український математичний журнал. — 2013. — Т. 65, № 5. — С. 654–669. — Бібліогр.: 29 назв. — англ. |
| series |
Український математичний журнал |
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2025-12-02T13:47:20Z |
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2025-12-02T13:47:20Z |
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| fulltext |
UDC 517.98
D. Ouchenane (Badji Mokhtar Univ., Algeria),
Kh. Zennir (Djillali Liabes Univ., Algeria),
M. Bayoud (Univ. 20 Aout 55, Algeria)
GLOBAL NONEXISTENCE OF SOLUTIONS OF A SYSTEM
OF NONLINEAR VISCOELASTIC WAVE EQUATIONS
WITH DEGENERATE DAMPING AND SOURCE TERMS
ГЛОБАЛЬНЕ НЕIСНУВАННЯ РОЗВ’ЯЗКIВ СИСТЕМИ НЕЛIНIЙНИХ
В’ЯЗКОЕЛАСТИЧНИХ ХВИЛЬОВИХ РIВНЯНЬ
IЗ ВИРОДЖЕНИМ ЗАТУХАННЯМ ТА ДЖЕРЕЛАМИ
The global existence and nonexistence of solutions of a system of nonlinear wave equations with degenerate damping and
source terms supplemented with initial and Dirichlet boundary conditions was shown by Rammaha and Sakuntasathien in
a bounded domain Ω ⊂ Rn, n = 1, 2, 3, in the case where the initial energy is negative. A global nonexistence result
on a solution with positive initial energy for a system of viscoelastic wave equations with nonlinear damping and source
terms was obtained by Messaoudi and Said-Houari. Our result extends these previous results. We prove that the solutions
of a system of wave equations with viscoelastic term, degenerate damping, and strong nonlinear sources acting in both
equations at the same time are globally nonexistent, provided that the initial data are large enough in a bounded domain
Ω of Rn, n ≥ 1, the initial energy is positive, and the strong nonlinear functions f1 and f2 satisfy appropriate conditions.
The main tool of the proof is based on methods used by Vitillaro and developed by Said-Houari.
Глобальне iснування та неiснування розв’язкiв системи нелiнiйних хвильових рiвнянь iз виродженим затуханням
та джерелами, доповненої початковими умовами та граничними умовами Дiрiхле, було встановлено Rammaha та
Sakuntasathien у обмеженiй областi Ω ⊂ Rn, n = 1, 2, 3, при вiд’ємнiй початковiй енергiї. Результат про глобальне
неiснування розв’язку системи нелiнiйних в’язкоеластичних хвильових рiвнянь iз нелiнiйним затуханням та джере-
лами при додатнiй початковiй енергiї було отримано у роботi Messaoudi та Said-Houari. Наш результат узагальнює цi
попереднi результати. Доведено, що розв’язки системи хвильових рiвнянь iз в’язкоеластичним членом, виродженим
затуханням та сильно нелiнiйними джерелами, що дiють одночасно в обох рiвняннях, глобально не iснують, якщо
початковi данi є достатньо великими в обмеженiй областi Ω в Rn, n ≥ 1, початкова енергiя є додатною, а сильно
нелiнiйнi функцiї f1 та f2 задовольняють вiдповiднi умови. Доведення базується на методах, що були використанi
у роботi Vitillaro та розвинутi у роботi Said-Houari.
1. Introduction. In this work we consider the following system of viscoelastic wave equations with
degenerate damping and strong nonlinear source terms:
utt −∆u+
t∫
0
g (t− s) ∆u (x, s) ds+
(
a |u|k + b |v|l
)
|ut|m−1ut = f1(u, v),
vtt −∆v +
t∫
0
h (t− s) ∆v (x, s) ds+
(
c |v|θ + d |u|%
)
|vt|r−1vt = f2(u, v),
(1.1)
where m, r > 0, k, l, θ, % ≥ 1 and the two functions f1 (u, v) and f2 (u, v) given by
f1(u, v) = a1|u+ v|2(ρ+1)(u+ v) + b1|u|ρu|v|(ρ+2),
f2(u, v) = a1|u+ v|2(ρ+1)(u+ v) + b1|u|(ρ+2)|v|ρv, a1, b1 > 0, ρ > −1.
(1.2)
c© D. OUCHENANE, KH. ZENNIR, M. BAYOUD, 2013
654 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
GLOBAL NONEXISTENCE OF SOLUTIONS OF A SYSTEM OF NONLINEAR VISCOELASTIC WAVE . . . 655
In (1.1), u = u (t, x) , v = v (t, x) , where x ∈ Ω is a bounded domain of Rn, n ≥ 1, with a
smooth boundary ∂Ω and t > 0, a, b, c, d > 0.
System (1.1) is supplemented with the following initial conditions:
(u(0), v(0)) = (u0, v0), (ut(0), vt(0)) = (u1, v1), x ∈ Ω, (1.3)
and boundary conditions
u(x) = v(x) = 0, x ∈ ∂Ω. (1.4)
Viscoelastic materials have properties between two types (elastic materials and viscous fluids).
This two types of materials are usually considered in basic texts on continuum mechanics. At each
material point of an elastic material the stress at the present time depends only on the present value
of the strain. On the other hand, for an incompressible viscous fluid the stress at a given point is a
function of the present value of the velocity gradient at that point. Such materials have memory: the
stress depends not only on the present values of the strain and /or velocity gradient, but also on the
entire temporal history of motion.
This type of problems arise usually in viscoelasticity, it has been considered first by Dafermos
[9], where the general decay was discussed. A related problems to (1.1) have attracted a great deal
of attention in the last decades, and many results have been appeared on the existence and long time
behavior of solutions. See in this directions [3 – 5, 8, 11, 15, 19 – 21] and references therein.
In the absence of viscoelastic term, some special cases of the single wave equations with nonlinear
damping and nonlinear source terms in the form
utt −∆u+ a|ut|m−1ut = b|u|p−1u, (1.5)
arise in quantum field theory which describe the motion of charged mesons in an electromagnetic
field. Equation (1.5) together with initial and boundary conditions of Dirichlet type, has been exten-
sively studied and results concerning existence, blow up and asymptotic behavior of smooth, as well
as weak solutions have been established by several authors over the past decades.
The study of single wave equation with the presence of different mechanisms of dissipation,
damping and nonlinear sources has been extensively studied and results concerning existence, nonex-
istence and asymptotic behavior of solutions have been established by several authors and many
results appeared in the literature over the past decades. See [2, 10, 12, 13, 16, 18, 23] and references
therein.
Concerning the system of equations, in [1] Agre and Rammaha studied the following system:
utt −∆u+ |ut|m−1ut = f1(u, v),
vtt −∆v + |vt|r−1vt = f2(u, v),
in Ω × (0, T ) with initial and boundary conditions and the nonlinear functions f1 and f2 satisfying
appropriate conditions. They proved under some restrictions on the parameters and the initial data
many results on the existence of a weak solution. They also showed that any weak solution with
negative initial energy blows up in finite time using the same techniques as in [10].
In [23], author considered the same problem treated in [1], and he improved the blow up result
obtained in [1], for a large class of initial data in which the initial energy can take positive values.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
656 D. OUCHENANE, KH. ZENNIR, M. BAYOUD
In the work [18], authors considered the nonlinear viscoelastic system:
utt −∆u+
t∫
0
g(t− s)∆u(x, s)ds+ |ut|m−1 ut = f1 (u, v) ,
vtt −∆v +
t∫
0
h(t− s)∆v(x, s)ds+ |vt|r−1 vt = f2 (u, v) ,
x ∈ Ω, t > 0, (1.6)
where
f1(u, v) = a|u+ v|2(ρ+1)(u+ v) + b|u|ρu|v|(ρ+2),
f2(u, v) = a|u+ v|2(ρ+1)(u+ v) + b|u|(ρ+2)|v|ρv,
and they prove a global nonexistence theorem for certain solutions with positive initial energy, the
main tool of the proof is a method used in [23].
Concerning the study of decay of solutions of systems of evolution equations, let us mention the
work of B. Said-Houari, S. A. Messaoudi and A. Guesmia in [25], where they treated the nonlinear
viscoelastic system in (1.6) and under some restrictions on the nonlinearity of the damping and the
source terms, they prove that, for certain class of relaxation functions and for some restrictions on
the initial data, the rate of decay of the total energy depends on those of the relaxation functions.
Recently, in [22] M. A. Rammaha and Sawanya Sakuntasathien focus on the global well-
posedness of the system of nonlinear wave equations
utt −∆u+
(
d |u|k + e |v|l
)
|ut|m−1ut = f1(u, v),
vtt −∆v +
(
d′ |v|θ + e′ |u|ρ
)
|vt|r−1vt = f2(u, v),
in a bounded domain Ω ⊂ Rn, n = 1, 2, 3, with Dirichlet boundary conditions. The nonlinearities
f1(u, v) and f2(u, v) act as a strong source in the system. Under some restriction on the parameters
in the system, they obtain several results on the existence and uniqueness of solutions. In addition,
they prove that weak solutions blow up in finite time whenever the initial energy is negative and the
exponent of the source term is more dominant than the exponents of both damping terms. This type
of problems are not only important from the theoretical point of view, but also arise in many physical
applications and describe a great deal of models in applied science, many questions in physics and
engineering give rise to problems that deal with system of nonlinear wave equations.
The presente paper is organized as follows:
In Section 2 we introduce and present some notation and prepare some material needed for our
proof. In Section 3 we state and prove our main result, where we prove a global nonexistence (blow
up for all time) of solution of system (1.1) – (1.4) with positive initial energy for some conditions on
the functions f1 and f2.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
GLOBAL NONEXISTENCE OF SOLUTIONS OF A SYSTEM OF NONLINEAR VISCOELASTIC WAVE . . . 657
2. Assumptions, notations and preliminaries. In this section, we introduce and present some
notations and some technical lemmas to be used throughout this paper. The constants ci, i =
= 0, 1, 2, . . . , used throughout this paper are positive generic constants, which may be different
in various occurrences.
We assume that the relaxation functions g, h : R+ −→ R+ are of class C1 and nonnicreasing
differentiable satisfying:
1−
∞∫
0
g(s)ds = l > 0, g(t) ≥ 0, g′(t) ≤ 0,
1−
∞∫
0
h(s)ds = k > 0, h(t) ≥ 0, h′(t) ≤ 0,
t ≥ 0. (2.1)
Lemma 2.1. There exists a function F (u, v) such that
F (u, v) =
1
2 (ρ+ 2)
[uf1 (u, v) + vf2 (u, v)] =
1
2 (ρ+ 2)
[
a1 |u+ v|2(ρ+2) + 2b1 |uv|ρ+2
]
≥ 0,
where
∂F
∂u
= f1(u, v),
∂F
∂v
= f2(u, v).
We introduce the “modified ” energy functional E(t) associated to our system
2E (t) = ‖ut‖22 + ‖vt‖22 + J (u, v)− 2
∫
Ω
F (u, v) dx, (2.2)
where
J (u, v) =
1−
t∫
0
g (s) ds
‖∇u‖22 +
1−
t∫
0
h (s) ds
‖∇v‖22 + (g ◦ ∇u) + (h ◦ ∇v)
and
(g ◦ u) (t) =
t∫
0
g (t− τ) ‖u (t)− u (τ)‖22 dτ,
(h ◦ v) (t) =
t∫
0
h (t− τ) ‖v (t)− v (τ)‖22 dτ.
Let us point out that the integral
∫
Ω
F (u, v) dx in (2.2) makes sense because H1
0 (Ω) ⊂
⊂ L2(ρ+2) (Ω) , for
− 1 < ρ if n = 1, 2,
− 1 < ρ ≤ 4− n
n− 2
if n ≥ 3.
(2.3)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
658 D. OUCHENANE, KH. ZENNIR, M. BAYOUD
Lemma 2.2 [23]. There exist two positive constants c0 and c1 such that
c0
2 (ρ+ 2)
(
|u|2(ρ+2) + |v|2(ρ+2)
)
≤ F (u, v) ≤ c1
2 (ρ+ 2)
(
|u|2(ρ+2) + |v|2(ρ+2)
)
.
The following technical lemma will play an important role in the sequel.
Lemma 2.3. Suppose that (2.3) holds. Then there exists η > 0 such that for any (u, v) ∈
∈ H1
0 (Ω)×H1
0 (Ω) the inequality
2(ρ+ 2)
∫
Ω
F (u, v) dx ≤ η
(
‖∇u‖22 + ‖∇v‖22
)ρ+2
(2.4)
holds.
Direct computations, using Minkowski, Hôlder’s and Young’s inequalities and the embedding
H1
0 (Ω) ↪→ L2(ρ+2)(Ω) yields the proof of this previous Lemma 2.3.
Lemma 2.4. Let ν > 0 be a real positive number and L(t) be a solution of the ordinary
differential inequality
dL(t)
dt
≥ ξL1+ν(t) (2.5)
defined in [0,∞).
If L(0) > 0, then the solution ceases to exist for t ≥ L(0)−νξ−νν−1.
Proof. Direct integration of (2.5) gives
L−ν (0)− L−ν (t) ≥ ξνt.
Thus we obtain the following estimate:
Lν (t) ≥
[
L−ν (0)− ξνt
]−1
. (2.6)
It is clear that the right-hand side of (2.6) is unbounded when
ξνt = L−ν (0) .
Lemma 2.4 is proved.
3. Blow up results.
Lemma 3.1. Suppose that (2.3) holds. Let (u, v) be the solution of the system (1.1) – (1.4) then
the energy functional is a nonincreasing function, that is for all t ≥ 0,
E′ (t) = −
∫
Ω
(
|u (t)|k + |v (t)|l
)
|ut (t)|m+1 dx−
∫
Ω
(
|v (t)|θ + |u (t)|%
)
|vt (t)|r+1 dx+
+
1
2
(
g′ ◦ ∇u
)
+
1
2
(
h′ ◦ ∇v
)
− 1
2
g (s) ‖∇u‖22 −
1
2
h (s) ‖∇v‖22 . (3.1)
Our main result reads as follows:
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
GLOBAL NONEXISTENCE OF SOLUTIONS OF A SYSTEM OF NONLINEAR VISCOELASTIC WAVE . . . 659
Theorem 3.1. Suppose that (2.3) holds. Assume further that
ρ > max
(
k +m− 3
2
,
l +m− 3
2
,
θ + r − 3
2
,
%+ r − 3
2
)
, (3.2)
and that there exists p, such that 2 < p < 2 (ρ+ 2), for which
max
∞∫
0
g (s) ds,
∞∫
0
h (s) ds
<
(p/2)− 1
(p/2)− 1 + (1/2p)
, (3.3)
holds. Then any solution of problem (1.1) – (1.4), with initial data satisfying
‖∇u0‖22 + ‖∇v0‖22 > α2
1 and E (0) < E2
blows up for all time, where the constants α1 and E2 are defined in (3.4).
We take a1 = b1 = 1 for convenience. We introduce the following:
B = η
1
2(ρ+2) , α1 = B
− (ρ+2)
(ρ+1) , E1 =
(
1
2
− 1
2 (ρ+ 2)
)
α2
1, (3.4)
E2 =
(
1
p
− 1
2 (ρ+ 2)
)
α2
1,
where η is the optimal constant in (2.4).
Lemma 3.2 [23]. Suppose that (2.3), (3.2) and (3.3) hold. Let (u, v) be a solution of (1.1) –
(1.4). Assume further that E (0) < E2 and
‖∇u0‖22 + ‖∇v0‖22 > α2
1.
Then there exists a constant α2 > α1 such that
J (t) > α2
2, (3.5)
and
2(ρ+ 2)
∫
Ω
F (u, v) dx ≥ (Bα2)2(ρ+2) ∀t ≥ 0.
Proof of Theorem 3.1. We suppose that the solution exists for all time and we reach to a
contradiction. For this purpose, we set
H (t) = E2 − E (t) . (3.6)
By using the definition of H(t), we get
H ′(t) = −E′(t) =
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660 D. OUCHENANE, KH. ZENNIR, M. BAYOUD
=
∫
Ω
(
|u (t)|k + |v (t)|l
)
|ut (t)|m+1 dx+
∫
Ω
(
|v (t)|θ + |u (t)|%
)
|vt (t)|r+1 dx−
−1
2
(
g′ ◦ ∇u
)
− 1
2
(
h′ ◦ ∇v
)
+
1
2
g (s) ‖∇u‖22 +
1
2
h (s) ‖∇v‖22 ≥ 0 ∀t ≥ 0.
Consequently, since E′(t) is absolutely continuous
H(0) = E2 − E(0) > 0.
Then
0 < H (0) ≤ H (t) = E2 −
1
2
(
‖ut‖22 + ‖vt‖22
)
− J (t)
2
+
+
1
2 (ρ+ 2)
[
‖u+ v‖2(ρ+2)
2(ρ+2) + 2‖uv‖ρ+2
ρ+2
]
.
From (2.1) and (3.5), we obtain
E2 −
1
2
(
‖ut‖22 + ‖vt‖22
)
− J (t)
2
< E2 −
1
2
α2
2 < E2 −
1
2
α2
1 <
< E1 −
1
2
α2
1 = − 1
2 (ρ+ 2)
α2
1 < 0 ∀t ≥ 0.
Hence, by the above inequality and Lemma 2.2, we have for all t ≥ 0
0 < H (0) ≤ H (t) ≤ 1
2 (ρ+ 2)
[
‖u+ v‖2(ρ+2)
2(ρ+2) + 2‖uv‖ρ+2
ρ+2
]
≤
≤ c1
2 (ρ+ 2)
(
‖u‖2(ρ+2)
2(ρ+2) + ‖v‖2(ρ+2)
2(ρ+2)
)
.
Then we define the functional
M(t) =
1
2
∫
Ω
(
u2 + v2
)
(x, t) dx.
We introduce
L (t) = H1−σ (t) + εM ′(t), (3.7)
for ε small to be chosen later and
0 < σ ≤ min
{
1
2
,
2ρ+ 3− (k +m)
2 (m+ 1) (ρ+ 2)
,
2ρ+ 3− (l +m)
2 (m+ 1) (ρ+ 2)
,
2ρ+ 3− (%+ r)
2 (r + 1) (ρ+ 2)
,
2ρ+ 3− (θ + r)
2 (r + 1) (ρ+ 2)
,
2ρ+ 2
4 (ρ+ 2)
}
. (3.8)
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We will show that L (t) satisfies a differential inequality in Lemma 2.4. By taking a derivative of
(3.7) and using (1.1), we obtain
L′ (t) = (1− σ)H−σ (t)H ′ (t) + ε
(
‖ut‖22 + ‖vt‖22
)
− ε
(
‖∇u‖22 + ‖∇v‖22
)
−
− ε
∫
Ω
u
(
|u (t)|k + |v (t)|l
)
|ut|m−1 utdx− ε
∫
Ω
v
(
|v (t)|θ + |u (t)|%
)
|vt|r−1 vtdx+
+ ε
∫
Ω
(uf1 (u, v) + vf2 (u, v)) dx+ ε
∫
Ω
∇u (t)
t∫
0
g (t− s)∇u (τ) dxds+
+ ε
∫
Ω
∇v (t)
t∫
0
h (t− s)∇v (τ) dxds.
Then
L′ (t) = (1− σ)H−σ (t)H ′ (t) + ε
(
‖ut‖22 + ‖vt‖22
)
− ε
(
‖∇u‖22 + ‖∇v‖22
)
−
− ε
∫
Ω
u
(
|u (t)|k + |v (t)|l
)
|ut|m−1 utdx− ε
∫
Ω
v
(
|v (t)|θ + |u (t)|%
)
|vt|r−1 vtdx +
+ ε
(
‖u+ v‖2(ρ+2)
2(ρ+2) + 2‖uv‖ρ+2
ρ+2
)
+ ε
t∫
0
g (s) ds
‖∇u‖22 +
t∫
0
h (s) ds
‖∇v‖22 +
+ ε
t∫
0
g (t− s)
∫
Ω
∇u (t) . [∇u (τ)−∇u (t)] dxds +
+ ε
t∫
0
h (t− s)
∫
Ω
∇v (t) . [∇v (τ)−∇v (t)] dxds.
By Cauchy – Schwarz and Young’s inequalities, we estimate
t∫
0
g (t− s)
∫
Ω
∇u (t) . [∇u (τ)−∇u (t)] dxds ≤
≤
t∫
0
g (t− s) ‖∇u‖2 ‖∇u (τ)−∇u (t)‖2 dτ ≤
≤ λ (g ◦ ∇u) +
1
4λ
t∫
0
g (s) ds
‖∇u‖22, λ > 0,
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662 D. OUCHENANE, KH. ZENNIR, M. BAYOUD
and
t∫
0
h (t− s)
∫
Ω
∇v (t) . [∇v (τ)−∇v (t)] dxds ≤
≤ λ (h ◦ ∇v) +
1
4λ
t∫
0
h (s) ds
‖∇v‖22, λ > 0.
Adding and substituting pE(t), using the definition of H(t), E2 lead to
L′ (t) ≥ (1− σ)H−σ (t)H ′ (t) + ε
(
1 +
p
2
) (
‖ut‖22 + ‖vt‖22
)
+
+ε
(p
2
− λ
)
[(g ◦ ∇u) + (h ◦ ∇v)] + pεH (t)− pεE2−
−ε
∫
Ω
u
(
|u (t)|k + |v (t)|l
)
|ut|m−1 utdx− ε
∫
Ω
v
(
|v (t)|θ + |u (t)|%
)
|vt|r−1 vtdx+
+ε
(
1− p
2 (ρ+ 2)
)(
‖u+ v‖2(ρ+2)
2(ρ+2) + 2‖uv‖ρ+2
ρ+2
)
+
+ε
(p
2
− 1
)
−
(
p
2
− 1 +
1
4λ
) ∞∫
0
g (s) ds
‖∇u‖22+
+ε
(p
2
− 1
)
−
(
p
2
− 1 +
1
4λ
) ∞∫
0
h (s) ds
‖∇v‖22, (3.9)
for some λ such that
a1 =
p
2
− λ > 0, a2 =
(p
2
− 1
)
−
(
p
2
− 1 +
1
4λ
)
max
∞∫
0
g (s) ds,
∞∫
0
h (s) ds
> 0.
Then, estimate (3.9) becomes
L′ (t) ≥ (1− σ)H−σ (t)H ′ (t) + ε
(
1 +
p
2
)(
‖ut‖22 + ‖vt‖22
)
+
+εa1
[
(g ◦ ∇u) + (h ◦ ∇v)
]
+ pεH (t)− pεE2−
−ε
∫
Ω
u
(
|u (t)|k + |v (t)|l
)
|ut|m−1 utdx−
−ε
∫
Ω
v
(
|v (t)|θ + |u (t)|%
)
|vt|r−1 vtdx+
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GLOBAL NONEXISTENCE OF SOLUTIONS OF A SYSTEM OF NONLINEAR VISCOELASTIC WAVE . . . 663
+ε
(
1− p
2 (ρ+ 2)
)(
‖u+ v‖2(ρ+2)
2(ρ+2) + 2‖uv‖ρ+2
ρ+2
)
+ εa2
(
‖∇u‖22 + ‖∇v‖22
)
. (3.10)
By taking c3 = 1 − p
ρ+ 2
− 2E2 (Bα2)−2(ρ+2) > 0, since α2 > B
−2(ρ+2)
ρ+1 . Therefore, (3.10) takes
the form
L′ (t) ≥ (1− σ)H−σ (t)H ′ (t) + ε
(
1 +
p
2
) (
‖ut‖22 + ‖vt‖22
)
+
+εa1 [(g ◦ ∇u) + (h ◦ ∇v)] + εa2
(
‖∇u‖22 + ‖∇v‖22
)
+ pεH (t) +
+εc3
(
‖u+ v‖2(ρ+2)
2(ρ+2) + 2‖uv‖ρ+2
ρ+2
)
− ε
∫
Ω
u
(
|u (t)|k + |v (t)|l
)
|ut|m−1 utdx−
−ε
∫
Ω
v
(
|v (t)|θ + |u (t)|%
)
|vt|r−1 vtdx.
We use the Young’s inequality as follows:
XY ≤ δαXα
α
+
δ−βY β
β
,
where X,Y ≥ 0, δ > 0, and α, β ∈ R+ such that
1
α
+
1
β
= 1, we get for all δ1 > 0
∣∣∣u |ut|m−1 ut
∣∣∣ ≤ δm+1
1
m+ 1
|u|m+1 +
m
m+ 1
δ
−(m+1)/m
1 |ut|m+1
and ∫
Ω
(
|u (t)|k + |v (t)|l
) ∣∣∣u |ut|m−1 ut
∣∣∣ dx ≤ δm+1
1
m+ 1
∫
Ω
(
|u (t)|k + |v (t)|l
)
|u|m+1 dx+
+
m
m+ 1
δ
−(m+1)/m
1
∫
Ω
(
|u (t)|k + |v (t)|l
)
|ut|m+1 dx.
Similarly, for all δ2 > 0∣∣∣v |vt|r−1 vt
∣∣∣ ≤ δr+1
2
r + 1
|v|r+1 +
r
r + 1
δ
−(r+1)/r
2 |vt|r+1 ,
which gives∫
Ω
(
|v (t)|θ + |u (t)|%
) ∣∣∣v |vt|r−1 vt
∣∣∣ dx ≤ δr+1
2
r + 1
∫
Ω
(
|v (t)|θ + |u (t)|%
)
|v|r+1 dx+
+
r
r + 1
δ
−(r+1)/r
2
∫
Ω
(
|v (t)|θ + |u (t)|%
)
|vt|r+1 dx.
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664 D. OUCHENANE, KH. ZENNIR, M. BAYOUD
Then
L′ (t) ≥ (1− σ)H−σ (t)H ′ (t) + ε
(
1 +
p
2
) (
‖ut‖22 + ‖vt‖22
)
+
+εa1 [(g ◦ ∇u) + (h ◦ ∇v)] + εa2
(
‖∇u‖22 + ‖∇v‖22
)
+ pεH (t) +
+εc3
(
‖u+ v‖2(ρ+2)
2(ρ+2) + 2‖uv‖ρ+2
ρ+2
)
− ε δ
m+1
1
m+ 1
∫
Ω
(
|u (t)|k + |v (t)|l
)
|u|m+1 dx−
−ε m
m+ 1
δ
−(m+1)/m
1
∫
Ω
(
|u (t)|k + |v (t)|l
)
|ut|m+1 dx−
−ε δ
r+1
2
r + 1
∫
Ω
(
|v (t)|θ + |u (t)|%
)
|v|r+1 dx−
−ε r
r + 1
δ
−(r+1)/r
2
∫
Ω
(
|v (t)|θ + |u (t)|%
)
|vt|r+1 dx.
Let us choose δ1 and δ2 such that
δ
−(m+1)/m
1 = M1H (t)−σ , δ
−(r+1)/r
2 = M2H (t)−σ , (3.11)
for M1 and M2 large constants to be fixed later. Thus, by using (3.11), we get
L′ (t) ≥ ((1− σ)−Mε)H−σ (t)H ′ (t) + ε
(
1 +
p
2
) (
‖ut‖22 + ‖vt‖22
)
+
+εa1 [(g ◦ ∇u) + (h ◦ ∇v)] + εa2
(
‖∇u‖22 + ‖∇v‖22
)
+ pεH (t) +
+εc3
(
‖u+ v‖2(ρ+2)
2(ρ+2) + 2‖uv‖ρ+2
ρ+2
)
−
−εM−m1 Hσm (t)
∫
Ω
(
|u (t)|k + |v (t)|l
)
|u|m+1 dx−
−ε m
m+ 1
δ
−(m+1)/m
1
∫
Ω
(
|u (t)|k + |v (t)|l
)
|ut|m+1 dx−
−εM−r2 Hσr (t)
∫
Ω
(
|v (t)|θ + |u (t)|%
)
|v|r+1 dx−
−ε r
r + 1
δ
−(r+1)/r
2
∫
Ω
(
|v (t)|θ + |u (t)|%
)
|vt|r+1 dx, (3.12)
where M = m/(m+ 1)M1 + r/(r + 1)M2. Consequently we have
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GLOBAL NONEXISTENCE OF SOLUTIONS OF A SYSTEM OF NONLINEAR VISCOELASTIC WAVE . . . 665∫
Ω
(
|u (t)|k + |v (t)|l
)
|u|m+1 dx = ‖u‖k+m+1
k+m+1 +
∫
Ω
|v|l |u|m+1 dx
and ∫
Ω
(
|v (t)|θ + |u (t)|%
)
|v|r+1 dx = ‖v‖θ+r+1
θ+r+1 +
∫
Ω
|u|% |v|r+1 dx.
Also by using Young’s inequality, we obtain∫
Ω
|v|l |u|m+1 ≤ l
l +m+ 1
δ
(l+m+1)/l
1 ‖v‖l+m+1
l+m+1 +
m+ 1
l +m+ 1
δ
−(l+m+1)/(m+1)
1 ‖u‖l+m+1
l+m+1 ,
∫
Ω
|u|% |v|r+1 ≤ %
%+ r + 1
δ
(%+r+1)/%
2 ‖u‖%+r+1
%+r+1 +
r + 1
%+ r + 1
δ
−(%+r+1)/(r+1)
2 ‖v‖%+r+1
%+r+1 .
Consequently
Hσm (t)
∫
Ω
(
|u (t)|k + |v (t)|l
)
|u|m+1 dx =
= Hσm (t) ‖u‖k+m+1
k+m+1 +
l
l +m+ 1
δ
(l+m+1)/l
1 Hσm (t) ‖v‖l+m+1
l+m+1 +
+
m+ 1
l +m+ 1
δ
−(l+m+1)/(m+1)
1 Hσm (t) ‖u‖l+m+1
l+m+1 (3.13)
and
Hσr (t)
∫
Ω
(
|v (t)|θ + |u (t)|%
)
|v|r+1 dx =
= Hσr (t) ‖v‖θ+r+1
θ+r+1 +
%
%+ r + 1
δ
%+r+1
%
2 Hσr (t) ‖u‖%+r+1
%+r+1 +
+
r + 1
%+ r + 1
δ
− (%+r+1)
r+1
2 Hσr (t) ‖v‖%+r+1
%+r+1 . (3.14)
Since (3.2) holds, we obtain by using (3.8)
Hσm (t) ‖u‖k+m+1
k+m+1 ≤ c5
(
‖u‖2σm(ρ+2)+k+m+1
2(ρ+2) + ‖v‖2σm(ρ+2)
2(ρ+2) ‖u‖k+m+1
k+m+1
)
,
Hσr (t) ‖v‖θ+r+1
θ+r+1 ≤ c6
(
‖v‖2σr(ρ+2)+θ+r+1
2(ρ+2) + ‖u‖2σr(ρ+2)
2(ρ+2) ‖v‖θ+r+1
θ+r+1
)
.
(3.15)
This implies
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666 D. OUCHENANE, KH. ZENNIR, M. BAYOUD
l
l +m+ 1
δ
l+m+1
l
1 Hσm (t) ‖v‖l+m+1
l+m+1 ≤
≤ c7
l
l +m+ 1
δ
l+m+1
l
1
(
‖v‖2σm(ρ+2)+l+m+1
2(ρ+2) + ‖u‖2σm(ρ+2)
2(ρ+2) ‖v‖l+m+1
l+m+1
)
,
%
%+ r + 1
δ
%+r+1
%
2 Hσr (t) ‖u‖%+r+1
%+r+1 ≤
≤ c8
%
%+ r + 1
δ
%+r+1
%
2
(
‖u‖2σr(ρ+2)+%+r+1
2(ρ+2) + ‖v‖2σr(ρ+2)
2(ρ+2) ‖u‖%+r+1
%+r+1
)
.
(3.16)
By using (3.8) and the algebraic inequality
zν ≤ (z + 1) ≤
(
1 +
1
a
)
(z + a) ∀z ≥ 0, 0 < ν ≤ 1, a ≥ 0, (3.17)
we have, for all t ≥ 0,
‖u‖2σm(ρ+2)+k+m+1
2(ρ+2) ≤ d
(
‖u‖2(ρ+2)
2(ρ+2) +H (0)
)
≤ d
(
‖u‖2(ρ+2)
2(ρ+2) +H (t)
)
,
‖v‖2σr(ρ+2)+θ+r+1
2(ρ+2) ≤ d
(
‖v‖2(ρ+2)
2(ρ+2) +H (t)
)
∀t ≥ 0,
(3.18)
where d = 1 + 1/H (0) . Similarly
‖v‖2σm(ρ+2)+l(m+1)
2(ρ+2) ≤ d
(
‖v‖2(ρ+2)
2(ρ+2) +H (0)
)
≤ d
(
‖v‖2(ρ+2)
2(ρ+2) +H (t)
)
,
‖u‖2σr(ρ+2)+%(r+1)
2(ρ+2) ≤ d
(
‖u‖2(ρ+2)
2(ρ+2) +H (t)
)
∀t ≥ 0.
(3.19)
Also, since
(X + Y )s ≤ C (Xs + Y s) , X, Y ≥ 0, s > 0, (3.20)
by using (3.8) and (3.17) we conclude
‖v‖2σm(ρ+2)
2(ρ+2) ‖u‖k+m+1
k+m+1 ≤ c9
(
‖v‖2(ρ+2)
2(ρ+2) + ‖u‖2(ρ+2)
k+m+1
)
≤ c10
(
‖v‖2(ρ+2)
2(ρ+2) + ‖u‖2(ρ+2)
2(ρ+2)
)
, (3.21)
similarly
‖u‖2σr(ρ+2)
2(ρ+2) ‖v‖θ+r+1
θ+r+1 ≤ c11
(
‖u‖2(ρ+2)
2(ρ+2) + ‖v‖2(ρ+2)
2(ρ+2)
)
, (3.22)
‖u‖2σm(ρ+2)
2(ρ+2) ‖v‖l+m+1
l+m+1 ≤ c12
(
‖u‖2(ρ+2)
2(ρ+2) + ‖v‖2(ρ+2)
2(ρ+2)
)
, (3.23)
and
‖v‖2σr(ρ+2)
2(ρ+2) ‖u‖%+r+1
%+r+1 ≤ c13
(
‖v‖2(ρ+2)
2(ρ+2) + ‖u‖2(ρ+2)
2(ρ+2)
)
. (3.24)
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GLOBAL NONEXISTENCE OF SOLUTIONS OF A SYSTEM OF NONLINEAR VISCOELASTIC WAVE . . . 667
Taking into account (3.13) – (3.24), then (3.12) takes the form
L′ (t) ≥ ((1− σ)−Mε)H−σ (t)H ′ (t) + 2ε
(
‖ut‖22 + ‖vt‖22
)
+
+ε
[
2− CM−m1
(
1 +
l
l +m+ 1
δ
l+m+1
l
1 +
m+ 1
l +m+ 1
δ
− (l+m+1)
m+1
1
)
−
− CM−r2
(
1 +
%
%+ r + 1
δ
%+r+1
%
2 +
r + 1
%+ r + 1
δ
− (%+r+1)
r+1
2
)]
H (t) +
+ε
[
c4 − CM−m1
(
1 +
l
l +m+ 1
δ
l+m+1
l
1 +
m+ 1
l +m+ 1
δ
− (l+m+1)
m+1
1
)
−
− CM−r2
(
1 +
%
%+ r + 1
δ
%+r+1
%
2 +
r + 1
%+ r + 1
δ
− (%+r+1)
r+1
2
)](
‖u‖2(ρ+2)
2(ρ+2) + ‖v‖2(ρ+2)
2(ρ+2)
)
. (3.25)
At this point, and for large values of M1 and M2, we can find positive constants Λ1 and Λ2 such that
(3.25) becomes
L′ (t) ≥ ((1− σ)−Mε)H−σ (t)H ′ (t) + 2ε
(
‖ut‖22 + ‖vt‖22
)
+
+εΛ1
(
‖u (t)‖2(ρ+2)
2(ρ+2) + ‖v (t)‖2(ρ+2)
2(ρ+2)
)
+ εΛ2H (t) . (3.26)
Once M1 and M2 are fixed (hence, Λ1 and Λ2), we pick ε small enough so that ((1− σ)−Mε) ≥ 0
and
L (0) = H1−σ (0) +
∫
Ω
[u0.ut + v0.vt] dx > 0.
Consequently, there exists Γ > 0 such that (3.26) becomes
L′ (t) ≥ εΓ
(
H (t) + ‖ut‖22 + ‖vt‖22 + ‖u‖2(ρ+2)
2(ρ+2) + ‖v‖2(ρ+2)
2(ρ+2)
)
. (3.27)
Thus, we have L (t) ≥ L (0) > 0, for all t ≥ 0.
Next, by Holder’s and Young’s inequalities, we estimate∫
Ω
u.ut (x, t) dx+
∫
Ω
v.vt (x, t) dx
1/(1−σ)
≤
≤ C
(
‖u‖τ/(1−σ)
2(ρ+2) + ‖ut‖s/(1−σ)
2 + ‖v‖τ/(1−σ)
2(ρ+2) + ‖vt‖s/(1−σ)
2
)
, (3.28)
for
1
τ
+
1
s
= 1. We takes s = 2 (1− σ) , to get
τ
1− σ
=
2
1− 2σ
. By using (3.6) and (3.17) we get
‖u‖2/(1−2σ)
2(ρ+2) ≤ d
(
‖u‖2(ρ+2)
2(ρ+2) +H (t)
)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
668 D. OUCHENANE, KH. ZENNIR, M. BAYOUD
and
‖v‖2/(1−2σ)
2(ρ+2) ≤ d
(
‖v‖2(ρ+2)
2(ρ+2) +H (t)
)
∀t ≥ 0.
Therefore, (3.28) becomes∫
Ω
uut (x, t) dx+
∫
Ω
vvt (x, t) dx
1/(1−σ)
≤
≤ c14
(
‖u‖2(ρ+2)
2(ρ+2) + ‖v‖2(ρ+2)
2(ρ+2) + ‖ut‖22 + ‖vt‖22 +H (t)
)
∀t ≥ 0.
Also, by noting that
L1/(1−σ) (t) =
H1−σ (t) + ε
∫
Ω
(u.ut + v.vt) (x, t) dx
1/(1−σ)
≤
≤ c15
H (t) +
∣∣∣∣∣∣
∫
Ω
(u.ut (x, t) + v.vt (x, t)) dx
∣∣∣∣∣∣
1/(1−σ)
≤
≤ c16
[
H (t) + ‖u‖2(ρ+2)
2(ρ+2) + ‖v‖2(ρ+2)
2(ρ+2) + ‖ut‖22 + ‖vt‖22
]
∀t ≥ 0, (3.29)
and combining with (3.29) and (3.27), we arrive at
L′ (t) ≥ a0L
1/(1−σ) (t) ∀t ≥ 0. (3.30)
Finally, a simple integration of (3.30) gives the desired result.
Acknowledgment. The authors are grateful to Prof. Hocine Sissaoui of the university of Annaba,
Algeria, for fruitful discussions and helpful comments.
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Received 16.11.11,
after revision — 10.05.12
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