Journal of Inequalities in Pure and
Applied Mathematics
ENERGY DECAY OF SOLUTIONS OF A WAVE EQUATION OF
p−LAPLACIAN TYPE WITH A WEAKLY NONLINEAR DISSIPATION
volume 7, issue 1, article 15,
2006.
ABBÈS BENAISSA AND SALIMA MIMOUNI
Université Djillali Liabès
Faculté des Sciences
Déepartement de Mathématiques
B. P. 89, Sidi Bel Abbès 22000, ALGERIA.
Received 03 November, 2005;
accepted 15 November, 2005.
Communicated by: C. Bandle
EMail: benaissa_abbes@yahoo.com
EMail: bbsalima@yahoo.fr
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
329-05
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Abstract
In this paper we study decay properties of the solutions to the wave equation of
p−Laplacian type with a weak nonlinear dissipative.
2000 Mathematics Subject Classification: 35B40, 35L70.
Key words: Wave equation of p−Laplacian type, Decay rate.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Preliminaries and Main Results . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
Abbès Benaissa and
Salima Mimouni
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
1.
Introduction
We consider the initial boundary problem for the nonlinear wave equation of
p−Laplacian type with a weak nonlinear dissipation of the type

0 l−2 0 0
0

 (|u | u ) − ∆p u + σ(t)g(u ) = 0 in Ω × [0, +∞[,

u = 0 on ∂Ω × [0, +∞[,
(P)



u(x, 0) = u0 (x), u0 (x, 0) = u1 (x) in Ω.
where ∆p u = div(|∇x u|p−2 ∇x u), p, l ≥ 2, g : R → R is a continuous nondecreasing function and σ is a positive function.
When p = 2, l = 2 and σ ≡ 1, for the case g(x) = δx (δ > 0), Ikehata
and Suzuki [5] investigated the dynamics, showing that for sufficiently small
initial data (u0 , u1 ), the trajectory (u(t), u0 (t)) tends to (0, 0) in H01 (Ω) × L2 (Ω)
as t → +∞. When g(x) = δ|x|m−1 x (m ≥ 1), Nakao [8] investigated the
decay property of the problem (P ). In [8] the author has proved the existence
of global solutions to the problem (P ).
For the problem (P ) with σ ≡ 1, l = 2, when g(x) = δ|x|m−1 x (m ≥ 1),
p
Yao [1] proved that the energy decay rate is E(t) ≤ (1 + t)− (mp−m−1) for t ≥ 0
by using a general method introduced by Nakao [8]. Unfortunately, this method
does not seem to be applicable in the case of more general functions σ and is
more complicated.
Our purpose in this paper is to give energy decay estimates of the solutions
to the problem (P ) for a weak nonlinear dissipation. We extend the results
obtained by Yao and prove in some cases an exponential decay when p > 2 and
the dissipative term is not necessarily superlinear near the origin.
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
Abbès Benaissa and
Salima Mimouni
Title Page
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
We use a new method recently introduced by Martinez [7] (see also [2]) to
study the decay rate of solutions to the wave equation u00 − ∆x u + g(u0 ) = 0 in
Ω × R+ , where Ω is a bounded domain of Rn . This method is based on a new
integral inequality that generalizes a result of Haraux [4].
Throughout this paper the functions considered are all real valued. We omit
the space variable x of u(t, x), ut (t, x) and simply denote u(t, x), ut (t, x) by
u(t), u0 (t), respectively, when no confusion arises. Let l be a number with
2 ≤ l ≤ ∞. We denote by k · kl the Ll norm over Ω. In particular, the L2 norm
is denoted by k · k2 . (·) denotes the usual L2 inner product. We use familiar
function spaces W01,p .
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
Abbès Benaissa and
Salima Mimouni
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
2.
Preliminaries and Main Results
First assume that the solution exists in the class
(2.1)
u ∈ C(R+ , W01,p (Ω)) ∩ C 1 (R+ , Ll (Ω)).
λ(x), σ(t) and g satisfy the following hypotheses:
(H1) σ : R+ → R+ is a non increasing function of class C 1 on R+ satisfying
Z +∞
(2.2)
σ(τ )dτ = +∞.
0
(H2) Consider g : R → R a non increasing C 0 function such that
g(v)v > 0 for all v 6= 0.
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
Abbès Benaissa and
Salima Mimouni
and suppose that there exist ci > 0; i = 1, 2, 3, 4 such that
(2.3)
1
c1 |v|m ≤ |g(v)| ≤ c2 |v| m if |v| ≤ 1,
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(2.4)
c3 |v|s ≤ |g(v)| ≤ c4 |v|r for all |v| ≥ 1,
where m ≥ 1, l − 1 ≤ s ≤ r ≤ n(p−1)+p
.
n−p
We define the energy associated to the solution given by (2.1) by the following
formula
l−1 0 l 1
ku kl + k∇x ukpp .
E(t) =
l
p
We first state two well known lemmas, and then state and prove a lemma that
will be needed later.
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
Lemma 2.1 (Sobolev-Poincaré inequality). Let q be a number with 2 ≤ q <
np
+∞ (n = 1, 2, . . . , p) or 2 ≤ q ≤ (n−p)
(n ≥ p + 1), then there is a constant
c∗ = c(Ω, q) such that
kukq ≤ c∗ k∇ukp
for
u ∈ W01,p (Ω).
Lemma 2.2 ([6]). Let E : R+ → R+ be a non-increasing function and assume
that there are two constants q ≥ 0 and A > 0 such that
Z +∞
E q+1 (t) dt ≤ AE(S), 0 ≤ S < +∞,
S
then we have
E(t) ≤ cE(0)(1 + t)
−1
q
∀t ≥ 0,
if q > 0
and
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
Abbès Benaissa and
Salima Mimouni
Title Page
E(t) ≤ cE(0)e−ωt ∀t ≥ 0,
if q = 0,
where c and ω are positive constants independent of the initial energy E(0).
Lemma 2.3 ([7]). Let E : R+ → R+ be a non increasing function and φ :
R+ → R+ an increasing C 2 function such that
φ(0) = 0
and
φ(t) → +∞ as t → +∞.
Assume that there exist q ≥ 0 and A > 0 such that
Z +∞
E(t)q+1 (t)φ0 (t) dt ≤ AE(S), 0 ≤ S < +∞,
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S
J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
then we have
E(t) ≤ cE(0)(1 + φ(t))
−1
q
∀t ≥ 0,
if q > 0
and
E(t) ≤ cE(0)e−ωφ(t) ∀t ≥ 0,
if q = 0,
where c and ω are positive constants independent of the initial energy E(0).
Proof of Lemma 2.3. Let f : R+ → R+ be defined by f (x) := E(φ−1 (x)). f is
non-increasing, f (0) = E(0) and if we set x := φ(t) we obtain
Z φ(T )
Z φ(T )
q+1
q+1
f (x) dx =
E φ−1 (x)
dx
φ(S)
φ(S)
T
Z
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
Abbès Benaissa and
Salima Mimouni
E(t)q+1 φ0 (t) dt
=
S
≤ AE(S) = Af (φ(S))
0 ≤ S < T < +∞.
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Setting s := φ(S) and letting T → +∞, we deduce that
Z +∞
f (x)q+1 dx ≤ Af (s) 0 ≤ s < +∞.
s
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By Lemma 2.2, we can deduce the desired results.
Our main result is the following
W01,p
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l
Theorem 2.4. Let (u0 , u1 ) ∈
× L (Ω) and suppose that (H1) and (H2)
hold. Then the solution u(x, t) of the problem (P ) satisfies
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
(1) If l ≥ m + 1, we have
Z
E(t) ≤ C(E(0)) exp 1 − ω
t
σ(τ ) dτ
∀t > 0.
0
(2) If l < m + 1, we have
E(t) ≤
C(E(0))
Rt
σ(τ ) dτ
0
p
! (mp−m−1)
∀t > 0.
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
Examples
1) If σ(t) =
1
(0 ≤ θ ≤ 1), by applying Theorem 2.4 we obtain
tθ
1−θ
E(t) ≤ C(E(0))e1−ωt
if θ ∈ [0, 1[, l ≥ m + 1,
(1−θ)p
E(t) ≤ C(E(0))t− mp−m−1
if 0 ≤ θ < 1, l < m + 1
and
p
E(t) ≤ C(E(0))(ln t)− (mp−m−1)
if θ = 1, l < m + 1.
Abbès Benaissa and
Salima Mimouni
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1
2) If σ(t) = θ
, where k is a positive integer and
t ln t ln2 t . . . lnk t
(
ln1 (t) = ln(t)
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lnk+1 (t) = ln(lnk (t)),
J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
by applying Theorem 2.4, we obtain
p
E(t) ≤ C(E(0))(lnk+1 t)− (mp−m−1)
(1−θ)p
if θ = 1, l < m + 1,
p
E(t) ≤ C(E(0))t− mp−m−1 (ln t ln2 t . . . lnk t) mp−m−1 if 0 ≤ θ < 1, l < m+1.
3) If σ(t) =
1
tθ (ln t)γ
, by applying Theorem 2.4, we obtain
(1−θ)p
γp
E(t) ≤ C(E(0))t− mp−m−1 (ln t) mp−m−1
(1−γ)p
E(t) ≤ C(E(0))(ln t)− mp−m−1
p
− mp−m−1
E(t) ≤ C(E(0))(ln2 t)
if 0 ≤ θ < 1, l < m + 1,
if θ = 1, 0 ≤ γ < 1, l < m + 1,
if θ = 1, γ = 1, l < m + 1.
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
Abbès Benaissa and
Salima Mimouni
Title Page
Proof of Theorem 2.4.
First we have the following energy identity to the problem (P )
Lemma 2.5 (Energy identity). Let u(t, x) be a local solution to the problem
(P ) on [0, ∞) as in Theorem 2.4. Then we have
Z Z t
E(t) +
σ(s)u0 (s)g(u0 (s)) ds dx = E(0)
Ω
for all t ∈ [0, ∞).
0
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
Proof of the energy decay. From now on, we denote by c various positive constants which may be different at different occurrences. We multiply the first
equation of (P ) by E q φ0 u, where φ is a function satisfying all the hypotheses of
Lemma 2.3 to obtain
Z T
Z
q 0
0=
E φ
u((|u0 |l−2 u0 )t − ∆p u + σ(t)g(u0 )) dx dt
S
Ω
T Z T
Z
Z
q 0
0 0 l−2
0 q−1 0
q 00
= E φ
uu |u | dx −
(qE E φ + E φ ) uu0 |u0 |l−2 dxdt
Ω
S
Ω
S
Z T
Z
Z T
Z 3l − 2
l − 1 02 1
q 0
0 2
q 0
p
E φ
|u | dxdt + 2
E φ
u + |∇u| dxdt
−
l
l
p
Ω
Ω
S
S
Z T
Z T
Z
2
+
E q φ0 k∇ukpp dxdt.
E q φ0 σ(t)ug(u0 ) dxdt + 1 −
p
S
S
Ω
T
(2.5) 2
E
T
Z
q 0
0 0 l−2
φ dt ≤ − E φ
uu |u | dx
q+1 0
Ω
S
Z
+
Abbès Benaissa and
Salima Mimouni
Title Page
We deduce that
Z
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
T
0
q 00
Z
φ + E φ ) uu0 |u0 |l−2 dxdt
S
Ω
Z
Z
Z T
Z
3l − 2 T q 0
0 l
q 0
+
E φ
|u | dxdt −
E φ
σ(t)ug(u0 ) dxdt.
l
S
Ω
S
Ω
(qE E
q−1 0
S
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0
Since E is nonincreasing, φ is a bounded nonnegative function on R+ (and we
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
denote by µ its maximum), using the Hölder inequality, we have
Z
+ p1
E(t)q φ0 uu0 |u0 |l−2 dx ≤ cµE(S)q+ l−1
l
∀t ≥ S.
Ω
Z
T
0
(qE E
S
q−1 0
Z
q 00
φ + E φ ) uu0 |u0 |l−2 dxdt
Ω
Z T
Z
q− 1l + p1
0
≤ cµ
−E (t)E(t)
dt + c
S
T
E(t)q+
l−1
+ p1
l
(−φ00 (t)) dt
S
+ p1
q+ l−1
l
≤ cµE(S)
.
Using these estimates we conclude from the above inequality that
Z T
2
(2.6)
E(t)1+q φ0 (t) dt
S
Z T
Z
1
3l
−
2
q+ l−1
+
q
0
≤ cE(S) l p +
E φ
|u0 |l dxdt
l
S
Ω
Z T
Z
−
E q φ0 σ(t)ug(u0 ) dxdt
S
Ω
Z
Z
3l − 2 T q 0
q+ l−1
+ p1
l
≤ cE(S)
+
E φ
|u0 |l dxdt
l
S
Ω
Z T
Z
q 0
−
E φ
σ(t)ug(u0 ) dxdt
|u0 |≤1
S
Z
−
S
T
E q φ0
Z
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
Abbès Benaissa and
Salima Mimouni
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σ(t)ug(u0 ) dxdt.
Page 11 of 19
|u0 |>1
J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
Define
Z
φ(t) =
t
σ(s) ds.
0
It is clear that φ is a non decreasing function of class C 2 on R+ . The hypothesis
(2.2) ensures that
(2.7)
φ(t) → +∞ as t → +∞.
Now, we estimate the terms of the right-hand side of (2.6) in order to apply the
results of Lemma 2.3:
Using the Hölder inequality, we get for l < m + 1
Z T
Z
q 0
E φ
|u0 |l dxdt
S
Ω
Z T
Z
1 0
q 0
≤C
E φ
u ρ(t, u0 ) dx dt
σ(t)
S
Ω
l
(m+1)
Z T
Z 1 0
0
q 0
0
dx dt
+C
E φ
u ρ(t, u )
σ(t)
S
Ω
Z T
Z T
0
l
φ0
q φ
0
0
q
≤C
E
(−E ) dt + C (Ω)
E
(−E 0 ) m+1 dt
l
σ(t)
S
S
σ m+1 (t)
l
0 m+1
Z T
l
φ
q+1
0
q 0 m+1−l
≤ CE (S) + C (Ω)
E φ m+1
(−E 0 ) m+1 dt.
σ(t)
S
Now, fix an arbitrarily small ε > 0 (to be chosen later). By applying Young’s
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
Abbès Benaissa and
Salima Mimouni
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
inequality, we obtain
T
Z
q 0
Z
|u0 |l dxdt
E φ
(2.8)
S
Ω
≤ CE
q+1
(m+1)
m + −l (m+1−l)
(S) + C (Ω)
ε
m+1
0
Z
T
m+1
E q m+1−l φ0 dt
S
+ C 0 (Ω)
1
l
E(S).
(m+1)
m+1ε l
If l ≥ m + 1, we easily obtain from (2.3) and (2.4)
Z T
Z
q 0
(2.9)
E φ
|u0 |l dxdt ≤ CE q+1 (S).
S
Abbès Benaissa and
Salima Mimouni
Ω
Next, we estimate the third term of the right-hand of (2.6). We get for l < m + 1
Z T
Z
q 0
(2.10)
E φ
σ(t)ug(u0 ) dxdt
|u0 |≤1
S
T
Z
E q φ0
≤ ε1
|u0 |≤1
S
Z
≤ cε1
kukpp dt + C(ε1 )
T
E
S
Z
q+1 0
T
Z
E q φ0
S
Z
φ dt + C(ε1 )
T
q 0
Z
E φ
S
|u0 |≤1
Z
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
(σg(u0 ))
|u0 |≤1
p
(σg(u0 )) p−1 dx.
p
p−1
dx
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
We now estimate the last term of the above inequality to get
Z T
Z
p
q 0
(2.11)
E φ
(σg(u0 )) p−1 dx dt
|u0 |≤1
T
S
Z
Z
q 0
≤
p
(u0 g(u0 )) (m+1)(p−1) dx dt
E φ
|u0 |≤1
S
T
Z
≤
E φ
σ
S
Z
≤ C(Ω)
Z
1
q 0
p
(m+1)(p−1)
T
|u0 |≤1
1
E q φ0
p
(σu0 g(u0 )) (m+1)(p−1) dx dt
p
(m+1)(p−1)
(−E 0 )
p
(m+1)(p−1)
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
dt.
σ
Set ε2 > 0; due to Young’s inequality, we obtain
Z T
Z
p
q 0
(2.12)
E φ
(σg(u0 )) p−1 dxdt
S
S
≤ C(Ω)
Abbès Benaissa and
Salima Mimouni
Title Page
|u0 |≤1
(m + 1)(p − 1) − p
ε
(m + 1)(p − 1) 2
(m+1)(p−1)
(m+1)(p−1)−p
Z
T
(m+1)(p−1)
E q (m+1)(p−1)−p φ0 dt
S
C(Ω) p
+
(m + 1)(p − 1)
1
(m+1)(p−1)
p
E(S),
ε2
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we chose q such that
Close
(m + 1)(p − 1)
= q + 1.
q
(m + 1)(p − 1) − p
thus we find q =
Contents
mp−m−1
p
m+1
and thus q m+1−l
= q + 1 + α with α =
Quit
(m+1)(p l−p−l)
.
p(m+1−l)
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
Using the Hölder inequality, the Sobolev imbedding and the condition (2.4),
we obtain
Z T
Z
q 0
E φ
σ(t)ug(u0 ) dxdt
|u0 |≥1
S
T
Z
Z
q 0
≤
r+1
|u|
E φ σ(t)
S
1
Z
(r+1)
|g(u )|
dx
T
≤c
E
q+ p1
0
φσ
1
(r+1)
Z
≤c
0
0
σu g(u ) dx
(t)
r
r+1
dx
dt
r
r+1
dt
|u0 |>1
S
Z
r+1
r
|u0 |>1
Ω
Z
0
T
1
1
r
E q+ p φ0 σ (r+1) (t)(−E 0 ) r+1 dt.
Abbès Benaissa and
Salima Mimouni
S
Applying Young’s inequality, we obtain
Z T
Z
q 0
(2.13)
E φ
σ(t)ug(u0 ) dxdt
S
|u0 |≥1
Z T
1
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Z
1
T
(E q+ p φ0 σ (r+1) (t))r+1 dt + c(ε3 )
(−E 0 ) dt
S
S
Z T
(p−1)(mr−1)
p
≤ ε3 µr+1 E
(0)
E q+1 φ0 dt + c(ε3 )E(S).
≤ ε3
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
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If l ≥ m + 1, the last inequality is also valid in the domain {|u0 | < 1} and with
m instead of r.
Choosing ε, ε1 , ε2 and ε3 small enough, we deduce from (2.6), (2.8), (2.10),
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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006
(2.12) and (2.13) for l < m + 1
Z T
l−1
1
E(t)1+q φ0 (t) dt ≤ CE(S)q+1 + C 0 E(S)q+ l + p + C 00 E(S)
S
+ C 000 E(0)
(p l−p−l)(m+1)
p l
E(S) + C 0000 E(0)
(m r−1)(p−1)
p r
E(S),
where C, C 0 , C 00 , C 000 , C 000 are different positive constants independent of E(0).
Choosing ε3 small enough, we deduce from (2.6), (2.9) and (2.13) for l ≥
m+1
Z T
(m2 −1)(p−1)
l−1
1
E(t)1+q φ0 (t) dt ≤ CE(S)q+1 + C 0 E(S)q+ l + p + C 00 E(0) p m E(S),
S
where C, C 0 , C 00 are different positive constants independent of E(0), we may
thus complete the proof by applying Lemma 2.3.
Remark 1. We obtain the same results for the following problem

(|u0 |l−2 u0 )0 − e−Φ(x) div(eΦ(x) |∇x u|p−2 ∇x u) + σ(t)g(u0 ) = 0 in Ω × [0, +∞[,



u = 0 on ∂Ω × [0, +∞[,



u(x, 0) = u0 (x), u0 (x, 0) = u1 (x) in Ω,
where Φ is a positive function such that Φ ∈ L∞ (Ω), in this case (u0 , u1 ) ∈
1,p
W0,Φ
× LlΦ , where
Z
1,p
1,p
Φ(x)
p
W0,Φ (Ω) = u ∈ W0 (Ω),
e
|∇x u| dx < ∞ ,
Ω
Z
l
l
Φ(x)
l
e
|u| dx < ∞ .
LΦ (Ω) = u ∈ L (Ω),
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
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Salima Mimouni
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Ω
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Thus the energy associated to the solution is given by the following formula
E(t) =
l − 1 Φ(x)/l 0 l 1 Φ(x)/p
ke
u kl + ke
∇x ukpp .
l
p
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
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Salima Mimouni
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References
[1] YAO-JUN YE, On the decay of solutions for some nonlinear dissipative
hyperbolic equations, Acta Math. Appl. Sin. Engl. Ser., 20(1) (2004), 93–
100.
[2] A. BENAISSA AND S. MOKEDDEM, Global existence and energy decay
of solutions to the Cauchy problem for a wave equation with a weakly nonlinear dissipation, Abstr. Appl. Anal., 11 (2004), 935–955.
[3] Y. EBIHARA, M. NAKAO AND T. NAMBU, On the existence of global
classical solution of initial boundary value problem for u00 − ∆u − u3 = f ,
Pacific J. of Math., 60 (1975), 63–70.
[4] A. HARAUX, Two remarks on dissipative hyperbolic problems, in: Research Notes in Mathematics, Pitman, 1985, p. 161–179.
[5] R. IKEHATA AND T. SUZUKI, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475–
491.
[6] V. KOMORNIK, Exact Controllability and Stabilization. The Multiplier
Method, Masson-John Wiley, Paris, 1994.
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
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Salima Mimouni
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[7] P. MARTINEZ, A new method to decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419–444.
[8] M. NAKAO, A difference inequality and its applications to nonlinear evolution equations, J. Math. Soc. Japan, 30 (1978), 747–762.
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[9] M. NAKAO, On solutions of the wave equations with a sublinear dissipative
term, J. Diff. Equat., 69 (1987), 204–215.
Energy Decay of Solutions of a
Wave Equation of p−Laplacian
Type with a Weakly Nonlinear
Dissipation
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Salima Mimouni
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