Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 222, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
ENERGY DECAY FOR DEGENERATE KIRCHHOFF
EQUATIONS WITH WEAKLY NONLINEAR DISSIPATION
MAMA ABDELLI, SALIM A. MESSAOUDI
Abstract. In this article we consider a degenerate Kirchhoff equation wave
equation with a weak frictional damping,
“Z
”γ
(|ut |l−2 ut )t −
|∇x u|2 dx ∆x u + α(t)g(ut ) = 0.
Ω
We prove general stability estimates using some properties of convex functions,
without imposing any growth condition at the frictional damping term.
1. Introduction
In this article, we consider the initial-boundary value problem for the nonlinear
Kirchhoff equation
Z
γ
l−2
(|ut | ut )t −
|∇u|2 dx ∆u + α(t)g(ut ) = 0, in Ω × (0, ∞)
(1.1)
Ω
u = 0,
0
u(x, 0) = u (x),
on ∂Ω × (0, ∞)
1
ut (x, 0) = u (x),
(1.2)
x ∈ Ω,
(1.3)
where l ≥ 2, γ ≥ 0 are given constants, Ω is a bounded domain in Rn with a smooth
boundary ∂Ω, and g and α are one-variable functions satisfying some conditions
to be specified later. This problem has been studied by many authors and several
existence, nonexistence, and decay results have appeared. For instance, when l = 2
and γ = 0, the problem was treated by Mustafa and Massaoudi [11]. By using
some properties of convex functions, they established a general decay result without
imposing any growth condition on g at the origin. Abdelli and Benaissa [1] treated
system (1.1)-(1.3) for g having a polynomial growth near the origin and established
energy decay results depending on α and g under appropriate relations between l
and γ. In a realted work, Amroun and Benaissa [3] constructed an exact solution
of (1.1)-(1.3) in the presence of a nonlinear source term and for α ≡ 0. They also
proved a finite-time blow-up result for some specific initial data. Benaissa and
Guesmia [5] proved the existence of global solution, as well as, a general stability
result for the following equation
(|u0 |l−2 u0 )0 − ∆φ u + α(t)g(u0 ) = 0,
in Ω × R+ ,
2000 Mathematics Subject Classification. 35B37, 35L55, 74D05, 93D15, 93D20.
Key words and phrases. Decay of solutions; nonlinear; degenerate; Kirchhoff equation.
c
2013
Texas State University - San Marcos.
Submitted September 18, 2013. Published October 11, 2013.
1
2
M. ABDELLI, S. A. MESSAOUDI
EJDE-2013/222
Pn
where ∆φ = i=1 ∂xi (φ(|∂xi |2 )∂xi ).
In this article, we use some technique from [11] to establish an explicit and
general decay result, depending on g and α. The proof is based on the multiplier
method and makes use of some properties of convex functions, the general Young
inequality and Jensen’s inequality. These convexity arguments were introduced
and developed by Lasiecka and co-workers ([7, 8, 9]) and used, with appropriate
modifications, by Liu and Zuazua [10], Alabau-Boussouira [2] and others.
The paper is organized as follows: in section 2, we give our hypotheses and
establish a useful lemma. In section 3, we state and prove our main result.
2. Preliminaries
To state and prove our result, we need the following hypotheses:
(H1) α : R+ → R+ is a nonincreasing differentiable function.
(H2) g : R → R is a nondecreasing C 0 function such that there exist ε, c1 , c2 > 0,
and a convex and increasing function G : R+ → R+ of class C 1 (R+ ) ∩
C 2 (]0, +∞[) satisfying G(0) = G0 (0) = 0 or G is linear on [0, ε] such that
c1 |s|l−1 ≤ |g(s)| ≤ c2 |s|p ,
|s|l + |g(s)|
l
l−1
≤ G−1 (sg(s)),
if |s| ≥ ε;
if |s| ≤ ε;
with p satisfying
n+2
, if n > 2;
n−2
l − 1 ≤ p < ∞, if n ≤ 2 .
l−1≤p≤
Now we define the energy associated to the solution of the system (1.1) -(1.3) by
E(t) =
l−1
1
2(γ+1)
kut kll +
k∇x uk2
l
1+γ
Lemma 2.1. Let u be the solution of (1.1)-(1.3). Then
Z
0
E (t) = −α(t)
ut g(ut ) dx ≤ 0.
(2.1)
(2.2)
Ω
Proof. Multiplying (1.1) by ut and integrating over Ω, using the boundary conditions, the assertion of the lemma follows.
3. Main Result
To prove our main result, first prove the following lemma.
Lemma 3.1. Assume that (H1), (H2) hold and that l ≥ 2(γ + 1). Then the
functional
Z
F (t) = M E(t) +
u|ut |l−2 ut dx,
Ω
defined along the solution of (1.1)-(1.3), satisfies the following estimate, for some
positive constants M, c, m:
Z
l
0
F (t) ≤ −mE(t) + c (|ut |l + |ug(ut )| l−1 ) dx
Ω
and F (t) ∼ E(t).
EJDE-2013/222
ENERGY DECAY
3
Proof. Using system (1.1)-(1.3), (2.1) and (2.2), we obtain
Z
Z
0
0
l
F (t) = M E (t) +
|ut | dx +
u(|ut |l ut )t dx
Ω
Ω
Z
Z
Z
γ Z
≤
|ut |l dx +
|∇u|2 dx
u∆u dx − α(t)
ug(ut ) dx
Ω
Ω
Ω
Ω
Z
Z
Z
≤
|ut |l dx −
|∇u|2(γ+1) dx − α(t)
ug(ut ) dx
Ω
Ω
ZΩ
≤ −mE(t) + c [|ut |l + |ug(ut )|] dx
Ω
To prove that F (t) ∼ E(t), we show that for some positive constants λ1 and λ2 ,
λ1 E(t) ≤ E(t) ≤ λ2 E(t) .
(3.1)
We use (2.1), Poincaré’s and Young’s inequalities with exponents
2n
recall that 2 ≤ l ≤ p + 1 ≤ n+2
, to obtain
Z
Z
Z
l−2
l
u|ut | ut dx ≤ Cε
|u| dx + ε
|ut |l dx
Ω
Ω
l
l−1
and
1
l
and
Ω
≤ Cε k∇ukl2 + εkut ||ll
l
≤ Cε E 2(γ+1) (t) + cεE(t)
≤ Cε E
l−2(γ+1)
2(γ+1)
(t)E(t) + cεE(t).
By noting that l ≥ 2(γ + 1) and using (2.2), we have
Z
l−2(γ+1)
u|ut |l−2 ut dx ≤ Cε E 2(γ+1) (0)E(t) + cεE(t),
Ω
and
Z
u|ut |l−2 ut dx ≥ −Cε
Z
Ω
|u|l dx − ε
Ω
≥ −Cε E
≥ −Cε E
Z
|ut |l dx
Ω
l−2(γ+1)
2(γ+1)
l−2(γ+1)
2(γ+1)
(t)E(t) − cεE(t)
(0)E(t) − cεE(t)
Then, for M large enough, we obtain (3.1). This completes the proof.
Taking 0 < ε1 < ε such that
sg(s) ≤ min{ε, G(ε)},
if |s| ≤ ε1 ,
(3.2)
and
(
if |s| ≥ ε1
c01 |s|l−1 ≤ |g(s)| ≤ c02 |s|p ,
l
l
−1
l−1
|s| + |g(s)|
≤ G (sg(s)), if |s| ≤ ε1 .
(3.3)
Considering the following partition of Ω,
Ω1 = {x ∈ Ω : |ut | ≤ ε1 },
Ω2 = {x ∈ Ω : |ut | > ε1 }
and using the embedding H01 (Ω) ,→ Lp+1 (Ω) and Hölder’s inequality, we obtain
Z
1 Z
Z
p+1
p/(p+1)
1
|ug(ut )| dx ≤
|u|p+1 dx
|g(ut )|1+ p dx
Ω2
Ω2
Ω2
4
M. ABDELLI, S. A. MESSAOUDI
≤ ckukH01 (Ω)
Z
EJDE-2013/222
1
|g(ut )|1+ p dx
p/(p+1)
Ω2
Using Poincaré’s inequality and (3.3) yields
Z
[|ut |l + |ug(ut )|] dx
Ω2
Z
p/(p+1)
Z
1/2 Z
1
≤c
|ut |l−1 |ut | dx + c
|∇u|2 dx
|g(ut )|1+ p dx
Ω
Ω
Ω2
Z 2
Z
1/2 Z
p/(p+1)
≤c
ut g(ut ) dx + c
|∇u|2 dx
ut g(ut ) dx
Ω2
0
Ω
≤ −cE (t) + cE
1
2(γ+1)
Ω2
0
p/(p+1)
(−E (t))
.
Then, we use Young’s inequality and the fact that p ≥ l − 1 ≥ 2γ + 1, for any δ > 0,
we have
Z
p+1
[|ut |l + |ug(ut )|] dx ≤ −cE 0 (t) + cδE 2(γ+1) (t) + Cδ (−E 0 (t))
Ω2
(3.4)
p+1
≤ cδE 2(γ+1) (t) − Cδ E 0 (t)
≤ cδE
p−(2γ+1)
2(γ+1)
(0)E(t) − Cδ E 0 (t)
Similarly, using (2.1) and Young’s inequality, we have, for any δ > 0,
Z
Z
Z
Z
l
[|ut |l + |ug(ut )|] dx ≤
|ut |l dx + δ
|u|l dx + Cδ
|g(ut )| l−1 dx
Ω1
Ω1
ZΩ1
ZΩ1
(3.5)
l
l
|g(ut )| l−1 dx
≤
|ut |l dx + cδE 2(γ+1) (t) + Cδ
Ω1
Ω1
By Lemma 3.1, (3.4) and (3.5), for δ small enough, the function L = F + Cδ E
satisfies
l−2(γ+1)
p−(2γ+1)
2
L0 (t) ≤ − m + cδE
(0) + cδE 2(γ+1) (0) E(t)
Z
Z
l
+
|ut |l dx + Cδ
|g(ut )| l−1 dx
(3.6)
Ω1
Z Ω1
l
≤ −dE(t) + c
|ut |l + |g(ut )| l−1 dx
Ω1
and
L(t) ∼ E(t).
(3.7)
Theorem 3.2. Assume that (H1), (H2) hold and l ≥ 2(γ + 1). Then there exist
positive constants k1 , k2 , k3 and ε0 such that the solution of (1.1)-1.3 satisfies
Z t
−1
E(t) ≤ k3 G1 k1
α(s) ds + k2
∀t ≥ 0,
(3.8)
0
where
Z
G1 (t) =
t
1
1
ds,
G2 (s)
G2 (t) = tG0 (ε0 t).
Here, G1 is strictly decreasing and convex on (0, 1] with limt→0 G1 (t) = +∞.
(3.9)
EJDE-2013/222
ENERGY DECAY
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Proof. Multiplying (3.6) by α(t), we have
α(t)L0 (t) ≤ −dα(t)E(t) + cα(t)
Z
l
|ut |l + |g(ut )| l−1 dx
(3.10)
Ω1
Case 1. G is linear on [0, ε], then we deduce that
Z
α(t)L0 (t) ≤ −dα(t)E(t) + cα(t)
ut g(ut ) dx = −dα(t)E(t) − cE 0 (t)
Ω1
Consequently, we arrive at
(αL + cE)0 (t) ≤ −dα(t)E(t).
Recalling that
αL + cE ∼ E,
we obtain
00
E(t) ≤ c0 e−c
Rt
0
(3.11)
α(s) ds
Thus, we have
00
E(t) ≤ c0 e−c
Rt
0
α(s) ds
00
= c0 G−1
1 (c
Z
t
α(s) ds)
0
by a simple computation.
Case 2. G is nonlinear on [0, ε]. In this case, we define
Z
1
ut g(ut ) dx.
I(t) =
|Ω1 | Ω1
and exploit Jensen’s inequality and the concavity of G−1 to obtain
Z
G−1 (I(t)) ≥ c
G−1 (ut g(ut )) dx.
Ω1
By using this inequality and (3.3), we obtain
Z
Z
l
G−1 (ut g(ut )) dx ≤ cα(t)G−1 (I(t)). (3.12)
[|ut |l +|g(ut )| l−1 ] dx ≤ α(t)
α(t)
Ω1
Ω1
Let us set H0 = αL+E and exploit (2.2), (3.10), (3.12), and α being nonincreasing,
to obtain
H00 (t) ≤ −dα(t)E(t) + cα(t)G−1 (I(t)) + E 0 (t)
(3.13)
≤ −dα(t)E(t) + cα(t)G−1 (I(t)),
and recall (3.7), to deduce that H0 ∼ E.
For ε0 < ε and c0 > 0, we define H1 by
E(t)
H1 (t) = G0 (ε0
)H0 (t) + c0 E(t).
E(0)
Then, we see easily that, for a1 , a2 > 0,
a1 H1 (t) ≤ E(t) ≤ a2 H1 (t),
0
0
(3.14)
00
By recalling that E ≤ 0, G > 0, G > 0 on (0, ε] and making use of (2.1) and
(3.13), we obtain
E 0 (t) 00
E(t)
E(t) 0
G (ε0
)H0 (t) + G0 (ε0
)H (t) + c0 E 0 (t)
E(0)
E(0)
E(0) 0
E(t)
E(t) −1
) + cα(t)G0 (ε0
)G (I(t)) + c0 E 0 (t).
≤ −dα(t)E(t)G0 (ε0
E(0)
E(0)
H10 (t) = ε0
(3.15)
6
M. ABDELLI, S. A. MESSAOUDI
EJDE-2013/222
Let G∗ be the convex conjugate of G in the sense of Young (see Arnold [4, p.
61-64]), then
G∗ (s) = s(G0 )−1 (s) − G[(G0 )−1 (s)],
if s ∈ (0, G0 (ε)],
(3.16)
and G∗ satisfies the generalized Young’s inequality
AB ≤ G∗ (A) + G(B),
if A ∈ (0, G0 (ε)], B ∈ (0, ε].
(3.17)
with A = G0 (ε0 E(t)/E(0)) and B = G−1 (I(t)), using (2.2), (3.2) and (3.15)–(3.16),
we obtain
E(t)
E(t) ) + cα(t)G∗ G0 (ε0
) + cα(t)I(t) + c0 E 0 (t)
H10 (t) ≤ −dα(t)E(t)G0 (ε0
E(0)
E(0)
E(t)
E(t) 0
E(t)
≤ −dα(t)E(t)G0 (ε0
) + cε0 α(t)
G (ε0
) − cE 0 (t) + c0 E 0 (t).
E(0)
E(0)
E(0)
Choosing c0 > c and ε0 small enough, we obtain
E(t) E(t) 0 E(t) G ε0
= −kα(t)G2
H10 (t) ≤ −kα(t)
E(0)
E(0)
E(0)
(3.18)
where G2 (t) = tG0 (ε0 t). Since
G02 (t) = G0 (ε0 t) + ε0 tG00 (ε0 t).
and G is convex on (0, ε], we find that G02 (t) > 0 and G2 (t) > 0 on (0, 1]. By setting
H1 (t)
H(t) = a1E(0)
(a1 is given in (3.14)), we easily see that, by (3.14), we have
H(t) ∼ E(t)
(3.19)
Using (3.18), we arrive at
H 0 (t) ≤ −k1 α(t)G2 (H(t))
By recalling (3.9), we deduce G2 (t) = −1/G01 (t), hence
H 0 (t) ≤ k1 α(t)
1
,
G01 (H(t))
which gives
[G1 (H(t))]0 = H 0 (t)G01 (H(t)) ≤ k1 α(t)
A simple integration leads to
Z
t
G1 (H(s)) ≤ k1
α(s) ds + G1 (H(0))
0
Consequently,
H(t) ≤
G−1
1 (k1
Z
t
α(s) ds + k2 )
(3.20)
0
Using (3.19) and (3.20) we obtain (3.8). The proof is complete.
Acknowledgments. The second author wants to thank KFUPM for its financial
support.
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ENERGY DECAY
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Mama Abdelli
Université Djillali Liabés, Laboratoire de Mathématique, B.P. 89. Sidi Bel Abbés 22000,
Algeria
E-mail address: abdelli mama@yahoo.fr
Salim A. Messaoudi
King Fahd University of Petroleum and Minerals, Department of Mathematics and
Statistics, Dhahran 31261, Saudi Arabia
E-mail address: messaoud@kfupm.edu.sa