ASYMPTOTIC BEHAVIOR FOR A NONLINEAR VISCOELASTIC
PROBLEM WITH A VELOCITY-DEPENDENT
MATERIAL DENSITY
NASSER-EDDINE TATAR
Received 22 September 2004 and in revised form 10 May 2005
We consider a nonlinear viscoelastic problem and prove that the solutions are uniformly
bounded and decay exponentially to zero as time goes to infinity. This is established under
weaker conditions on the relaxation function than the usually used ones. In particular,
we remove the assumptions on the derivative of the kernel. In fact, our kernels are not
necessarily differentiable.
1. Introduction
The problem we would like to investigate is the following:
ρ
ut utt − ∆u − ∆utt +
t
0
g(t − s)∆u(s)ds − γ∆ut = 0 in Ω × (0, ∞),
u = 0 on Γ × (0, ∞),
(1.1)
ut (x,0) = u1 (x) in Ω,
u(x,0) = u0 (x),
where Ω is a bounded domain in Rn , n ≥ 1, with a smooth boundary Γ. The real number
ρ is assumed to satisfy 0 < ρ ≤ 2/(n − 2) if n ≥ 3 or ρ > 0 if n = 1,2. The function g(t) is
positive and will be specified further below.
This model appears in viscoelasticity. We are in the case where the material density depends on ut (see [5, 11]). In [1], Cavalcanti et al. studied this nonlinear problem (ρ > 0)
and proved well posedness as well as a uniform decay result. It has been shown that solutions go to zero in an exponential manner provided that the kernel g(t) is also exponentially decaying to zero. Namely, the following assumptions were assumed:
(H1) g : R+ → R+ is a bounded C 1 -function such that
1−
∞
0
g(s)ds = l > 0,
(1.2)
(H2) there exist positive constants ξ1 , ξ2 such that
−ξ1 g(t) ≤ g (t) ≤ −ξ2 g(t)
for all t ≥ 0.
Copyright © 2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:10 (2005) 1497–1506
DOI: 10.1155/IJMMS.2005.1497
(1.3)
1498
Asymptotic behavior for a viscoelastic problem
These two assumptions are in fact frequently used also in the linear case (ρ = 0) (see [3,
4, 5, 6, 7, 8] and also [10, 13]). In [9], the present author with Messaoudi have improved
the result in [1] by showing that the same asymptotic behavior occurs also for the case
γ = 0. This means that the convolution term produces a weak dissipation which is able
to drive solutions to the equilibrium state in an exponential manner. We do not need
the strong damping. In [6], the present author with Furati proved that for “sufficiently
small” g and g , we also have exponential decay (in case ρ > 0). Namely, we need eαt g(t)
and eαt g (t) to have “small” L1 -norms for some α > 0. The conditions in (H2) are not
imposed. In particular, g is not necessarily always negative.
Here in this work, we intend to improve further this latter result by removing the
condition on g . To this end, we combine the multiplier technique with some appropriate
estimations and some new “Lyapunov-type” functionals. These functionals are somewhat
similar in spirit to the one introduced by the author in [12].
The plan of the paper is as follows. In the next section, we state an existence theorem,
introduce our functionals, and prove some useful propositions for our result. Section 3 is
devoted to the exponential decay theorem.
2. Preliminaries
We start by stating an existence result due to Cavalcanti et al. [1] (see also [2]).
∞
Theorem 2.1. Assume that the kernel g : R+ → R+ satisfies 1 − 0 g(s)ds = l > 0. Let u0 ,u1
∈ H01 (Ω) and γ ≥ 0. Then, problem (1.1) possesses at least one weak solution u : Ω × (0, ∞)
→ R in the class
u ∈ L∞ 0, ∞;H01 (Ω) ,
u ∈ L∞ 0, ∞;H01 (Ω) ,
u ∈ L2 0, ∞;H01 (Ω) .
(2.1)
We point out here that the differentiability of g is not needed to prove local existence.
In this paper, we consider γ = 0. We may assume that γ = 1.
The (classical) energy associated to problem (1.1) is defined by
E(t) :=
1
ρ+2
Ω
ρ+2
ut dx + 1
2
Ω
|∇u|2 dx +
1
2
Ω
∇ut 2 dx.
(2.2)
If we differentiate E(t) with respect to t along solutions of (1.1), we get
E (t) =
Ω
∇u t
t
0
g(t − s)∇u(s)dsdx −
Ω
∇ut 2 dx.
(2.3)
This expression is of an undefined sign, and therefore the boundedness (and the dissipativity) of the energy functional E(t) is not clear. In the prior works, the authors defined
(g ∇u)(t) :=
t
Ω 0
2
g(t − s)∇u(t) − ∇u(s) dsdx
(2.4)
Nasser-Eddine Tatar 1499
and observed that
Ω
∇u t
t
1
1
g(t − s)∇u(s)dsdx = (g ∇u)(t) − (g ∇u) (t)
2
2
0
t
1 d
∇ut 2 dx − 1 g(t) ∇ut 2 dx.
g(s)ds
+
2 dt
2
0
Ω
Ω
(2.5)
Then, considering the modified energy functional
t
ρ+2
1
ut dx + 1 1 − g ds
ρ+2 Ω
2
0
2
1
1 ∇ut dx + (g ∇u)(t),
+
2 Ω
2
Ᏹ(t) =
Ω
|∇u|2 dx
(2.6)
it appears that
Ᏹ (t) = −
Ω
∇ut 2 dx + 1 (g ∇u)(t) − 1 g(t)
2
2
Ω
|∇u|2 dx.
(2.7)
At this point, they use the fact that g (t) ≤ 0 to obtain uniform boundedness. In our case,
we do not have this assumption. To overcome this, a new functional has been proposed
in [6]. An exponential decay result has been obtained under some “smallness” condition
on g(t) and g (t). It is our objective here to remove the smallness condition on g (t). In
fact, even the differentiability of g is not required. We will need the assumptions
(G1) g : R + → R+ is a bounded continuous function such that
1−
∞
0
g(s)ds = l > 0,
(2.8)
(G2) g(t)eαt ∈ L1 (0, ∞) for some α > 0.
We will use repeatedly the following inequality.
Lemma 2.2. For any a,b ∈ R and δ > 0,
ab ≤ δa2 +
1 2
b.
4δ
(2.9)
We denote
ḡ :=
+∞
0
g(s)ds,
ḡα :=
+∞
0
eαs g(s)ds.
(2.10)
Next, we prove the uniform boundedness of the classical energy.
Proposition 2.3. Assume that (G1) and (G2) hold. If g is such that ḡα ≤ α/2, then
E(t) ≤ E(0)
for all t ≥ 0.
(2.11)
1500
Asymptotic behavior for a viscoelastic problem
Proof. We have
Ω
∇u t
t
t
g(t − s)∇u(s)dsdx =
0
0
+
t
0
1 d
∇ut ∇udx =
2 dt
Ω
g ds
g(s)ds
t
0
Ω
t
∇u t
0
Ω
∇ut ∇udx
1
|∇u| dx − g(t)
2
Ω
2
g ds
(2.12)
g(t − s) ∇u(s) − ∇u(t) dsdx,
Ω
|∇u|2 dx.
(2.13)
From (2.12) and Lemma 2.2 with δ = 1/4, we find
Ω
∇u t
t
0
g(t − s)∇u(s)dsdx ≤
1
4
∇ut 2 dx − 1 g(t)
|∇u|2 dx
2
Ω
t
1 d
g(s)ds
|∇u|2 dx .
+ ḡ(g ∇u)(t) +
2 dt
Ω
0
(2.14)
Ω
Considering
e(t) :=
1
ρ+2
Ω
t
ρ+2
1 ut dx + 1 1 − g ds
∇ut 2 dx,
|∇u|2 dx +
2
2
Ω
0
Ω
(2.15)
a simple computation shows, with the help of (2.14), that
e (t) ≤ −
3
4
Ω
|∇u|2 dx + ḡ(g ∇u)(t).
(2.16)
Next, we introduce the functional
Φ(t) :=
t
Ω 0
2
Gα (t − s)∇u(t) − ∇u(s) dsdx =: Gα ∇u (t)
(2.17)
with
Gα (t) := e−αt
+∞
t
eαs g(s)ds
(2.18)
for some α > 0. A differentiation of (2.17) yields
Φ (t) = −αΦ(t) − (g ∇u)(t) + 2
Ω
∇u t
t
0
Gα (t − s) ∇u(t) − ∇u(s) dsdx.
(2.19)
By Lemma 2.2 with δ = 1/8λ, for some λ > 0 to be determined, we have
Ω
∇u t
t
0
Gα (t − s) ∇u(t) − ∇u(s) dsdx
1
≤
8λ
Ω
∇ut 2 dx + 2λ
t
0
Gα (s)ds
Gα ∇u (t).
(2.20)
Nasser-Eddine Tatar 1501
Notice that
t
Gα (s)ds ≤
0
1
α
∞
eαs g(s)ds =
0
ḡα
.
α
(2.21)
Therefore,
1
4λ
Φ (t) ≤ −αΦ(t) − (g ∇u)(t) +
≤ − α−
Ω
∇ut 2 dx + 4λḡα Φ(t)
4λḡα
1
Φ(t) − (g ∇u)(t) +
α
4λ
α
Ω
∇ut 2 dx.
(2.22)
We define
V (t) := e(t) + λΦ(t).
(2.23)
Clearly, by (2.16) and (2.22), we have
V (t) ≤ −
1
2
Ω
∇ut 2 dx + (ḡ − λ)(g ∇u)(t) − λ α − 4λḡα Φ(t).
α
(2.24)
If ḡα ≤ α/2, then it is possible to choose λ so that λ ≥ ḡα (notice that ḡα > ḡ) and λ ≤ α2 /4ḡα .
Hence, V (t) ≤ 0. Consequently, e(t) and thereafter E(t) are uniformly bounded for all
t ≥ 0 by e(0).
This proposition will be used in a crucial manner in our main result. However, the
functional V (t) is still not suitable to work with. We introduce
Ψ(t) : =
1
ρ+1
χ(t) : =
Ω
Ω
ρ
ut ut udx +
∆ut −
ρ t
u t u t
ρ+1
0
Ω
∇u∇ut dx,
(2.25)
Gα (t − s) u(t) − u(s) dsdx.
Then, we form the expression
W(t) := V (t) + ε Ψ(t) + χ(t) ,
t ≥ 0,
(2.26)
for some ε > 0 to be determined later.
The next proposition will show, in particular, that the result we will derive for W(t)
will also hold for the classical energy.
Proposition 2.4. There exist an ε0 , m, and M > 0 such that
mE(t) ≤ W(t) ≤ M E(t) + Φ(t) ,
for all λ ≥ ḡα and 0 < ε ≤ ε0 .
t ≥ 0,
(2.27)
1502
Asymptotic behavior for a viscoelastic problem
Proof. We begin by the left inequality. Observe first that by the embedding H01 (Ω) L2(ρ+1) (Ω) for 0 < ρ ≤ 2/(n − 2) if n ≥ 3 or ρ > 0 if n = 1,2, we can write
Ω
2(ρ+1)
u t dx ≤ Ce
Ω
∇ut 2 dx
ρ+1
,
(2.28)
where Ce > 0 is the embedding constant (the subscript “e” is for embedding). Further, in
virtue of Proposition 2.3, we get
Ω
2(ρ+1)
ρ
u t dx ≤ Ce 2e(0)
Ω
∇ut 2 dx = C
Ω
∇ut 2 dx,
(2.29)
where C = Ce (2e(0))ρ . This relation, together with Lemma 2.2, implies that
Ω
ρ
ut ut udx ≤ δ2
2(ρ+1)
Cp
u t dx +
|∇u|2 dx
4δ2 Ω
C
2 ∇ut 2 dx + p
≤ Cδ
|∇u|2 dx,
4δ2 Ω
Ω
Ω
(2.30)
δ2 > 0,
where C p is the Poincaré constant (the subscript “p” is for Poincaré), and
Ω
ρ
u t u t
t
≤ δ3 C
0
Gα (t − s) u(t) − u(s) dsdx
Ω
∇ut 2 dx + C p ḡα Gα ∇u (t),
4δ3 α
Gathering (2.30), (2.31), (2.20) with δ1 and
Ω
∇u∇ut dx ≤ δ4
Ω
∇ut 2 dx + 1
4δ4
(2.31)
δ3 > 0.
Ω
|∇u|2 dx,
δ4 > 0,
(2.32)
we obtain from (2.25) and (2.26) that
1
W(t) ≥
ρ+2
Ω
ρ+2
ut dx + 1 l − ε
2
Cp
1
+
2 δ2 (ρ + 1) δ4
Ω
|∇u|2 dx
δ2 + δ3 C
1
∇ut 2 dx
− ε δ 1 + δ4 +
2
ρ+1
Ω
Cp
εḡα 1
+ λ−
+
Φ(t).
4α δ1 δ3 (ρ + 1)
(2.33)
+
δ4 = 3/4l, and ε suffiTaking for instance δ1 = 1/5, δ2 = 3C p /4(ρ + 1)l, δ3 = (ρ + 1)/5C,
ciently small,
2
l (ρ + 1)2 l 2α(ρ + 1)2
ε ≤ ε0 := min
,
,
,
5
3 3C p C (ρ + 1)2 + C p C
(2.34)
we find that W(t) ≥ mE(t), for all t ≥ 0, for some positive constant m. The right-hand
side inequality may be proved easily by taking for instance all the δi , i = 1,2,3,4, equal
to 1/2 and summing up the inequalities in (2.30), (2.31), (2.34), and (2.20) with their
respective coefficients in the expression of W(t).
Nasser-Eddine Tatar 1503
3. Long-time behavior
In this section, we state and prove our main result. Observe that assuming the hypotheses
in Proposition 2.3, we have uniqueness of the weak solution. The solution corresponding
to E(0) = 0 is the trivial one and is included in our next result.
Theorem 3.1. Assume that the kernel g satisfies (G1) and (G2). Then, the √
weak solution of
(1.1) decays exponentially to zero, in the energy norm, provided that ḡα ≤ α 2/4.
Proof. We differentiate W(t) (see (2.26)) along solutions of (1.1), we obtain from (2.26)
and (2.24) that
W (t) ≤ −
1
2
∇ut 2 dx + (ḡ − λ)(g ∇u)(t) − λ α − 4λḡα Φ(t) + εΨ (t) + εχ (t),
α
Ω
(3.1)
with
Ψ (t) = −
+
χ (t) =
Ω
∇u
t
0
+
+
Ω
Ω
0
∇u t
α
ρ+1
1
+
ρ+1
−
1
ρ+1
Ω
∇ut 2 dx +
Ω
∇u t
t
∇u
1
ρ+1
0
g(t − s)∇u(s)dsdx −
Ω
Ω
ρ+2
ut dx,
∇u∇ut dx
(3.2)
0
t
0
Ω
Ω
t
0
Gα (t − s) ∇u(t) − ∇u(s) dsdx
g(t − s) ∇u(t) − ∇u(s) dsdx −
ρ
u t u t
ρ
u t u t
t
0
Gα (t − s) ∇u(t) − ∇u(s) dsdx
g(t − s)∇u(s)ds
Ω
|∇u| dx +
t
Ω
t
Gα (t − s) ∇u(t) − ∇u(s) dsdx
+ (1 + α)
−
2
t
0
t
0
0
Gα (s)ds
Ω
∇ut 2 dx
(3.3)
Gα (t − s) u(t) − u(s) dsdx
g(t − s) u(t) − u(s) dsdx
Gα (s)ds
t
Ω
ρ+2
ut dx.
We estimate some terms in the expressions of Ψ (t) and χ (t) separately. We denote by
t
G0 the value G0 := 00 Gα (s)ds for some t0 > 0 (selected so that E(t0 ) > 0, and thereafter by Proposition 2.4, W(t0 ) > 0). Applying Lemma 2.2 with δ = l/4, δ = l/4(1 + ḡ),
δ = G0 /4(1 + α), and δ = G0 /4, we obtain
Ω
∇u
t
0
≤ ḡ
g(t − s)∇u(s)dsdx
Ω
|∇u|2 dx +
Ω
∇u
t
0
g(t − s) ∇u(s) − ∇u(t) dsdx
1504
Asymptotic behavior for a viscoelastic problem
≤
Ω
∇u
t
0
l
+ ḡ
4
Ω
Ω
∇u t
t
0
Ω
∇u t
t
0
|∇u|2 dx +
Ω
1 + ḡ ḡα Gα ∇u (t),
lα
Ω
∇ut 2 dx + (1 + α)ḡα Gα ∇u (t),
αG0
g(t − s) ∇u(t) − ∇u(s) dsdx
G0
4
≤
Gα (t − s) ∇u(t) − ∇u(s) dsdx
G0
4(1 + α)
≤
Gα (t − s) ∇u(t) − ∇u(s) dsdx
l
≤
4(1 + ḡ)
ḡ
l
|∇u|2 dx + (g ∇u)(t),
Ω
∇ut 2 dx + ḡ (g ∇u)(t),
G0
(3.4)
respectively.
We also have
t
Ω
0
t
g(t − s)∇u(s)ds
0
Gα (t − s) ∇u(t) − ∇u(s) dsdx
t
2
1 g(t − s) ∇u(s) − ∇u(t) ds dx
≤
2 Ω
0
(3.5)
t
2
1 Gα (t − s) ∇u(t) − ∇u(s) ds
+
dx
2 Ω
t
+
0
0
g(s)ds
Ω
∇u
t
0
Gα (t − s) ∇u(t) − ∇u(s) dsdx.
Therefore,
t
Ω
0
≤
t
g(t − s)∇u(s)ds
ḡ
(g ∇u)(t) + ḡ
2
+
0
Ω
Gα (t − s) ∇u(t) − ∇u(s) dsdx
∇u
t
0
Gα (t − s) ∇u(t) − ∇u(s) dsdx
(3.6)
ḡα Gα ∇u (t).
2α
By Lemma 2.2 again with δ = l/4, we find
Ω
∇u∇ut dx ≤
l
4
Ω
|∇u|2 dx +
1
l
Ω
∇ut 2 dx.
(3.7)
Nasser-Eddine Tatar 1505
Finally, by virtue of the embedding stated at the beginning of the proof of Proposition 2.4
(see (2.29)), we can deduce that
Ω
ρ
u t u t
t
0
Gα (t − s) u(t) − u(s) dsdx
p
CC
(ρ + 1)G0
≤
ḡα Gα ∇u (t) +
(ρ + 1)G0
4α
Ω
(3.8)
∇ut 2 dx.
Here we have used Lemma 2.2 with δ = αC/(ρ
+ 1)G0 . With δ = C/(ρ
+ 1)G0 , we find that
Ω
ρ
u t u t
t
0
g(t − s) u(t) − u(s) dsdx
p ḡ
CC
(ρ + 1)G0
≤
(g ∇u)(t) +
(ρ + 1)G0
4
Ω
(3.9)
∇ut 2 dx.
Taking into account all the above estimates (3.4), (3.6), (3.7), (3.8), (3.9), (3.2), and (3.3)
in (3.1), we entail that for t ≥ t0 ,
1 C + C p
1
W (t) ≤ −
−ε 1+ +
2
l 2(ρ + 1)
Ω
1 1 1
− λ − ḡ − εḡ
+
+ +
2
− λ α−
l
G0
∇ut 2 dx − εl
4
C p C
(ρ + 1)2 G0
Ω
|∇u|2 dx
(g ∇u)(t) −
εG0
ρ+1
p
4λḡα
εḡα (1 + ḡ)2 (1 + α)2 1 α2 CC
+ +
+
−
α
α
l
G0
2
G0
We must point out here that, to avoid a contradiction, the term in
appears in the derivative of Ψ(t) (see (3.2)) has been estimated by
ε
ρ+1
Ω
ρ+2
ut dx ≤
Ω
ρ+2
ut dx
Gα ∇u (t).
(3.10)
ρ+2 dx
Ω |ut |
ε
1 ut 2(ρ+1) dx + C p
∇ut 2 dx
ρ+1 2 Ω
2 Ω
ε ∇ut 2 dx.
≤
C + Cp
2(ρ + 1)
Ω
which
(3.11)
From (3.10), it is clear that for sufficiently small ε and ḡα ≤ α2 /8λ, there exists C1 > 0
such that
W (t) ≤ −C1 E(t) + Φ(t) .
(3.12)
The right-hand side inequality in Proposition 2.4 implies that
W (t) ≤ −
C1
W(t).
M
(3.13)
From this, we infer that
W(t) ≤ W t0 e−C1 (t−t0 )/M ,
t ≥ t0 .
(3.14)
1506
Asymptotic behavior for a viscoelastic problem
Then, the left-hand side inequality in Proposition 2.4 allows us to conclude that
E(t) ≤
W t0 −C1 (t−t0 )/M
,
e
m
This completes the proof of the theorem.
t ≥ t0 .
(3.15)
Acknowledgments
The author would like to thank the anonymous referee for valuable comments. The author is also grateful for the financial support and the facilities provided by King Fahd
University of Petroleum and Minerals.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. Ferreira, Existence and uniform decay for
a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001),
no. 14, 1043–1053.
M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho, and J. A. Soriano, Existence and
uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential
Integral Equations 14 (2001), no. 1, 85–116.
M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential
Equations 2002 (2002), no. 44, 1–14.
H. Engler, Weak solutions of a class of quasilinear hyperbolic integro-differential equations describing viscoelastic materials, Arch. Rational Mech. Anal. 113 (1990), no. 1, 1–38.
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in
Applied Mathematics, vol. 12, SIAM, Pennsylvania, 1992.
K. M. Furati and N.-E. Tatar, Uniform boundedness and stability for a viscoelastic problem, to
appear in Appl. Math. Comp.
W. J. Hrusa, Global existence and asymptotic stability for a semilinear hyperbolic Volterra equation
with large initial data, SIAM J. Math. Anal. 16 (1985), no. 1, 110–134.
M. Medjden and N.-E. Tatar, On the wave equation with a temporal nonlocal term, to appear in
ANZIAM J.
S. A. Messaoudi and N.-E. Tatar, Global existence and asymptotic behavior for a nonlinear viscoelastic problem, Math. Sci. Res. J. 7 (2003), no. 4, 136–149.
J. Milota, J. Nečas, and V. Šverák, On weak solutions to a viscoelasticity model, Comment. Math.
Univ. Carolin. 31 (1990), no. 3, 557–565.
M. Renardy, W. J. Hrusa, and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman
Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific &
Technical, Harlow; John Wiley & Sons, New York, 1987.
N.-E. Tatar, On a problem arising in isothermal viscoelasticity, Int. J. Pure Appl. Math. 8 (2003),
no. 1, 1–12.
Q. Tiehu, Asymptotic behavior of a class of abstract semilinear integrodifferential equations and
applications, J. Math. Anal. Appl. 233 (1999), no. 1, 130–147.
Nasser-Eddine Tatar: Department of Mathematical Sciences, College of Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
E-mail address: tatarn@kfupm.edu.sa