Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 72, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
UNIFORM DECAY OF SOLUTIONS FOR COUPLED
VISCOELASTIC WAVE EQUATIONS
JIANGHAO HAO, LI CAI
Abstract. In this article, we consider a system of two coupled viscoelastic
equations with Dirichlet boundary conditions. By using the perturbed energy
method, we obtain a general decay result which depends on the behavior of
the relaxation functions and source terms.
1. Introduction
In this article, we study the coupled system of quasi-linear viscoelastic equations
Z t
|ut |ρ utt − ∆u − ∆utt +
g1 (t − s)∆u(s)ds + f1 (u, v) = 0, (x, t) ∈ Ω × (0, ∞),
0
Z t
|vt |ρ vtt − ∆v − ∆vtt +
g2 (t − s)∆v(s)ds + f2 (u, v) = 0, (x, t) ∈ Ω × (0, ∞),
0
u = v = 0,
(x, t) ∈ ∂Ω × (0, ∞),
u(x, 0) = u0 (x),
ut (x, 0) = u1 (x),
x ∈ Ω̄,
v(x, 0) = v0 (x),
vt (x, 0) = v1 (x),
x ∈ Ω̄,
(1.1)
where Ω is a bounded domain in Rn (n ≥ 1) with smooth boundary ∂Ω, ρ satisfies
2
, n ≥ 3,
n−2
ρ > 0, n = 1, 2.
0<ρ≤
(1.2)
The functions u0 , u1 , v0 , v1 are given initial data. The functions g1 , g2 , f1 , f2 will
be specified later.
The study of the asymptotic behavior of viscoelastic problems has attracted
lots of interest of researchers. The pioneer work of Dafermos [4] studied a onedimensional viscoelastic problem, established some existence and asymptotic stability results for smooth monotone decreasing relaxation functions. Muñoz Rivera
[18] considered equations for linear isotropic viscoelastic solids of integral type, and
established exponential decay and polynomial decay in a bounded domain and in
2010 Mathematics Subject Classification. 35L05, 35L20, 35L70, 93D15.
Key words and phrases. Coupled viscoelastic wave equations; relaxation functions;
uniform decay.
c
2016
Texas State University.
Submitted August 21, 2015. Published March 15, 2016.
1
2
J. HAO, L. CAI
EJDE-2016/72
the whole space respectively. Messaoudi [12] considered a nonlinear viscoelastic
wave equation with source and damping terms
Z t
g(t − s)∆u(s)ds + ut |ut |m−1 = u|u|p−1 .
(1.3)
utt − ∆u +
0
He established blow-up result for solutions with negative initial energy and m < p,
and gave a global existence result for arbitrary initial if m ≥ p. This work was later
improved by Messaoudi [13].
Liu [10] considered the equation with initial-boundary value conditions
Z t
utt − ∆u +
g(t − τ )∆u(τ )dτ + a(x)|ut |m ut + b|u|r u = 0,
(1.4)
0
he established exponential or polynomial decay result which depends on the rate
of the decay of the relaxation function g. Song et al [23] studied the problem (1.4)
with replacing a(x)|ut |m ut by a(x)ut , they obtained general decay result.
Cavalcanti et al [3] discussed the wave equation
Z t
utt − ∆u +
g(t − τ )div[a(x)∇u(τ )]dτ + b(x)f (ut ) = 0
(1.5)
0
on a compact Riemannian manifold (M, g) subject to a combination of locally
distributed viscoelastic and frictional dissipations. It is shown that the solutions
decay according to the law dictated by the decay rates corresponding to the slowest
damping.
Muñoz Rivera and Naso [19] studied a viscoelastic systems with nondissipative
kernels, and showed that if the kernel function decays exponentially to zero, then
the solution decays exponentially to zero. On the other hand, if the kernel function
decays polynomially as t−p , then the corresponding solution also decays polynomially to zero with the same rate of decay.
Wu [24] considered the equation
Z t
|ut |ρ utt − ∆u − ∆utt +
g(t − s)∆u(s)ds + ut = |u|p−2 u,
(1.6)
0
he also improved some results to obtain the decay rate of the energy under the
suitable conditions.
Cavalcanti et al [2] discussed a quasilinear initial-boundary value problem of
equation
Z t
ρ
|ut | utt − ∆u − ∆utt +
g(t − s)∆u(τ )dτ − γ∆ut = bu|u|p−2 ,
(1.7)
0
with Dirichlet boundary condition, where ρ > 0, γ ≥ 0, p ≥ 2, b = 0. An exponential decay result for γ > 0 and b = 0 has been obtained. For γ = 0 and b > 0,
Messaoudi and Tatar [16], [17] showed that there exists an appropriate set, called
stable set, such that if the initial data are in stable set, the solution continuous to
live there forever, and the solution approaches zero with an exponential or polynomial rate depending on the decay rate of relaxation function. For other related
single wave equation, we refer the reader to [7, 14, 21].
Han and Wang [6] studied the initial-boundary value problem for a coupled
system of nonlinear viscoelastic equations
Z t
utt − ∆u +
g1 (t − τ )∆u(τ )dτ + |ut |m−1 ut = f1 (u, v), (x, t) ∈ Ω × (0, T ),
0
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UNIFORM DECAY OF SOLUTIONS
Z
vtt − ∆v +
3
t
g2 (t − τ )∆v(τ )dτ + |vt |m−1 vt = f2 (u, v),
(x, t) ∈ Ω × (0, T ),
0
u = v = 0,
(x, t) ∈ ∂Ω × (0, T ),
u(x, 0) = u0 (x), ut (x, 0) = u1 (x),
x ∈ Ω,
v(x, 0) = v0 (x), vt (x, 0) = v1 (x),
x ∈ Ω.
(1.8)
Existence of local and global solutions, uniqueness, and blow up in finite time were
obtained when f1 , f2 , g1 , g2 and the initial values satisfy some conditions.
Messaoudi and Said-Houari [15] dealt with the problem (1.8) and proved a global
nonexistence of solutions for a large class of initial data for which the initial energy
takes positive values. Also, Said-Houari et al [22] discussed (1.8) and proved a
general decay result.
Liu [9] studied the coupled equations
Z t
utt − ∆u +
g(t − s)∆u(x, s)ds + f1 (u, v) = 0,
0
(1.9)
Z t
vtt − ∆v +
h(t − s)∆v(x, s)ds + f2 (u, v) = 0,
0
he proved that the decay rate of the solution energy is similar to that of relaxation
functions which is not necessarily of exponential or polynomial type. Others similar
problems were considered in [1, 20].
Motivated by the above researches, we consider the system (1.1). Liu [8] already
considered the system (1.1), and obtained the exponential or polynomial decay of
the solutions energy depending on the decay rate of the relaxation functions. In
[8], the relaxation functions gi (t) (i = 1, 2) satisfy gi0 (t) ≤ −ξi gipi (t) for all t ≥ 0,
pi ∈ [1, 3/2) and some constants ξ1 , ξ2 . In this paper, the conditions have been
replaced by gi0 (t) ≤ −ξi (t)gi (t) where ξi (t) are positive non-increasing functions.
This allow us to obtain a general decay rate than just exponential or polynomial
type. We use the perturbed energy method to obtain a general decay of solutions
energy. The rest of this article is organized as follows. Some preparation and main
result are given in Section 2. In Section 3, we give the proof of our main result.
2. Preliminaries and statement of main results
We denote the norm in Lρ (Ω) by k · kρ , 1 ≤ ρ < ∞. The Dirichlet norm in H01 (Ω)
is k∇ · k2 . C and Ci denote general constants, which may be different in different
estimates.
Throughout this paper, we use the following notation,
Z t
(φ ◦ ψ)(t) =
φ(t − τ )kψ(t) − ψ(τ )k22 dτ.
0
To state our main result, we need the following assumptions.
(A1) gi : R+ → R+ , i = 1, 2, are differentiable functions such that
Z +∞
gi (0) > 0, 1 −
gi (s)ds = li > 0,
0
and there exist non-increasing functions ξ1 , ξ2 : R+ → R+ satisfying
gi0 (t) ≤ −ξi (t)gi (t),
t ≥ 0.
4
J. HAO, L. CAI
EJDE-2016/72
(A2) There exists nonnegative function F (u, v) such that
∂F (u, v)
∂F (u, v)
, f2 (u, v) =
,
∂u
∂v
and there exist constants C, d > 0 such that
f1 (u, v) =
uf1 (u, v) + vf2 (u, v) ≥ CF (u, v),
p
p
|f1 (u, v)| ≤ d(|u + v|p−1 + |u| 2 −1 |v| 2 ),
p
p
|f2 (u, v)| ≤ d(|u + v|p−1 + |u| 2 |v| 2 −1 ),
where p > 2 if n = 1, 2 and 2 < p ≤
2(n−1)
n−2
if n ≥ 3.
By using the Galerkin method, as in [11], we can obtain the existence of a local
weak solution to (1.1). We omit the proof here.
Theorem 2.1. Assume that (A1), (A2) hold. For the initial data (u0 , v0 , u1 , v1 ) ∈
(H01 (Ω))4 , there exists at least one weak local solution (u, v) such that for some
T > 0,
u, v ∈ L∞ (0, T ; H01 (Ω)),
ut , vt ∈ L∞ (0, T ; H01 (Ω)),
utt , vtt ∈ L2 (0, T ; H01 (Ω)).
We introduce the energy functional of system (1.1),
1
1
1
1
2
2
E(t) =
kut kρ+2
kvt kρ+2
ρ+2 +
ρ+2 + k∇ut k2 + k∇vt k2
ρ+2
ρ+2
2
2
Z t
1
1
1
1−
g1 (s)ds k∇uk22
+ (g1 ◦ ∇u) + (g2 ◦ ∇v) +
(2.1)
2
2
2
0
Z
Z
t
1
+
1−
g2 (s)ds k∇vk22 +
F (u, v)dx.
2
0
Ω
It is easy to prove that
1
1
1
1
E 0 (t) = (g10 ◦ ∇u)(t) + (g20 ◦ ∇v)(t) − g1 (t)k∇u(t)k22 − g2 (t)k∇vk22 ≤ 0. (2.2)
2
2
2
2
Then we have
k∇ut k22 + k∇vt k22 + l1 k∇uk22 + l2 k∇vk22 ≤ 2E(0).
(2.3)
Our main result reads as follows.
Theorem 2.2. Assume that (A1), (A2) hold. Let (u0 , v0 , u1 , v1 ) ∈ (H01 (Ω))4 be
given, and (u, v) be the solution to (1.1). Then for any t1 > 0 there exist positive
constants C and α such that for all t ≥ t1 ,
−α
E(t) ≤ Ce
Rt
t1
ξ(τ )dτ
,
where ξ(t) = min {ξ1 (t), ξ2 (t)}.
3. Decay result
To prove the general decay result, we define the perturbed modified energy functional
L(t) = M E(t) + εI(t) + J(t),
where M and ε are positive constants to be specified later and
Z
Z
Z
Z
1
1
ρ
ρ
|ut | ut udx +
|vt | vt vdx +
∇ut ∇udx +
∇vt ∇vdx,
I(t) =
ρ+1 Ω
ρ+1 Ω
Ω
Ω
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UNIFORM DECAY OF SOLUTIONS
5
J(t) = J1 (t) + J2 (t),
where
Z
|ut |ρ ut t
g1 (t − τ ) (u(t) − u(τ )) dτ dx,
ρ+1
Ω
0
Z Z
|vt |ρ vt t
J2 (t) =
∆vt −
g2 (t − τ ) v(t) − v(τ ) dτ dx.
ρ+1
Ω
0
Firstly, we have the following lemmas.
J1 (t) =
Z ∆ut −
Lemma 3.1 ([5]). Under assumption (A1), if (u, v) is the solution of (1.1), then
the following hold for i = 1, 2:
Z Z t
2
gi (t − τ ) (∇u(t) − ∇u(τ )) dτ dx ≤ Ci (gi ◦ ∇u),
(3.1)
Ω
0
Z Z t
2
0
0
−gi (t − τ ) (∇u(t) − ∇u(τ )) dτ dx ≤ −Ci (gi ◦ ∇u).
(3.2)
Ω
0
Lemma 3.2. Let (A1), (A2) hold and (u, v) be the solution of (1.1). Then
C1
1
l1
(g1 ◦ ∇u) +
kut kρ+2
I 0 (t) ≤ − k∇uk22 +
ρ+2
2
4δ
ρ+1
l2
C2
1
2
+ k∇ut k22 − k∇vk22 +
(g2 ◦ ∇v) +
kvt kρ+2
(3.3)
ρ+2 + k∇vt k2
2
4δ
ρ+1
Z
−C
F (u, v)dx,
Ω
in which δ = min{ l21 , l22 }.
Proof. Differentiating I(t) and using (1.1), we obtain
Z
Z t
1
ρ+2
0
2
kut kρ+2 − k∇uk2 +
I (t) =
∇u(t)
g1 (t − τ )∇u(τ ) dτ dx
ρ+1
Ω
0
Z
Z t
1
2
+
kvt kρ+2
∇v(t)
g2 (t − τ )∇v(τ ) dτ dx
ρ+2 − k∇vk2 +
ρ+1
Ω
0
Z
− (uf1 + vf2 )dx + k∇ut k22 + k∇vt k22 .
(3.4)
Ω
Using (3.1) and Young’s inequality, we can estimate the third term of (3.4) as
follows
Z
Z t
∇u(t)
g1 (t − τ )∇u(τ ) dτ dx
Ω
0
Z
Z t
=
∇u
g1 (t − τ ) (∇u(τ ) − ∇u(t) + ∇u(t)) dτ dx
Ω
0
Z t
Z
Z t
= k∇uk22
g1 (τ )dτ +
∇u
g(t − τ ) (∇u(τ ) − ∇u(t)) dτ dx
0
Ω
0
Z Z t
Z t
2
1
g1 (t − τ )|∇u(τ ) − ∇u(t)|dτ dx
≤ k∇uk22
g1 (τ )dτ + δk∇uk22 +
4δ Ω
0
0
Z t
C
1
≤ k∇uk22
g1 (τ )dτ + δk∇uk22 +
(g1 ◦ ∇u).
4δ
0
(3.5)
6
J. HAO, L. CAI
Similarly, we obtain
Z
t
Z
∇v(t)
Ω
≤
EJDE-2016/72
g2 (t − τ )∇v(τ ) dτ dx
0
k∇vk22
Z
t
g2 (τ )dτ +
0
δk∇vk22
C2
+
(g2 ◦ ∇v).
4δ
From (3.4)–(3.6), we obtain
Z t
C1
1
0
I (t) ≤ − 1 −
g1 (τ )dτ − δ k∇uk22 +
(g1 ◦ ∇u) +
kut kρ+2
ρ+2
4δ
ρ
+
1
0
Z
t
C2
+ k∇ut k22 − 1 −
g2 (τ )dτ − δ k∇vk22 +
(g2 ◦ ∇v)
4δ
0
Z
1
2
kvt kρ+2
(uf1 + vf2 )dx.
+
ρ+2 + k∇vt k2 −
ρ+1
Ω
We can choose δ = min{ l21 , l22 }. From (3.7) and (A1), (3.3) follows.
Lemma 3.3. Under assumptions (A1), (A2), we have
J 0 (t) ≤ δ + 2δ(1 − l2 )2 + 2Cδ k∇vk22 + δ + 2δ(1 − l1 )2 + 2Cδ k∇uk22
3C2
3C1
+ 2δC2 (g2 ◦ ∇v) +
+ 2δC1 (g1 ◦ ∇u)
+
4δ
4δ
C2
C1
C2
C1
0
+
+
+
(g2 ◦ ∇v) +
(g10 ◦ ∇u)
4δ
4δ(ρ + 1)
4δ
4δ(ρ + 1)
Z t
δ
ρ
(2E(0)) k∇vt k22
−
g2 (s)ds − δ −
ρ+1
0
Z t
δ
ρ
−
g1 (s)ds − δ −
(2E(0)) k∇ut k22
ρ+1
0
Z
Z
1 t
1 t
−
g2 (s)ds kvt kρ+2
−
g
(s)ds
kut kρ+2
1
ρ+2
ρ+2 .
ρ+1 0
ρ+1 0
(3.6)
(3.7)
(3.8)
Proof. Differentiating J1 (t) and using (1.1), we obtain
Z
Z t
J10 (t) =
∇u(t)
g1 (t − τ ) (∇u(t) − ∇u(τ )) dτ dx
Ω
0
Z Z t
Z t
−
g1 (t − τ )∇u(τ )dτ
g1 (t − τ ) (∇u(t) − ∇u(τ )) dτ dx
Ω
0
0
Z
Z t
Z t
+
f1 (u, v)
g1 (t − τ ) (u(t) − u(τ )) dτ dx −
g1 (s)ds k∇ut k22
Ω
0
0
Z
Z t
0
−
∇ut
g1 (t − τ ) (∇u(t) − ∇u(τ )) dτ dx
(3.9)
Ω
0
Z
Z t
1
ρ
|ut | ut
g10 (t − τ ) (u(t) − u(τ )) dτ dx
−
ρ+1 Ω
0
Z
1 t
−
g1 (s)ds kut kρ+2
ρ+2 .
ρ+1 0
EJDE-2016/72
UNIFORM DECAY OF SOLUTIONS
7
By using Young’s inequality and (3.1), we obtain that for some δ > 0,
Z
Z t
C1
∇u(t)
g1 (t − τ ) (∇u(t) − ∇u(τ )) dτ dx ≤ δk∇uk22 +
(g1 ◦ ∇u). (3.10)
4δ
Ω
0
For the second term of (3.9), employing Young’s inequality, (A1) and (3.1), we have
for some δ > 0,
Z Z t
Z t
g1 (t − τ )∇u(τ )dτ
g1 (t − τ ) (∇u(t) − ∇u(τ )) dτ dx
Ω
0
0
Z Z t
Z t
2
≤δ
g1 (t − τ ) (∇u(τ ) − ∇u(t)) dτ +
g1 (t − τ )|∇u(t)|dτ dx
+
Ω
0
1
4δ
Z Z
0
2
t
g1 (t − τ ) (∇u(t) − ∇u(τ )) dτ
Ω
dx
0
(3.11)
Z Z t
2
1
≤ (2δ + )
g1 (t − τ )|∇u(t) − ∇u(τ )|dτ dx
4δ Ω
0
Z Z t
2
+ 2δ
g1 (t − τ )dτ |∇u(t)|2 dx
Ω
0
1
≤ (2δ + )C1 (g1 ◦ ∇u) + 2δ(1 − l1 )2 k∇uk22 .
4δ
Thanks to Young’s inequality, Sobolev embedding theorem and (3.1), for some δ > 0
we have
Z
Z t
f1 (u, v)
g1 (t − τ ) (u(t) − u(τ )) dτ dx
Ω
0
Z
Z Z t
2
1
2
≤δ
f1 (u, v)dx +
g(t − τ ) (u(t) − u(τ )) dτ dx
4δ Ω
0
ZΩ
C1
2
≤δ
f1 (u, v)dx +
(g1 ◦ u)
4δ
Ω
Z
C1
≤δ
f12 (u, v)dx +
(g1 ◦ ∇u).
4δ
Ω
Using (A2) and the Sobolev embedding theorem and (2.3), we have
Z
Z
Z
2
2(p−1)
f1 (u, v)dx ≤ C
|u + v|
dx +
|u|p−2 |v|p dx
Ω
Ω
Ω
2(p−1)
2(p−1)
2(p−2)
≤ C kuk2(p−1) + kvk2(p−1) + kukn(p−2) + kvk2pnp
n−1
2 p−2
2
2 p−2
2
≤ C k∇uk2
k∇uk2 + k∇vk2
k∇vk2
p−3
p−1
+ C k∇uk22
k∇uk22 + k∇vk22
k∇vk22
2E(0) p−2
2E(0) p−2
≤C
k∇uk22 +
k∇vk22
l1
l2
2E(0) p−3
2E(0) p−1
+C
k∇uk22 +
k∇vk22
l1
l2
2
2
≤ C k∇uk2 + k∇vk2 .
8
J. HAO, L. CAI
EJDE-2016/72
Then we obtain
Z
Z
t
g1 (t − τ ) (u(t) − u(τ )) dτ dx
f1 (u, v)
0
Ω
≤ Cδ
k∇uk22
+
k∇vk22
C1
+
(g1 ◦ ∇u).
4δ
(3.12)
The fifth term of (3.9) yields
Z
Z
∇ut
Ω
t
g10 (∇u(t) − ∇u(τ )) dτ dx ≤ δk∇ut k22 +
0
C1 0
(g ◦ ∇u).
4δ 1
(3.13)
We estimate the sixth term of (3.9) by using Young’s inequality, Sobolev embedding
theorem and (2.3) as follows
Z
Z t
1
|ut |ρ ut
g10 (t − τ ) (u(t) − u(τ )) dτ dx
ρ+1 Ω
0
C1
δ
2(ρ+1)
kut k2(ρ+1) +
(g 0 ◦ ∇u)
≤
ρ+1
4δ(ρ + 1) 1
δ
C1
ρ
≤
(2E(0)) k∇ut k22 +
(g 0 ◦ ∇u).
ρ+1
4δ(ρ + 1) 1
(3.14)
Inserting (3.10)–(3.14) into (3.9), we obtain
J10 (t) ≤ δ + 2δ(1 − l1 )2 + Cδ k∇uk22 + Cδk∇vk22
3C
C
C1
1
1
+ 2δC1 (g1 ◦ ∇u) +
+
+
(g10 ◦ ∇u)
4δ
4δ
4δ(ρ + 1)
Z t
δ
ρ
(2E(0)) k∇ut k22
−
g1 (s)ds − δ −
ρ+1
0
Z t
1
−
g1 (s)ds kut kρ+2
ρ+2 .
ρ+1 0
(3.15)
In the same way, we conclude that
J20 (t) ≤ δ + 2δ(1 − l2 )2 + Cδ k∇vk22 + Cδk∇uk22
C
3C
C2
2
2
+ 2δC2 (g2 ◦ ∇v) +
+
(g20 ◦ ∇v)
+
4δ
4δ
4δ(ρ + 1)
Z t
δ
ρ
−
g2 (s)ds − δ −
(2E(0)) k∇vt k22
ρ+1
0
Z
1 t
−
g2 (s)ds kvt kρ+2
ρ+2 .
ρ+1 0
Combining the estimates (3.15) and (3.16), we can obtain (3.8).
(3.16)
EJDE-2016/72
UNIFORM DECAY OF SOLUTIONS
9
Proof of Theorem 2.1. It is not difficult to find positive constants a1 , a2 such that
a1 E(t) ≤ L(t) ≤ a2 E(t). Differentiating L(t), we have
Z
1 t
L0 (t) ≤ −
g1 (s)ds − ε kut kρ+2
ρ+2
ρ+1 0
Z t
δ
ρ
−
g1 (s)ds − δ −
(2E(0)) − ε k∇ut k22
ρ+1
0
Z t
1
−
g2 (s)ds − ε kvt kρ+2
ρ+2
ρ+1 0
Z
t
δ
ρ
−
g2 (s)ds − δ −
(2E(0)) − ε k∇vt k22
ρ+1
0
3C
(3.17)
C
ε
1
1
+ 2δC1 +
(g1 ◦ ∇u)
+
4δ
4δ
3C
C2 ε 2
+ 2δC2 +
(g2 ◦ ∇v)
+
4δ
4δ
M
l1
−
g1 (t) + ε − δ − 2δ(1 − l1 )2 − 2Cδ k∇uk22
2
2
M
l2 −
g2 (t) + ε − δ − 2δ(1 − l2 )2 − 2Cδ k∇vk22
2
Z2
− Cε
F (u, v)dx.
Ω
For any t0 > 0 we can pick ε, δ > 0 small enough, M so large such that for t > t0
there exist constants η1 , η2 , η3 , η4 > 0, and
ρ+2
2
2
L0 (t) ≤ −η1 (kut kρ+2
ρ+2 + kvt kρ+2 ) − η2 (k∇ut k2 + k∇vt k2 )
+ η3 ((g1 ◦ ∇u) + (g2 ◦ ∇v)) −
η4 (k∇uk22
+
k∇vk22 )
Z
− εC
F (u, v)dx.
Ω
(3.18)
Then, we can choose t1 > t0 such that η, C > 0 and (3.18) takes the form
L0 (t) ≤ −ηE(t) + C ((g1 ◦ ∇u) + (g2 ◦ ∇v)) ,
t ≥ t1 .
(3.19)
Multiplying (3.19) by ξ(t), by using (A1) we have
ξ(t)L0 (t)
Z Z t
≤C
ξ1 (t − τ )g1 (t − τ )|∇u(t) − ∇u(τ )|2 dτ dx
Ω 0
Z Z t
+C
ξ2 (t − τ )g2 (t − τ )|∇v(t) − ∇v(τ )|2 dτ dx − ηξ(t)E(t)
Ω 0
Z Z t
≤ −C
g10 (t − τ )|∇u(t) − ∇u(τ )|2 dτ dx
Ω 0
Z Z t
−C
g20 (t − τ )|∇v(t) − ∇v(τ )|2 dτ dx − ηξ(t)E(t)
Ω
(3.20)
0
≤ −CE 0 (t) − ηξ(t)E(t).
where ξ(t) = min {ξ1 (t), ξ2 (t)}. Thanks to (A1), we obtain
d
(ξ(t)L(t) + CE(t)) ≤ −ηξ(t)E(t),
dt
t ≥ t1 .
(3.21)
10
J. HAO, L. CAI
EJDE-2016/72
By defining the functional
F (t) := ξ(t)L(t) + CE(t) ∼ E(t),
(3.22)
we have
F 0 (t) ≤ −αξ(t)F (t).
Then integrating over (t1 , t), we have
F (t) ≤ F (t1 )e
−α
Rt
t1
By using (3.22) again, the decay result follows.
ξ(τ )dτ
(3.23)
.
Acknowledgements. The authors want to thank the anonymous referees for their
valuable comments and suggestions which lead to the improvement of this paper.
This work was partially supported by NNSF of China (61374089), NSF of Shanxi
Province (2014011005-2), Shanxi Scholarship council of China (2013-013), Shanxi
international science and technology cooperation projects (2014081026).
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Jianghao Hao (corresponding author)
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
E-mail address: hjhao@sxu.edu.cn
Li Cai
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
E-mail address: 2829742149@qq.com