Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 22, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE AND DECAY OF SOLUTIONS TO A
VISCOELASTIC PLATE EQUATION
SALIM A. MESSAOUDI, SOH EDWIN MUKIAWA
Abstract. In this article we study the fourth-order viscoelastic plate equation
Z t
g(t − τ )∆2 u(τ )dτ = 0
utt + ∆2 u −
0
in the bounded domain Ω = (0, π)×(−`, `) ⊂ R2 with non traditional boundary
conditions. We establish the well-posedness and a decay result.
1. Introduction
This article is devoted to the well-posedness and the decay rate of the energy
functional for the fourth-order viscoelastic plate problem
Z t
utt + ∆2 u −
g(t − τ )∆2 u(τ )dτ = 0, Ω × (0, T )
0
u(0, y, t) = uxx (0, y, t) = 0,
for (y, t) ∈ (−`, `) × (0, T )
u(π, y, t) = uxx (π, y, t) = 0,
for (y, t) ∈ (−`, `) × (0, T )
uyy (x, ±l, t) + σuxx (x, ±l, t) = 0,
uyyy (x, ±l, t) + (2 − σ)uxxy (x, ±l, t) = 0,
u(x, y, 0) = u0 (x, y),
(1.1)
for (x, t) ∈ (0, π) × (0, T )
for (x, t) ∈ (0, π) × (0, T )
ut (x, y, 0) = u1 (x, y),
in Ω
where Ω = (0, π)×(−`, `), 0 < σ < 12 and g is a positive and nonincreasing function.
This type of problems models the motion of a viscoelastic plate. The fundamental
work of Ferrero and Gazzola [14] in 2013, where they modeled a suspension bridge
as a rectangular plate with the same boundary conditions as (1.1), suggests the
investigation of the viscoelastic material used in construction. Al-Gwaiz et al [1]
also investigated the bending and stretching energies of the rectangular plate model
suggested in [14]. Contributions on the analysis of a suspension bridge have also
come from Mckenna and Walter [24], Mckenna et al [15], Ma and Zhong [23] and
Bochicchio et al [6].
The existence, decay and blow up properties of viscoelastic problems has attracted a lot of attention since the pioneer work by Dafermos [12, 13] in 1970.
Hence, a considerable number of results for models similar to (1.1), for both second
2010 Mathematics Subject Classification. 35L35, 37B25, 34D20, 74H20, 74H25.
Key words and phrases. Existence; decay; plate viscoelastic; fourth order.
c
2016
Texas State University.
Submitted October 12, 2015. Published January 13, 2016.
1
2
S. A. MESSAOUDI, S. E. MUKIAWA
EJDE-2016/22
and fourth order have been established. We begin with the result of Messaoudi
[25], where he considered
Z t
utt − ∆u +
g(t − τ )∆u(τ )dτ = 0, in Ω × (0, +∞)
(1.2)
0
with general conditions on the relaxation function g, and proved a general decay
result that is not necessarily of exponential or polynomial type. His result generalized and improved many results in literature such as [4, 5, 2, 3, 10]. Rivera et al
[27] considered the fourth-order equation
Z t
utt + γ∆utt + ∆2 u −
g(t − τ )∆2 u(τ )dτ = 0, in Ω × (0, T )
(1.3)
0
together with initial and dynamical boundary conditions and proved that the sum
of the first and second energies decays exponentially (polynomially) if the kernel
g decays exponentially (polynomially). Mustafa and Ghassan [28] considered the
plate equation
utt + ∆2 u = 0, in Ω × (0, +∞)
(1.4)
with viscoelastic damping localized on a part of the boundary and established a
decay result. For more results related to the plate equation, we refer the reader to
Messaoudi [26], Kang [17], Santos and Junior [29], Lagnese [19], Horn and Lasiecka
[16], Lasiecka [20], and Lasiecka et al [21], Cabanillas et al [8] Lasiecka et al [9].
The aim of this work is to take advantage of the techniques used in [25] and
the new model in [14] to establish a global existence and general decay results for
problem (1.1). We organize this work as follows. In section 2, we present some
important and fundamental materials to be used in establishing our main results.
In section 3, we state and prove the global existence result. Finally, in section 4 we
state and prove the general decay result.
2. Preliminaries
In this section, we present some fundamental materials needed for the proof of
our main results. For this, we assume the following conditions on the relaxation
function g.
(A1) g : R+ → R+ is a differentiable function such that
Z +∞
g(0) > 0, 1 −
g(s)ds = l0 > 0.
(2.1)
0
(A2) There exists a differentiable function ξ satisfying
g 0 (t) ≤ −ξ(t)g(t),
ξ(t) > 0,
ξ 0 (t) ≤ 0,
t≥0
Z +∞
∀t > 0,
ξ(s)ds = +∞
(2.2)
0
The following three functions satisfy (A1)–(A2).
g1 (t) =
g2 (t) =
a
,
(1 + t)p
ae−t
,
(1 + t)
a > 0,
p
p > 1, a > 0, g3 (t) = ae−b(1+t) ,
0 < p ≤ 1, a, b > 0.
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EXISTENCE AND DECAY OF SOLUTIONS
3
We introduce the space
H∗2 (Ω) = {w ∈ H 2 (Ω) : w = 0 on {0, π} × (−`, `)},
with the inner product
Z
(u, v)H∗2 = [(∆u∆v + (1 − σ)(2uxy vxy − uxx vyy − uyy vxx )] dx dy,
(2.3)
(2.4)
Ω
and set H(Ω) as the dual of H∗2 (Ω). For completeness, we state some results from
Ferrero and Gazzola [14].
Lemma 2.1 ([14]). Assume 0 < σ < 1/2. Then the norm k · kH∗2 (Ω) given by
k · k2H 2 (Ω) = (·, ·)H∗2 is equivalent to the usual H 2 (Ω)-norm. Moreover, H∗2 (Ω) is a
∗
Hilbert space when endowed with the scalar product (u, v)H∗2 .
Theorem 2.2 ([14]). Assume 0 < σ < 1/2 and let f ∈ L2 (Ω). Then there exists a
unique function u ∈ H∗2 (Ω) such that
Z
Z
[∆u∆v + (1 − σ)(2uxy vxy − uxx vyy − uyy vxx )] dx dy =
f v,
(2.5)
Ω
for all v ∈
Ω
H∗2 (Ω).
The function u ∈ H∗2 (Ω) satisfying (2.5) is called the weak solution of the stationary problem
∆2 u = f,
u(0, y) = uxx (0, y) = u(π, y) = uxx (π, y) = 0,
(2.6)
uyy (x, ±l) + σuxx (x, ±l) = uyyy (x, ±l) + (2 − σ)uxxy (x, ±l) = 0.
Lemma 2.3 ([30]). Let u ∈ H∗2 (Ω) and assume 1 ≤ p < +∞. Then, there exists a
positive constant Ce = Ce (Ω, p) > 0 such that
kukpp ≤ Ce kukpH 2 (Ω) .
∗
Let us also introduce the energy functional associated to problem (1.1),
Z
Z t
Z
1
1
1 t
u2t +
1−
g(s)ds kuk2H∗2 (Ω) +
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ.
E(t) =
2 Ω
2
2 0
0
(2.7)
3. Well-posedness
In this section, we show that problem (1.1) has a unique global weak solution.
Definition 3.1. A function
u ∈ C([0, T ), H∗2 (Ω)) ∩ C 1 ([0, T ), L2 (Ω)) ∩ C 2 ([0, T ), H(Ω))
is called a weak solution of (1.1) if
Z
Z t
utt w + (u, w)H∗2 (Ω) −
g(t − τ )(u(τ ), w)H∗2 (Ω) dτ = 0,
Ω
∀w ∈ H∗2 (Ω),
0
u(0) = u0 ,
(3.1)
(3.2)
ut (0) = u1 .
H∗2 (Ω) × L2 (Ω).
Theorem 3.2. Let (u0 , u1 ) ∈
Assume that (A1), (A2) hold. Then
problem (1.1) has a unique weak global solution
u ∈ C([0, T ), H∗2 (Ω)),
ut ∈ C([0, T ), L2 (Ω)),
utt ∈ C([0, T ), H(Ω))
(3.3)
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S. A. MESSAOUDI, S. E. MUKIAWA
EJDE-2016/22
Proof. We use the Galerkin approximation method. Let {wj }∞
j=1 be a basis of the
separable space H∗2 (Ω) and Vm = span{w1 , w2 , . . . , wm } be a finite subspace of
H∗2 (Ω) spanned by the first m vectors. Let
um
0 (x, y) =
m
X
aj wj (x, y)
and um
1 (x, y) =
j=1
m
X
bj wj (x, y)
j=1
be sequences in H∗2 (Ω) and L2 (Ω) respectively, such that
2
um
0 → u0 in H∗ (Ω),
2
um
1 → u1 in L (Ω).
(3.4)
We seek a solution of the form
um (x, y, t) =
m
X
cj (t)wj (x, y),
j=1
which satisfies the approximate problem
Z
m
um
tt (x, y, t)wj + (u (x, y, t), wj )H∗2 (Ω)
Ω
Z t
−
g(t − τ )(um (x, y, τ ), wj )H∗2 (Ω) dτ = 0,
∀wj ∈ Vm , j = 1, 2, . . . , m.
(3.5)
0
um (0) = um
0 ,
m
um
t (0) = u1 .
We note that (3.5) leads to system of ODEs with m unknown functions cj , j =
1, 2, . . . , m. Thus, using ODE theory (see [11]), we obtain functions
cj : [0, tm ) → R, j = 1, 2, . . . , m,
which satisfy (3.5) for almost every t ∈ (0, tm ), 0 < tm < T . Therefore, we obtain a
local solution um of (3.5) in a maximal interval [0, tm ), tm ∈ (0, T ]. Next, we show
that tm = T and that the local solution is uniformly bounded independent of m
and t. For this, we multiply (3.5) by c0j (t) and sum over j = 1, 2, . . . , m, to obtain
Z t
1
d h1 m 2
kut kL2 (Ω) + (1 −
g(s)ds)kum k2H∗2 (Ω)
dt 2
2
0
Z
i
1 t
g(t − τ )kum (t) − um (τ )k2H∗2 (Ω) dτ
+
2 0
Z t
1
1
=
g 0 (t − τ )kum (t) − um (τ )k2H∗2 (Ω) dτ − g(t)kum k2H∗2 (Ω)
2 0
2
It follows from (2.7) that
Z
d m
1 t 0
1
E (t) =
g (t − τ )kum (t) − um (τ )k2H∗2 (Ω) dτ − g(t)kum k2H∗2 (Ω) ≤ 0, (3.6)
dt
2 0
2
by assumptions (A1) and (A2). Integrating (3.6) over (0, t), t ∈ (0, tm ) and noting
m
2
2
that (um
0 ) and (u1 ) are bounded in H∗ (Ω) and L (Ω) respectively (as convergent
sequences (3.4)), we obtain
1 m 2
1
2
(3.7)
E m (t) ≤ E m (0) = kum
1 kL2 (Ω) + ku0 kH∗2 (Ω) ≤ C
2
2
where C is a positive constant independent of m and t. Therefore,
Z t
1 m 2
1
ku k 2
+ (1 −
g(s)ds)kum k2H∗2 (Ω)
2 t L (Ω) 2
0
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EXISTENCE AND DECAY OF SOLUTIONS
+
1
2
Z
0
5
t
g(t − τ )kum (t) − um (τ )k2H∗2 (Ω) dτ ≤ C.
This implies
1
l0
2
sup kum
t kL2 (Ω) +
2 t∈(0,tm )
2
sup kum k2H∗2 (Ω) ≤ C.
(3.8)
t∈(0,tm )
So, the approximate solution is uniformly bounded independent of m and t. Therefore, we can extend tm to T . Moreover, we obtain from (3.8) that
(um )is a bounded sequence in L∞ ((0, T ), H∗2 (Ω)),
∞
2
(um
t ) is a bounded sequence in L ((0, T ), L (Ω)).
(3.9)
Thus, there exists a subsequence (uk ) of (um ) such that
uk * u weakly star in L∞ ((0, T ), H∗2 (Ω)) and weakly in L2 ((0, T ), H∗2 (Ω))
ukt * ut weakly star in L∞ ((0, T ), L2 (Ω)) and weakly in L2 ((0, T ), L2 (Ω))
(3.10)
Using that H∗2 (Ω) is compactly embedded in L2 (Ω) (remember that Ω is bounded
and H∗2 (Ω) ⊂ H 2 (Ω)), we can extract a subsequence (ul ) of (uk ) such that
ul → u strongly in L2 ((0, T ), L2 (Ω)),
ul → u a.e in Ω × (0, T ).
Now, replacing (um ) by (ul ) in (3.5) and integrating over (0, t) we obtain
Z
Z t
Z tZ s
ult wj +
(ul , wj )H∗2 (Ω) dt −
g(s − τ )(ul (τ ), wj )H∗2 (Ω) dτ ds
Ω
0
0
0
Z
=
ul1 wj , ∀j ≤ l.
(3.11)
Ω
Letting l → +∞, we obtain
Z
Z t
Z tZ s
ut wj +
(u, wj )H∗2 (Ω) dt −
g(s − τ )(u(τ ), wj )H∗2 (Ω) dτ ds
Ω
0
0
0
Z
=
u1 wj , ∀j ≥ 1.
(3.12)
Ω
This implies
Z
Z t
Z tZ s
ut w = −
(u, w)H∗2 (Ω) dt +
g(s − τ )(u(τ ), w)H∗2 (Ω) dτ ds
Ω
0
0
Z0
+
u1 w, ∀w ∈ H∗2 (Ω).
(3.13)
Ω
Now, observe that the terms in the right-hand side of (3.13) are absolutely continuous since they are functions of t defined by integrals over (0, t), hence differentiable
almost everywhere. Thus, differentiating (3.13), we obtain that for a.e t ∈ (0, T ),
Z
Z t
utt w + (u, w)H∗2 (Ω) −
g(t − τ )(u(τ ), w)H∗2 (Ω) dτ = 0
(3.14)
Ω
0
for all w ∈ L2 ((0, T ), H∗2 (Ω)). To handle the initial conditions, we note that
ul * u weakly in L2 ((0, T ), H∗2 (Ω))
ult * ut weakly in L2 ((0, T ), L2 (Ω))
(3.15)
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S. A. MESSAOUDI, S. E. MUKIAWA
EJDE-2016/22
Thus, using Lions’ Lemma [22], we obtain
ul → u in C([0, T ), L2 (Ω)).
(3.16)
Therefore, ul (x, y, 0) makes sense and ul (x, y, 0) → u(x, y, 0) in L2 (Ω). Also we
have that
ul (x, y, 0) = ul0 (x, y) → u0 (x, y) in H∗2 (Ω).
Hence
u(x, y, 0) = u0 (x, y).
(3.17)
As in [14, 18], let φ ∈ C0∞ (0, T ) and replacing (um ) by (ul ), we obtain from (3.5)
and for any j ≤ l that
Z T
−
(ult (t), wj )L2 (Ω) φ0 (t)dt
0
(3.18)
Z T
Z TZ t
(ul (t), wj )H∗2 (Ω) φ(t)dt +
=−
0
0
0
g(t − τ )(ul (τ ), wj )H∗2 (Ω) φ(t)dτ dt.
As l → +∞, we obtain that
Z T
−
(ut (t), wj )L2 (Ω) φ0 (t)dt
0
Z
=−
0
T
T
Z
(u(t), wj )H∗2 (Ω) φ(t)dt +
0
Z
t
g(t − τ )(u(τ ), wj )H∗2 (Ω) φ(t)dτ dt,
0
for all j ≥ 1. This implies
Z T
−
(ut (t), w)L2 (Ω) φ0 (t)dt
0
Z
=−
for all w ∈
2
T
T
Z
(u(t), w)H∗2 (Ω) φ(t)dt +
0
H∗2 (Ω). This
2
ut ∈ L ([0, T ), L (Ω)),
0
2
Z
0
t
g(t − τ )(u(τ ), w)H∗2 (Ω) φ(t)dτ dt,
means utt ∈ L ([0, T ), H(Ω)). Thus,
utt ∈ L2 ([0, T ), H(Ω)) =⇒ ut ∈ C([0, T ), H(Ω)). (3.19)
So, ult (x, y, 0) makes sense (see [18, p.116]). It follows that
ult (x, y, 0) → ut (x, y, 0)
in H(Ω).
But
ult (x, y, 0) = ul1 (x, y) → u1 (x, y)
in L2 (Ω).
Hence
ut (x, y, 0) = u1 (x, y).
(3.20)
For the uniqueness, suppose u and ū satisfy (3.14), (3.17) and (3.20). Then v = u−ū
satisfies
Z
Z
t
Ω
vtt w + (v, w)H∗2 (Ω) −
0
g(t − τ )(v(τ ), w)H∗2 (Ω) dτ = 0,
∀w ∈ L2 ((0, T ), H∗2 (Ω)),
(3.21)
v(0) = vt (0) = 0.
Replacing w by vt in (3.21), we obtain
Z
i Z t
1
d h1
2
2
v + kvkH∗2 (Ω) −
g(t − τ )(v(τ ), vt (t))H∗2 (Ω) dτ = 0.
dt 2 Ω t
2
0
(3.22)
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EXISTENCE AND DECAY OF SOLUTIONS
We have that
Z t
g(t − τ )(v(τ ), vt (t))H∗2 (Ω) dτ
J1 = −
0
Z t
Z t
g(s)ds(vt (t), v(t))H∗2 (Ω)
g(t − τ )(vt (t), v(t) − v(τ ))H∗2 (Ω) dτ −
=
0
0
Z t
Z t
d 1
d 1
g(t − τ )
=
g(s)ds
kv(t) − v(τ )k2H∗2 (Ω) dτ −
kv(t)k2H∗2 (Ω)
dt
2
dt
2
0
0
Z
d 1 t
g(t − τ )kv(t) − v(τ )k2H∗2 (Ω) dτ
=
dt 2 0
Z
1 t 0
−
g (t − τ )kv(t) − v(τ )k2H∗2 (Ω) dτ
2 0
Z
1
d 1 t
g(s)dskv(t)k2H∗2 (Ω) + g(t)kv(t)k2H∗2 (Ω) .
−
dt 2 0
2
Inserting (3.23) into (3.22) and taking note of (2.7), we obtain
Z
dẼ(t)
1 t 0
1
=
g (t − τ )kv(t) − v(τ )k2H∗2 (Ω) dτ − g(t)kv(t)k2H∗2 (Ω) ≤ 0,
dt
2 0
2
7
(3.23)
(3.24)
by (A1) and (A2). Integrating (3.24) over (0, t), we obtain
Ẽ(t) ≤ Ẽ(0) = 0.
(3.25)
This implies
kvt k2L2 (Ω) + kvk2H∗2 (Ω) = 0.
Therefore, u = ū. The proof is complete.
4. Decay of solutions
In this section, we discuss the stability of solution of problem (1.1). Let us begin
by defining the Lyapunov functional
F (t) = E(t) + 1 Ψ(t) + 2 χ(t),
where 1 and 2 are positive constants to be specified later and
Z
Ψ(t) =
uut ,
Ω
Z
Z t
χ(t) = −
ut
g(t − τ )(u(t) − u(τ ))dτ dx dy.
Ω
(4.2)
0
Lemma 4.1. Assume (A1), (A2) hold. Then the
(2.7), satisfies
Z
1 t 0
dE(t)
=
g (t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ −
dt
2 0
energy functional, defined in
1
g(t)kuk2H∗2 (Ω) ≤ 0.
2
Proof. By using (3.14) and the density of H∗2 (Ω) in L2 (Ω) we obtain
Z
Z t
utt w + (u, w)H∗2 (Ω) −
g(t − τ )(u(τ ), w)H∗2 (Ω) dτ = 0
Ω
(4.1)
(4.3)
(4.4)
0
for all w ∈ L2 ([0, T ), L2 (Ω)). Repeating exactly the same arguments as in (3.22)(3.24), we obtain the result.
8
S. A. MESSAOUDI, S. E. MUKIAWA
EJDE-2016/22
Lemma 4.2. For every u ∈ H∗2 (Ω), we have
Z Z t
2
g(t − τ )(u(t) − u(τ ))dτ dx dy
Ω
0
Z t
≤ Ce (1 − l0 )
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ,
(4.5)
0
where Ce > 0 is the embedding constant introduced in Lemma 2.3.
Proof. Since g is positive, we have
Z Z t
2
2 Z Z t p
p
g(t − τ ) g(t − τ )(u(t) − u(τ ))dτ
g(t − τ )(u(t) − u(τ ))dτ =
Ω
0
Ω
0
By applying Cauchy-Schwarz, (A1) and Lemma 2.3, we obtain
Z Z t
g(t − τ )(u(t) − u(τ ))dτ )2
(
Ω 0
Z Z t
Z t
≤ (
g(s)ds)(
g(t − τ )(u(t) − u(τ ))2 dτ )
Ω 0
0
Z t
≤ Ce (1 − l0 )
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ.
0
Lemma 4.3. For 1 and 2 small enough, there exists two positive constants α1
and α2 such that
α1 F (t) ≤ E(t) ≤ α2 F (t)
(4.6)
The proof of the above lemma uses similar techniques as in [25, Lemma 3.3]; we
omit it here.
Lemma 4.4. Under assumptions (A1), (A2), the functional
Z
Ψ(t) =
uut
Ω
satisfies, along the solution of (1.1),
Z
Z
1 − l0 t
l0
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ.
Ψ0 (t) ≤
u2t − kuk2H∗2 (Ω) +
2
2l0
0
Ω
Proof. By using (3.14) and replacing w by u, direct differentiations yield
Z
Z t
Ψ0 (t) =
u2t − kuk2H∗2 (Ω) +
g(t − τ )(u(t), u(τ ))H∗2 (Ω) dτ.
Ω
(4.7)
(4.8)
0
By using Cauchy-Schwarz and Young’s inequalities, we estimate the third term
Z t
J2 =
g(t − τ )(u(t), u(τ ))H∗2 (Ω) dτ,
0
for any η > 0, as follows
Z t
J2 ≤
g(t − τ )ku(t)kH∗2 (Ω) ku(τ )kH∗2 (Ω) dτ
0
Z
2
1
1 t
2
g(t − τ )(ku(t) − u(τ )kH∗2 (Ω) + ku(t)kH∗2 (Ω) )dτ
≤ ku(t)kH∗2 (Ω) +
2
2 0
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EXISTENCE AND DECAY OF SOLUTIONS
9
Z
2
1
1 t
g(t − τ )ku(t) − u(τ )kH∗2 (Ω) dτ
ku(t)k2H∗2 (Ω) +
2
2 0
Z t
1
g(t − τ )ku(t)kH∗2 (Ω) )dτ )2
+ (
2 0
Z t
Z t
+
g(t − τ )ku(t) − u(τ )kH∗2 (Ω) dτ
g(t − τ )ku(t)kH∗2 (Ω) dτ .
0
0
By using Lemma 4.2, we obtain
Z t
1
1
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ
(1 + (1 − l0 )2 )kuk2H∗2 (Ω) + (1 − l0 )
2
2
0
Z t
Z t
η
1
+ (
g(t − τ )ku(t)kH∗2 (Ω) dτ )2 + (
g(t − τ )ku(t) − u(τ )kH∗2 (Ω) dτ )2
2 0
2η 0
1
≤ (1 + (1 − l0 )2 (1 + η))kuk2H∗2 (Ω)
2
Z t
1
1
+ (1 − l0 )(1 + )
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ.
2
η 0
(4.9)
Now, substituting (4.9) in (4.8), we obtain
J2 ≤
Ψ0 (t) ≤
Z
1
u2t + ((1 − l0 )2 (1 + η) − 1)kuk2H∗2 (Ω)
2
Ω
Z t
1
1
+ (1 − l0 )(1 + )
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ,
2
η 0
We choose η =
l0
1−l0
(4.10)
∀η > 0.
and obtain the result.
Lemma 4.5. Assume conditions (A1) and (A2) hold. Then the functional
Z
χ(t) = −
Z
Ω
t
g(t − τ )(u(t) − u(τ ))dτ dx dy
ut
(4.11)
0
satisfies, along the solution of (1.1),
χ0 (t) ≤
Z
t
Z
δ
g(s)ds
u2t + (1 + 2(1 − l0 )2 )kuk2H∗2 (Ω)
2
2
0
Ω
Z
Ce g(0) t 0
g (t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ
−
2δ
0
Z t
1
+ (1 − l0 )(δ + )
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ, ∀δ > 0.
δ 0
δ
−
(4.12)
10
S. A. MESSAOUDI, S. E. MUKIAWA
EJDE-2016/22
Proof. By differentiating (4.11) and using (3.14), with u instead of w, we obtain
Z
Z t
Z t
Z
2
χ (t) = −
g(s)ds
ut −
ut
g 0 (t − τ )(u(t) − u(τ )) dτ dx dy
0
Ω
Ω
0
Z
Z t
−
utt
g(t − τ )(u(t) − u(τ )) dτ dx dy
Ω
0
Z
Z t
Z t
Z
=−
g(s)ds
u2t −
ut
g 0 (t − τ )(u(t) − u(τ ))dτ dx dy
0
Ω
Ω
0
Z t
+ u(t),
g(t − τ )(u(t) − u(τ ))dτ
H∗2 (Ω)
0
Z
Z t
t
g(t − τ )(u(t) − u(τ ))dτ 2 dτ.
g(t − τ ) u(τ ),
−
0
H∗ (Ω)
0
0
(4.13)
By using Cauchy-Schwarz inequality, Young’s inequality and Lemma 4.2 for −g 0
instead of g, we estimate the terms in the right-hand side of (4.13). Thus, for the
term
Z
Z t
J3 = −
ut
g 0 (t − τ )(u(t) − u(τ ))dτ dx dy,
Ω
0
we have that for any δ > 0,
Z
δ
u2 +
J3 ≤
2 Ω t
Z
δ
≤
u2 +
2 Ω t
Z
δ
≤
u2 −
2 Ω t
Z Z t
2
1
−g 0 (t − τ )(u(t) − u(τ ))dτ dx dy
2δ Ω
0
Z Z t
Z t
1
0
−g (s)ds
−g 0 (t − τ )(u(t) − u(τ ))2 dτ dx dy
2δ Ω
0
0
Z
Ce g(0) t 0
g (t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ.
2δ
0
(4.14)
For the term
Z
J4 = (u(t),
0
t
g(t − τ )(u(t) − u(τ )dτ )H∗2 (Ω) ,
we have
Z
J4 ≤ ku(t)kH∗2 (Ω)
t
g(t − τ )ku(t) − u(τ )kH∗2 (Ω) dτ
Z t
δ
1
≤ ku(t)k2H∗2 (Ω) + (
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ )2
2
2δ 0
Z
δ
(1 − l0 ) t
2
≤ ku(t)kH∗2 (Ω) +
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ.
2
2δ
0
0
Similarly, for the term
Z
J5 = −
t
Z
g(t − τ )(u(τ ),
0
0
t
g(t − τ )(u(t) − u(τ ))dτ )H∗2 (Ω) dτ,
(4.15)
EJDE-2016/22
EXISTENCE AND DECAY OF SOLUTIONS
we obtain
Z t
Z t
J5 ≤
g(t − τ )ku(τ )kH∗2 (Ω) dτ
g(t − τ )ku(t) − u(τ )kH∗2 (Ω) dτ
0
0
Z
2
δ t
g(t − τ )(ku(t) − u(τ )kH∗2 (Ω) + ku(t)kH∗2 (Ω) )dτ
≤
2 0
Z
2
1 t
g(t − τ )ku(t) − u(τ )kH∗2 (Ω) dτ
+
2δ 0
Z t
2
δ
≤
g(t − τ )ku(t) − u(τ )kH∗2 (Ω) dτ
2 0
Z
2
δ t
g(t − τ )ku(t)kH∗2 (Ω) dτ
+
2 0
Z t
Z t
+ δ(
g(t − τ )ku(t) − u(τ )kH∗2 (Ω) dτ )
g(t − τ )ku(t)kH∗2 (Ω) dτ
0
0
Z
(1 − l0 ) t
+
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ
2δ
0
Z t
1
≤ (δ + )(1 − l0 )
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ
2δ
0
+ δ(1 − l0 )2 kuk2H∗2 (Ω) .
11
(4.16)
By substituting (4.14)–(4.16) in (4.13), we obtain (4.12), for any δ > 0.
Theorem 4.6. Let (u0 , u1 ) ∈ H∗2 (Ω) × L2 (Ω). Assume g and ξ satisfy (A1) and
(A2). Then, for any t0 > 0, there exist positive constants K and λ such that the
solution of (1.1) satisfies
−λ
E(t) ≤ Ke
Rt
t0
ξ(s)ds
,
∀t ≥ t0 .
(4.17)
Proof. Since g is positive, continuous, and g(0) > 0, then for any t ≥ t0 we have
Z t
Z t0
g(s)ds ≥
g(s)ds = g0 > 0.
0
0
Combination of (4.3), (4.7) and (4.12), gives that for any t ≥ t0 ,
F 0 (t)
Z
l
δ
δ
1 0
≤ − 2 (g0 − ) − 1
u2t −
− 2 (1 + 2(1 − l0 )2 ) kuk2H∗2 (Ω)
2
2
2
Ω
Z t
1
Ce g(0)
+
− 2
g 0 (t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ
2
2δ
0
(1 − l )
Z t
1
1
0
+
+ 2 (δ + )(1 − l0 )
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ.
2l0
δ
0
Now, we choose δ small enough such that
δ
g0
4δ
g0
g0 − > ,
(1 + 2(1 − l0 )2 ) < .
2
2
l0
4
By using (4.19), we easily check that any 1 and 2 , satisfying
2 g0
2 g0
< 1 <
,
16
2
(4.18)
(4.19)
(4.20)
12
S. A. MESSAOUDI, S. E. MUKIAWA
EJDE-2016/22
will make
l
δ
δ
1 0
β1 = 2 (g0 − ) − 1 > 0, β2 =
− 2 (1 + 2(1 − l0 )2 ) > 0.
2
2
2
Next, we pick 1 and 2 small enough such that (4.6) and (4.20) remain valid and
further we have
1
Ce g(0)
1 (1 − l0 )
1
− 2
> 0,
+ 2 (δ + )(1 − l0 ) > 0.
2
2δ
2l0
δ
Thus, (4.18) becomes
Z t
Z
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ
F 0 (t) ≤ −β1
u2t − β2 kuk2H∗2 (Ω) + C̃
0
Ω
Z t
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ, ∀t ≥ t0 .
≤ −βE(t) + C
(4.21)
0
Multiplying (4.21) by ξ(t) and using the facts that ξ is decreasing and
Z
1 t 0
g 0 (t) ≤ −ξ(t)g(t), E 0 (t) ≤
g (t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ,
2 0
(4.22)
we arrive at
Z t
ξ(t)F 0 (t) ≤ −βξ(t)E(t) + Cξ(t)
g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ
0
Z t
≤ −βξ(t)E(t) + C
ξ(t − τ )g(t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ
0
Z t
≤ −βξ(t)E(t) + C
−g 0 (t − τ )ku(t) − u(τ )k2H∗2 (Ω) dτ
0
≤ −βξ(t)E(t) − CE 0 (t),
∀t ≥ t0 .
This gives
(ξ(t)F (t) + CE(t))0 − ξ 0 (t)F (t) ≤ −βξ(t)E(t), ∀t ≥ t0 .
Consequently,
(ξ(t)F (t) + CE(t))0 ≤ −βξ(t)E(t),
∀t ≥ t0 .
(4.23)
Let
L = ξF + CE ∼ E,
(4.24)
since F ∼ E and 0 ≤ ξ(t) ≤ ξ(0). Then (4.23) and (4.24) lead to
L0 (t) ≤ −λξ(t)L(t),
∀t ≥ t0 .
(4.25)
A simple integration in (t0 , t) yields
L(t) ≤ L(t0 )e
−λ
Rt
t0
ξ(s)ds
,
∀t ≥ t0 .
(4.26)
Again, recalling (4.24), we obtain
−λ
E(t) ≤ Ke
This completes the proof.
Rt
t0
ξ(s)ds
,
∀t ≥ t0 .
(4.27)
Acknowledgements. The authors thank KFUPM for its continuous support and
also thank Prof. Kirane for his careful reading and precious suggestions.
EJDE-2016/22
EXISTENCE AND DECAY OF SOLUTIONS
13
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Salim A. Messaoudi
Department of Mathematics and Statistics, King Fahd University of Petroleum and
Minerals, Dhahran 31261, Saudi Arabia
E-mail address: messaoud@kfupm.edu.sa
Soh Edwin Mukiawa
Department of Mathematics and Statistics, King Fahd University of Petroleum and
Minerals, Dhahran 31261, Saudi Arabia
E-mail address: g201206120@kfupm.edu.sa