Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 179, pp. 1–16.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
UNIFORM DECAY FOR A LOCAL DISSIPATIVE
KLEIN-GORDON-SCHRÖDINGER TYPE SYSTEM
MARILENA N. POULOU, NIKOLAOS M. STAVRAKAKIS
Abstract. In this article, we consider a nonlinear Klein-Gordon-Schrödinger
type system in Rn , where the nonlinear term exists and the damping term is
effective. We prove the existence and uniqueness of a global solution and its
exponential decay. The result is achieved by using the multiplier technique.
1. Introduction
We consider the nonlinear system of Klein-Gordon-Schrödinger type with locally
distributed damping
x ∈ Ω, t > 0,
iψt + κ∆ψ + iαψ = φψχω ,
φtt − ∆φ + φ + λ(x)φt = − Re(F (x) · ∇ψ),
ψ=φ=0
ψ(0) = ψ0 ,
H01 (Ω)
n
2
x ∈ Ω, t > 0,
on Γ × (0, ∞),
φ(0) = φ0 ,
H01 (Ω)
(1.2)
(1.3)
φt (0) = φ1
2
(1.1)
H01 (Ω),
(1.4)
where ψ0 ∈
∩ H (Ω), φ0 ∈
∩ H (Ω), φ1 ∈
Ω is a bounded
domain of R , n ≤ 2, κ > 0, α > 0, Γ is the smooth boundary of Ω, ω is an open
subset of Ω such that meas(ω) > 0 and λ ∈ W 1,∞ (Ω) is a nonnegative function.
In what follows χω represents the characteristic function of ω; that is, χ = 1 in
ω and χ = 0 in Ω \ ω. So that the nonlinearity term φψ exists, where the damping
λ(x)φt takes palce and reciprocally.
Systems of Klein-Gordon-Schrödinger type have been studied for many years.
For example in [3, 4, 8, 5, 9, 10] the authors studied problems such as existence
and uniqueness of solutions, exponential decay, the existence of a global attractor
and its finite dimensionality in one or higher dimensions, in bounded or unbounded
domains.
The majority of works in the literature deals with linear dissipative terms, acting
on both equations. Very few is known about the polynomial decay of a KleinGordon-Schrödinger type system. In [2] the author proves the polynomial decay
when dealing with a localized dissipation in the wave equation and in [1] the authors
prove a similar result when dealing with a KGS system, idea that inspired this work.
2000 Mathematics Subject Classification. 35L70, 35B40.
Key words and phrases. Klein-Gordon-Schrödinger system; localized damping;
existence and uniqueness; energy decay.
c
2012
Texas State University - San Marcos.
Submitted June 7, 2012. Published October 17, 2012.
1
2
M. N. POULOU, N. M. STAVRAKAKIS
EJDE-2012/179
The rest of this article is divided into four sections. In Section 2, the basic
notation is given and the main assumptions made are quoted. In Section 3, the existence and uniqueness of global solutions are proved. Finally in Section 4, integral
inequalities for the energy of the system are proved using the multiplier method
combined with integral inequalities that can be found in [6] (See also [7]).
2. Notation and assumptions
Let us introduce some notation that will be used throughout this work. Denote
by H s (Ω) both the standard real and complex Sobolev spaces on Ω. For simplicity
reasons sometimes we use H s , Ls for H s (Ω), Ls (Ω). Let k · k, (·, ·) denote the norm
and inner product in L2 (Ω) respectively, as well as the symbol · denotes the inner
product in Rn . Finally, C is a general symbol for any positive constant.
Let x0 ∈ Rn , n ≤ 2 and n(x) be the unit exterior normal vector at x ∈ Γ,
m(x) = x − x0 , x ∈ Rn , n ≤ 2 and
R(x0 ) := sup m(x) = sup |x − x0 |.
x ∈Ω
We set the norms
Z
kukpp =
|u|p dx,
Ω
kukpΓ,p =
Z
(2.1)
x ∈Ω
|u(x)|p dΓ,
kuk∞ = ess supx∈Ω |u(x)|.
Γ
Some of the basic tools used are: the embedding inequality
for all u ∈ H01 (Ω);
kuk4 ≤ c1 k∇uk2 ,
Gagliardo-Nirenberg inequality for n = 1,
3/4
1/4
kuk4 ≤ c2 kuk2 k∇uk2 ,
for all u ∈ H01 (Ω);
Gagliardo-Nirenberg inequality for n = 2,
kuk4 ≤ c3 kuk1/2 k∇uk1/2 ,
for all u ∈ H01 (Ω);
and Young’s inequality
ab ≤
1 0
1 p
a + 0 bp ,
p
p
for all a, b ≥ 0.
Assumption 2.1. Let λ ∈ W 1,∞ (Ω) be a nonnegative function such that λ(x) ≥
λ0 > 0, a.e. in ω. If λ(x) ≥ λ0 > 0 in Ω, then χω ≡ 1 in Ω (See [9]).
Assumption 2.2. Let ω be a neighborhood of Γ(x0 ), where Γ(x0 ) := {x ∈ Γ :
m(x) · n(x) > 0}.
Assumption 2.3. We assume that F ∈ C 1 (Ω) and F ∈ L∞ (Ω) with kF k∞ =
M < +∞.
Assumption 2.4. Let there be a neighborhood ω̂ of Γ(x0 ) such that ω̂ ∩ Ω ⊂ ω
and a vector field h ∈ (C 1 (Ω))n , such that h = n on Γ(x0 ) h · n ≥ 0 a.e. in Γ, h = 0
on Ω \ ω̂.
We conclude this section with the following lemma which will play essential role
when establishing the asymptotic behavior of solutions in Section 4.
EJDE-2012/179
UNIFORM DECAY
3
+
Lemma 2.5 ([6, Lemma 9.1]). Let E : R+
0 → R0 be a non-increasing function and
assume that there exist two constants p > 0 and c > 0 such that
Z +∞
E (p+1)/2 (t)dt ≤ cE(s), 0 ≤ s < +∞.
s
Then, for all t ≥ 0,
(
cE(0)(1 + t)−2(p−1)
E(t) ≤
cE(0)e1−wt
if p > 1,
if p = 1,
where c and w are positive constants.
3. Existence and uniqueness of solutions
In this section we derive a priori estimates for the solutions of the Klein-GordonSchrödinger system (1.1) - (1.4). Let us represent by wn a basis in H01 (Ω) ∩ H 2 (Ω)
formed by the eigenfunctions of −∆, by Vm the subspace of H01 (Ω) ∩ H 2 (Ω) generated by the first m vectors and by
m
m
X
X
gim (t)wi , φm (t) =
him wi ,
ψm (t) =
i=1
i=1
where (ψm (t), φm (t)) is the solution of the Cauchy problem
i(ψt,m , u) + κ(∆ψ, u) + iα(ψm , u) = (φm ψm χω , u) ∀u ∈ Vm ,
(φtt,m , v) − (∆φm , v) + (φm , v) + (λ(x)φt,m , v) = − Re(F (x) · ∇ψm , v),
(3.1)
∀v ∈ Vm ,
(3.2)
with
ψm (x, 0) = ψ0m → ψ0 ,
φ(x, 0) = φ0m → φ0
φt,m (0) = φ1m → φ1
in H01 (Ω) ∩ H 2 (Ω),
in H01 (Ω).
(3.3)
Next we have the following existence and uniqueness result.
Theorem 3.1. Let (ψ0 , φ0 , φ1 ) ∈ (H01 (Ω) ∩ H 2 (Ω))2 × H01 (Ω) and Assumption 2.1
and 2.3 hold. Then, there exists a unique solution for the problem (1.1)-(1.4) such
that
ψ ∈ L∞ (0, ∞; H01 (Ω) ∩ H 2 (Ω)),
ψt ∈ L∞ (0, ∞; L2 (Ω)),
φ ∈ L∞ (0, ∞; H01 (Ω) ∩ H 2 (Ω)),
φt ∈ L∞ (0, ∞; H01 (Ω)),
φtt ∈ L∞ (0, ∞; L2 (Ω)),
ψ(x, 0) = ψ0 (x),
φ(x, 0) = φ0 (x),
φt,0 (x, 0) = φ1 (x),
x ∈ Ω.
Proof. The main idea is to use the Galerkin Method. Setting as u = ψ̄m (t) in (3.1)
and by integrating and taking the imaginary part of the equation we obtain
1 d
kψm (t)k2 + αkψm k2 = 0.
(3.4)
2 dt
Applying Gronwall’s Lemma produces
kψm (t)k ≤ kψm (0)ke−2αt .
(3.5)
kψm k ≤ R
(3.6)
Therefore,
for all t > 0.
4
M. N. POULOU, N. M. STAVRAKAKIS
EJDE-2012/179
Next, by setting u = −ψ̄t,m in (3.1) and by integrating and taking the real part,
(3.1) becomes
Z
Z
Z
κ d
|∇ψm |2 + α Im
ψm ψ̄t,m = − Re φm ψm ψ̄t,m .
2 dt Ω
Ω
ω
For the right hand side of the equation above we have
Z
Z
Z
1
1 d
φm |ψm |2 =
φt,m |ψm |2 + Re φm ψ ψ̄t,m .
2 dt ω
2 ω
ω
But from equation (3.1) we also obtain
Z
Z
Z
α Im
ψm ψ̄t,m dx = κα
|∇ψm |2 dx + α φm |ψm |2 dx.
Ω
Ω
ω
Therefore,
Z
Z
|∇ψm |2 + κα
|∇ψm |2 dx + α φm |ψm |2 dx
Ω
Ω
ω
Z
Z
1
1 d
φm |ψm |2 +
φt,m |ψm |2 .
=−
2 dt ω
2 ω
κ d
2 dt
Z
(3.7)
Next, letting v = φt,m , equation (3.2) gives
Z
i
1 dh
2
2
2
2
kφt,m k + k∇φm k + kφm k + λ0 kφt,m k ≤ − (F (x) · ∇ψm )φt,m dx.
2 dt
Ω
Now, by adding (3.4), (3.7) and the above inequality, we have
Z
i
1 dh
2
2
2
2
2
kψm k + κk∇ψm (t)k + kφt,m k + k∇φm k + kφm k +
φm |ψm |2 dx
2 dt
ω
Z
2
2
2
2
+ αkψm k + λ0 kφt,m k + καk∇ψm k + α φm |ψm | dx
ω
Z
1
2
φt,m |ψm |
−
2 ω
Z
≤ − (F (x) · ∇ψm )φt,m dx.
Ω
Evaluating these integrals by using Assumption 2.3, Gagliardo-Nirenberg inequality
and Young’s inequality, we obtain
Z
1 Z
αZ
C2
2
2
φt,m |ψm | dx ≤
|φt,m | dx +
|∇ψm |2 dx,
2 ω
4 Ω
4α Ω
Z
Z
αZ
M2
2
|φt,m | dx +
|∇ψm |2 dx,
(F (x) · ∇ψm )φt,m dx ≤
2 Ω
2α Ω
Ω
Z
Z
Z
αC 2
2
2
|φm | dx +
|∇ψm |2 dx.
α φm |ψm | dx ≤ α
4
Ω
ω
Ω
Integrating the above expression over (0, t) and applying Gronwall’s Lemma we
obtain the first estimate
Z
kψm k2 + κk∇ψm k2 + kφt,m k2 + k∇φm k2 + kφm k2 +
φm |ψm |2 dx ≤ L1 , (3.8)
ω
where L1 is a positive constant independent of m ∈ N. Let
2
2
2
2
2
Z
E(t) = kψm k + κk∇ψm k + kφt,m k + k∇φm k + kφm k +
ω
φm |ψm |2 dx
EJDE-2012/179
UNIFORM DECAY
5
evaluating the integral
Z
κ
1
|φm kψm |2 dx ≤ kφm kkψm k24 ≤ Ckφm kk∇ψm kkψm k ≤ k∇φm k2 + k∇ψm k2 +C
2
2
ω
one can deduce that
E(t) ≥ kψm k2 +
κ
1
k∇ψm k2 + kφt,m k2 + k∇φm k2 + kφm k2 + C,
2
2
E(t) ≤ kψm k2 +
3κ
3
k∇ψm k2 + kφt,m k2 + k∇φm k2 + kφm k2 + C.
2
2
and
so from (3.8) we have
kψm k2 + k∇ψm k2 + kφt,m k2 + k∇φm k2 + kφm k2 ≤ L1 + C.
Next, let u = ∆ψ̄t,m + α∆ψ̄m , in (3.1). Taking the real part and integrating over
Ω produces
1 d
κk∆ψm k2 + καk∆ψm k2
2 dt Z
= Re
ω
(3.9)
Z
φm ψm ∆ψ̄t,m dx + α Re
φm ψm ∆ψ̄m dx.
ω
Furthermore, by letting v = −∆φt,m in (3.2) and integrating, we also obtain
1 d
k∇φt,m k2 + k∆φm k2 + k∇φm k2 + λ0 k∇φt,m k2
2 dt Z
≤ Re
(3.10)
(F (x) · ∇ψm )∆φt,m dx.
Ω
Analyzing the right hand side of (3.9) produces the equation
Z
Re φm ψm ∆ψ̄t,m dx
ω
Z
Z
Z
d
=
Re φm ψm ∆ψ̄m dx − Re φt,m ψm ∆ψ̄m dx − Re φm ψt,m ∆ψ̄m dx,
dt
ω
ω
ω
while by ψt,m = −i(−∆ψm − iαψm − φm ψm χω ),
Z
Z
− Re φm ψt,m ∆ψ̄m dx = Re iφm [−∆ψm − iαψm − φm ψm ]∆ψ̄m dx
ω
ω
Z
Z
= α Re φm ψm ∆ψ̄m dx + Im φ2m ψm ∆ψ̄m dx.
ω
ω
Substituting the expressions above into (3.9) yields
Z
1 d
κk∆ψm k2 − 2 Re φm ψm ∆ψ̄m dx + καk∆ψm k2
2 dt
ω
Z
Z
Z
2
= 2α φm ψm ∆ψ̄m dx + Im φm ψm ∆ψ̄m dx − Re φt,m ψm ∆ψ̄m dx.
ω
ω
ω
(3.11)
6
M. N. POULOU, N. M. STAVRAKAKIS
EJDE-2012/179
Next, adding (3.10) and (2.1) gives
Z
1 d
2
κk∆ψm k − 2 Re φm ψm ∆ψ̄m dx + k∇φt,m k2 + k∆φm k2 + k∇φm k2
2 dt
ω
+ καk∆ψm k2 + λ0 k∇φt,m k2
Z
Z
≤ 2α φm ψm ∆ψ̄m dx + Im φ2m ψm ∆ψ̄m dx
ω
ω
Z
Z
− Re φt,m ψm ∆ψ̄m dx + Re (F (x) · ∇ψm )∆φt,m dx.
ω
Ω
(3.12)
Estimating the integrals on the right hand side of (3.12) using the Sobolev embedding theorem and Young’s Inequality
Z
1
Re φm ψm ∆ψ̄m dx ≤ kφm k4 kψm k4 k∆ψm k ≤ k∆ψm k2 + Ck∇φm k2 k∇ψm k2 ,
4
Zω
1
Im φ2m ψm ∆ψ̄m dx ≤ kφm k26 kψm k6 k∆ψm k ≤ k∆ψm k2 + Ck∇φm k4 k∇ψm k2 ,
4
Zω
1
− Re φ0m ψm ∆ψ̄m dx ≤ kφ0m k4 kψm k4 k∆ψm k ≤ k∆ψm k2 + Ck∇φ0m k2 k∇ψm k2 .
4
ω
Now evaluating the last term of (3.10),
Z
(F (x) · ∇ψm )∆φt,m dx
Ω
Z
=−
∇(F (x) · ∇ψm )∇φt,m dx
ZΩ
Z
= − (F (x) · ∆ψm )∇φt,m dx − (∇F (x) · ∇ψm )∇φt,m dx
Ω
ZΩ
− (∇ψm × (∇ × F (x)))∇φt,m dx
Ω
and taking into consideration Assumption 2.3 produces
Z
− (F (x) · ∆ψm )∇φt,m dx ≤ Ck∆ψm kk∇φt,m k,
Z Ω
− (∇F (x) · ∇ψm )∇φt,m dx ≤ Ck∇ψm kk∇φt,m k,
Z Ω
− (∇ψm × (∇ × F (x)))∇φt,m dx ≤ Ck∇ψm kk∇φt,m k.
Ω
Integrating over (0, t) and applying Gronwall’s Lemma we obtain the second estimate
k∆ψm k2 + k∇φt,m k2 + k∆φm k2 + k∇φm k2 ≤ L2 ,
(3.13)
where L2 is a positive constant independent of m ∈ N. The rest of the proof follows
the same basic steps as the one of [5, Theorem 3.1].
The energy associated to the problem is defined by
Z
1
2
2
E(t) :=
kψk + κk∇ψk +
φ|ψ|2 dx + kφt k2 + k∇φk2 + kφk2 .
2
ω
(3.14)
EJDE-2012/179
UNIFORM DECAY
7
4. Exponential decay
Let {ψ(t), φ(t), φt (t)} in H01 (Ω) ∩ H 2 × H01 (Ω) ∩ H 2 (Ω) × H01 (Ω) be a solution
of the (1.1)- (1.4).
2
M
Lemma 4.1. Let κ, α, λ0 be large and be small enough. Then for β := κα − 2λ
−
0
1
>
0
the
first
order
energy
satisfies
the
inequality
2
Z
Z
Z
Z
3
E 0 (t) ≤ −α
|ψ|2 dx − β
|∇ψ|2 dx + c(α, c1 , )
λ(x)|φt |2 dx.
|φ|2 dx −
8 Ω
Ω
Ω
Ω
Proof. Substituting into (3.1) u = −ψ̄t , taking the real part, next substituting also
v = φt , into (3.2) and integrating both over Ω, we have
Z
Z
Z
κ d
|∇ψ|2 dx + α Im
ψ ψ̄t dx = − Re φψ ψ̄t dx,
2 dt Ω
ω
ZΩ
Z
Z
1 d
2
2
2
2
(|φt | dx + |∇φ| + |φ| )dx +
λ(x)|φt | dx = − Re (F (x) · ∇ψ)φt dx.
2 dt Ω
Ω
Ω
The right hand side of the first equation becomes
Z
Z
Z
1 d
1
2
2
φ|ψ| dx =
φt |ψ| dx + Re φψ ψ̄t dx.
2 dt ω
2 ω
ω
Also from (3.1) we have
Z
Z
Z
ψ ψ̄t dx = κα
|∇ψ|2 dx + α φ|ψ|2 dx.
α Im
Ω
Ω
ω
Taking u = ψ̄, in (3.1) integrating over Ω and taking the imaginary part yields
Z
Z
1 d
|ψ|2 dx + α
|ψ|2 dx = 0.
2 dt Ω
Ω
Adding the above equations gives
Z
Z
Z
Z
d
E(t) + α
|ψ|2 dx + κα
|∇ψ|2 dx + α φ|ψ|2 dx +
λ(x)|φt |2 dx
dt
Ω
Ω
ω
Ω
Z
Z
1
2
=
φt |ψ| dx − Re (F (x) · ∇ψ)φt dx.
2 ω
Ω
Evaluating the integrals by using Assumption 2.3, Gagliardo-Nirenberg inequality
and Young’s inequality produces
Z
1Z
1 Z
c2
φt |ψ|2 dx ≤
λ(x)|φt |2 dx + 1
|∇ψ|2 dx,
2 ω
8 Ω
2λ0 Ω
Z
Z
1Z
M2
λ(x)|φt |2 dx +
|∇ψ|2 dx,
(F (x) · ∇ψ)φt dx ≤
2
2λ
0
Ω
Ω
Z
Z Ω
Z
1
2
2
|φ| dx +
|∇ψ|2 dx.
α φ|ψ| dx ≤ c(α, c1 , )
4 Ω
ω
Ω
Hence for κ, α, λ0 large, small enough and β as above, it holds that
Z
Z
Z
Z
3
λ(x)|φt |2 dx,
E 0 (t) ≤ −α
|ψ|2 dx − β
|∇ψ|2 dx + c(α, c1 , )
|φ|2 dx −
8 Ω
Ω
Ω
Ω
(4.1)
which concludes the proof of the Lemma.
8
M. N. POULOU, N. M. STAVRAKAKIS
EJDE-2012/179
Lemma 4.2. Let {ψ, ψ, φt } be solutions of (1.1) - (1.4), β > 0 and Assumptions
2.1 - 2.3 hold. Then there exists T0 > 0 such that if T > T0 we have
Z T
Z
Z
∂φ 2
2
1
1 T
dΓdt + |χ| + C0 E(s),
E 2 (t)dt ≤
E(t)
(m · n)
12 s
2 s
∂n
α
Γ(x0 )
where
Z
1h
|ψ|2 + κ|∇ψ|2 + 4αφt (m · ∇φ) + αδφ 4φt + 2λ(x)φ dx
E(t)
4
Ω
Z
iT
2
+
φ|ψ| dx
,
χ :=
s
ω
0 < s < T < +∞ and C0 depends on α, β, λ, R, M, κ, n, c1 .
Proof. Multiplying (3.1) by E(t)ψ̄, taking the imaginary part and integrating produces
Z T
Z
Z
Z
iT 1 Z T
1h
E 0 (t)
|ψ|2 dx + α
E(t)
|ψ|2 dx dt = 0.
E(t)
|ψ|2 dx −
2
2 s
s
Ω
s
Ω
Ω
Next, multiplying (3.1) by − 21 E(t)ψ̄t , taking the real part and adding the previous
equation produces
Z
Z
iT κ Z T
Z
1h
(|ψ|2 + κ|∇ψ|2 )dx +
φ|ψ|2 dx
−
E 0 (t)
|∇ψ|2 dx dt
E(t)
4
4
s
Ω
ω
s
Ω
Z
Z
Z
Z
α T
κα T
2
2
+
E(t)
|ψ| dx dt +
E(t)
|∇ψ| dx dt
2 s
2 s
Ω
Ω
Z
Z
Z
Z
α T
1 T 0
2
+
E(t) φ|ψ| dx dt −
E (t)
|ψ|2 dx dt
2 s
4 s
ω
Ω
Z
Z
Z
Z
1 T 0
1 T
=
E (t) φ|ψ|2 dx dt +
E(t) φt |ψ|2 dx dt
2 s
2 s
ω
ω
(4.2)
Multiplying the second equation by αE(t)(q · ∇φ), where q ∈ (W 1,∞ (Ω))n , integrating by parts and using Green’s identity we obtain
Z
Z T
Z
h
iT
αE(t)
φt (q · ∇φ)dx − α
E 0 (t)
φt (q · ∇φ) dx dt
Z
s
Z
Ω
T
Z
∂φ ∂qk ∂φ
dx dt
∂x
i ∂xi ∂xk
s
Ω
s
Ω
Z
Z
Z
Z
∂φ 2
α T
α T
2
−
E(t)
div q|∇φ| dx dt −
E(t) (q · n)
dΓdt
2 s
2 s
∂n
Ω
Γ
Z T
Z
Z T
Z
φ(q · ∇φ) dx dt + α
E(t)
λ(x)φt (q · ∇φ) dx dt
+α
E(t)
+
α
2
s
Ω
T
Z
div q|φt |2 dx dt + α
E(t)
Z
s
T
= −α
Ω
s
Ω
Z
(F (x) · ∇ψ)(q · ∇φ) dx dt.
E(t)
s
E(t)
Ω
Adding relations (4.2) and (4.3), we obtain
Z
Z
iT
1h
2
2
E(t)
(|ψ| + κ|∇ψ| + 4αφt (q · ∇φ))dx +
φ|ψ|2 dx)
4
s
Ω
ω
(4.3)
EJDE-2012/179
α
2
Z
κ
−
4
Z
+
UNIFORM DECAY
T
Z
s
T
E 0 (t)
s
Z T
α
2
Z
α
2
Z
1
−
4
Z
1
=
2
Z
+
+
|∇ψ|2 dx dt + α
T
Z
Ω
Z
|ψ|2 dx dt +
Ω
T
Z
s
T
ω
E 0 (t)
Z
s
κα
2
φ|ψ|2 dx dt −
E(t)
Z
λ(x)φt (q · ∇φ) dx dt
E(t)
s
E(t)
1
2
T
Z
Ω
Z
|∇ψ|2 dx dt
E(t)
s
Z T
Ω
E 0 (t)
Z
s
φ|ψ|2 dx dt
ω
|ψ|2 dx dt
Ω
T
Z
2
ω
T
Z
Z
φt |ψ| dx dt − α
E(t)
s
α
2
T
Z
Ω
s
φ(q · ∇φ) dx dt
E(t)
s
φt (q · ∇φ) dx dt + α
T
Ω
∂φ ∂qk ∂φ
dx dt
∂xi ∂xi ∂xk
Z
Z
s
Z
E(t)
s
Z
Ω
E 0 (t)
T
Z
Ω
−α
+
div q[|φt |2 − |∇φ|2 ] dx dt + α
E(t)
9
T
Z
(F (x) · ∇ψ)(q · ∇φ) dx dt
E(t)
s
Ω
Z
E(t)
s
∂φ 2
(q · n)
dΓdt.
∂n
Γ
(4.4)
Considering that q(x) = m(x) = x − x0 , relation (4.4) becomes
Z
iT
Z
1h
2
2
(|ψ| + κ|∇ψ| + 4αφt (m · ∇φ))dx +
φ|ψ|2 dx)
E(t)
4
s
Ω
ω
Z T
Z
Z T
Z
αn
E(t) [|φt |2 − |∇φ|2 ] dx dt
+α
E(t)
|∇φ|2 dx dt +
2 s
Ω
s
Ω
Z
Z T
Z T
Z
κ
α
−
E 0 (t)
|∇ψ|2 dx dt +
E(t)
|ψ|2 dx dt
4 s
2
Ω
s
Ω
Z T
Z
Z T
Z
+α
E(t)
φ(m · ∇φ) dx dt + α
E(t)
λ(x)φt (m · ∇φ) dx dt
s
Ω
T
s
Ω
Z
Z
1 T 0
0
−α
E (t)
φt (q · ∇φ) dx dt −
E (t)
|ψ|2 dx dt
4 s
s
Ω
Ω
Z
Z
κα T
+
E(t)
|∇ψ|2 dx dt
2 s
Ω
Z
Z
Z
Z
α T
1 T
≤−
E(t) φ|ψ|2 dx dt +
E(t) φt |ψ|2 dx dt
2 s
2 s
ω
ω
Z T
Z
Z T
Z
∂φ 2
1
α
+
E 0 (t) φ|ψ|2 dx dt +
E(t)
(m · n)
dΓdt
2 s
2 s
∂n
ω
Γ(x0 )
Z T
Z
−α
E(t) (F (x) · ∇ψ)(m · ∇φ) dx dt.
Z
Z
s
(4.5)
Ω
Now multiplying (3.2) by αE(t)ξφ, where ξ ∈ W 1,∞ (Ω), and integrating we obtain
Z T
Z
Z
h
λ(x)φ iT
dx − α
E(t)
ξ|φt |2 dx dt
αE(t)
φξ φt +
2
s
Ω
s
Ω
Z T
Z
Z T
Z
+α
E(t)
φ(∇ξ · ∇φ) dx dt + α
E(t)
ξ|∇φ|2 dx dt
s
Ω
s
Ω
10
M. N. POULOU, N. M. STAVRAKAKIS
T
Z
+α
Z
ξ|φ|2 dx dt +
E(t)
s
Ω
Z
T
α
2
Z
E(t)
s
E 0 (t)
Z
s
Z
= −α
T
EJDE-2012/179
Ω
Z
ξφ∇ψ dx dt + α
ω
ξλ(x)|φ|2 dx dt
T
Z
0
E (t)
s
ξφt φ dx dt.
(4.6)
Ω
Taking ξ = 2δ ∈ R and adding (4.5) and (4.6) produces the expression
Z T
Z T
Z
Z
n
n
2
χ + α( − 2δ)
E(t)
E(t)
|φt | dx dt + α(1 + 2δ − )
|∇φ|2 dx dt
2
2 s
s
Ω
Ω
Z
Z
Z T
Z
κ T 0
2
E (t)
|∇ψ| dx dt + α
E(t)
−
φ(m · ∇φ) dx dt
4 s
Ω
s
Ω
Z
Z
Z T
Z
1 T 0
E (t)
|ψ|2 dx dt − α
E 0 (t)
φt (m · ∇φ) dx dt
−
4 s
Ω
s
Ω
Z
Z
Z T
Z
α T
E(t)
|ψ|2 dx dt
+ 2αδ
E(t)
|φ|2 dx dt +
2
s
Ω
s
Ω
Z T
Z
Z
Z
κα T
+α
E(t)
λ(x)φt (m · ∇φ) dx dt +
E(t)
|∇ψ|2 dx dt
2
s
Ω
s
Ω
Z
Z T
Z
Z T
+ αδ
E 0 (t)
λ(x)|φ|2 dx dt + 2αδ
E(t) φ∇ψ dx dt
s
Ω
T
s
ω
Z
Z
α T
E 0 (t) φ|ψ|2 dx dt −
E(t) φ|ψ|2 dx dt
2 s
s
ω
ω
Z
Z
Z T
Z
1 T
+
E(t) φt |ψ|2 dx dt + 2δ
E 0 (t)
φt φ dx dt
2 s
ω
s
Ω
Z T
Z
−α
E(t) (F (x) · ∇ψ)(m · ∇φ) dx dt
1
≤+
2
Z
α
+
2
Z
Z
s
Ω
T
Z
E(t)
s
∂φ 2
(m · n)
dΓdt.
∂n
Γ(x0 )
(4.7)
Choosing δ =
n−1
4 ,
if n = 2 or δ ∈ (0, 1/2), if n = 1 relation (4.7) becomes
Z
Z
Z T
Z
κ T 0
2
χ+α
E (t)dt −
E (t)
|∇ψ| dx dt + α
E(t)
φ(m · ∇φ) dx dt
4 s
Ω
s
Ω
s
Z T
Z
Z T
Z
−α
E 0 (t)
φt (m · ∇φ) dx dt + α
E(t)
λ(x)φt (m · ∇φ) dx dt
Z
s
T
2
Ω
Z
T
s
Ω
Z
Z
α(n − 1) T 0
E 0 (t)
|ψ|2 dx dt +
E (t)
λ(x)|φ|2 dx dt
4
Ω
Ω
s
s
Z
Z
Z
Z
α T
α(n − 1) T
−
E(t)
|φ|2 dx dt +
E(t)
|φ|2 dx dt
2 s
2
Ω
s
Ω
Z
Z
α(n − 1) T
+
E(t) φ∇ψ dx dt
2
s
ω
Z
Z
Z
Z
1 T 0
1 T
2
≤
E (t) φ|ψ| dx dt +
E(t) φt |ψ|2 dx dt
2 s
2 s
ω
ω
1
−
4
Z
EJDE-2012/179
T
Z
−α
Z
s
11
Z
(F (x) · ∇ψ)(m · ∇φ) dx dt
E(t)
s
α
+
2
UNIFORM DECAY
(4.8)
Ω
T
Z
Z
∂φ 2
α(n − 1) T 0
dΓdt +
E(t)
E (t)
(m · n)
φt φ dx dt.
∂n
2
s
Γ(x0 )
Ω
Z
In order for the proof to be completed we need to estimate some of the terms
RT
R
appearing in (4.8). Let I1 = − 41 s E 0 (t) Ω (κ|∇ψ|2 + |ψ|2 ) dx dt. Then taking into
consideration (3.14),
Z
Z
1 T 2
1
1 T 0
|E (t)|E(t)dt ≤ −
(E (t))0 dt ≤ E(0)E(s).
|I1 | ≤
4 s
8 s
8
R
α(n−2) R T
E(t) Ω |φ|2 dx dt. Then taking into consideration (3.14),
Let I2 =
2
s
Z
(n − 2) T
(n − 2)
E(0)E(s).
E(t)E 0 (t)dt ≤
|I2 | ≤ −
αλ0
2αλ0
s
RT
R
Let I3 = α s E(t) Ω φ(m · ∇φ) dx dt. Then taking into consideration (3.14), (4.1)
and Young’s inequality gives
Z
Z
Z T
Z
R 2 α2 T
2
|I3 | ≤
E(t)
|φ| dx dt + E(t)
|∇φ|2 dx dt
4
s
Ω
s
Ω
Z T
R2
≤
E(0)E(s) + 2
E 2 (t)dt.
4λ0
s
R
RT
Let I4 = −α s E 0 (t) Ω φt (m · ∇φ) dx dt. Then taking into consideration (3.14),
Z T
αR
|E 0 (t)|E(t)dt ≤
E(0)E(s).
|I4 | ≤ αR
2
s
R
RT
Let I5 = α s E(t) Ω λ(x)φt (m · ∇φ) dx dt. Then taking into consideration (3.14)
and (4.1) and Young’s inequality produces
Z T
α2 R2 kλk∞
E(0)E(t) + 2
E 2 (t)dt.
|I5 | ≤
3
s
RT 0 R
Let I6 = α(n−1)
E (t) Ω λ(x)|φ|2 dx dt. Then taking into consideration (3.14),
4
s
Z
α(n − 1)kλk∞ T 0
α(n − 1)kλk∞
|I6 | ≤
|E (t)|E(t)dt ≤
E(0)E(s).
2
4
s
RT
R
Let I7 = α(n−1)
E(t) ω φ∇ψ dx dt. Then taking into consideration (3.14), (4.1)
2
s
and Young’s inequality produces
Z
Z
Z T
α2 (n − 1)2 T
2
|I7 | ≤
E(t)
|φ| dx dt + 2
E 2 (t)dt
16
s
Ω
s
Z T
(n − 1)2
≤
E(0)E(s) + 2
E 2 (t)dt.
8λ0
s
R
RT
Let I8 = 21 s E 0 (t) ω φ|ψ|2 dx dt. Then from relation (3.14) and Young’s inequality
we obtain
Z
Z
c1 T 0
c1 (κ + 1)
|I8 | ≤
|E (t)|
|φ||∇ψ|2 dx dt ≤
E(0)E(s).
2 s
4
Ω
12
M. N. POULOU, N. M. STAVRAKAKIS
RT
R
Let I9 = 12 s E(t) ω φt |ψ|2 dx dt.
Young’s inequality we obtain
|I9 | ≤
c21
16
Z
T
Then taking into consideration (3.14) and
Z
E(t)
s
EJDE-2012/179
|∇ψ|2 dx dt + 2
Ω
T
Z
E 2 (t)dt
s
c2
≤ 1 E(0)E(s) + 2
32β
T
Z
E 2 (t)dt.
s
RT
R
Let I10 = −α s E(t) Ω (F (x) · ∇ψ)(m · ∇φ) dx dt. Then using (3.14), (4.1) and
Young’s inequality we obtain
|I10 | ≤
α2 R 2 M 2
4
T
Z
Z
E(t)
s
|∇ψ|2 dx dt + 2
Ω
α2 R 2 M 2
E(0)E(s) + 2
≤
8β
Let I11 =
α(n−1)
2
RT
s
E 0 (t)
R
ω
Z
T
E 2 (t)dt
s
Z
T
E 2 (t)dt.
s
φt φ dx dt. Then relation (3.14) implies
α(n − 1)
|I11 | ≤
2
T
Z
|E 0 (t)|E(t)dt ≤
s
α(n − 1)
E(0)E(s).
4
Hence, the following inequality holds, with = 1/12,
2
12
T
Z
E 2 (t)dt ≤
s
1
2
Z
T
Z
E(t)
s
∂φ 2
1
(m · n)
dΓdt + |χ| + C0 E(s),
∂n
α
Γ(x0 )
(4.9)
where
C0 =
h α(n − 1)2 + 8α3 R2 kλk λ + 6αR2 + (n − 2) 3c2 + 12α2 R2 M 2
∞ 0
+ 1
λ0
8β
2α(n − 1)(1 + kλk∞ ) + 2c1 (κ + 1) + 1 + 4αR i
E(0).
+
8
Lemma 4.3. Let the assumptions in Lemma 4.2 and Assumption 2.4 hold. Then
there exists T0 > 0 such that if T > T0 , we have
1
24
Z
T
E 2 (t)dt
s
1
R
RC1
3
≤ |χ| + |Y | + (C0 +
)E(s) + kh|W 1,∞ R
α
α
α
2
Z
T
Z
E(t)
s
|∇φ|2 dx dt
ω̂
where
Z
Z
iT
1h
2
2
Y := E(t)
|ψ| + κ|∇ψ| + 4αφt (h · ∇φ) dx +
φ|ψ|2 dx
4
s
Ω
ω̂
and C1 depends on α, β, λ, R, M, κ, h, c1 , c2 .
EJDE-2012/179
UNIFORM DECAY
13
Proof. For q = h expression (4.4) becomes
Z
Z
α T
E(t) div h[|φt |2 − |∇φ|2 ] dx dt
Y −α
E (t) φt (h · ∇φ) dx dt +
2 s
ω̂
s
ω̂
Z
Z T
Z
Z
κ T 0
∂φ ∂hk ∂φ
+α
dx dt −
E (t)
E(t)
|∇ψ|2 dx dt
4 s
s
Ω
ω̂ ∂xi ∂xi ∂xk
Z T
Z
Z T
Z
+α
E(t) φ(h · ∇φ) dx dt + α
E(t) λ(x)φt (h · ∇φ) dx dt
Z
T
s
ω̂
T
α
2
Z
α
2
Z
Z
0
Z
s
Z
T
ω̂
Z
κα
E(t)
|∇ψ|2 dx dt
2
s
s
Ω
Ω
Z
Z
Z T
Z
α
1 T 0
E (t)
|ψ|2 dx dt +
E(t) φ|ψ|2 dx dt
−
4 s
2 s
Ω
ω
Z
Z
Z T
Z
1 T 0
1
−
E (t) φ|ψ|2 dx dt −
E(t) φt |ψ|2 dx dt
2 s
2 s
ω
ω
Z T
Z
+α
E(t) (F (x) · ∇ψ)(h · ∇φ) dx dt
+
s
≥
|ψ|2 dx dt +
E(t)
ω̂
T
Z
E(t)
s
∂φ 2
(h · n)
dΓdt.
∂n
Γ(x0 )
(4.10)
Evaluating some of the terms in the previous expression, we have
Z
α Z T
E(t) div h[|φt |2 − |∇φ|2 ] dx dt
2 s
ω̂
Z
Z
αkhkW 1,∞
αkhkW 1,∞ T
≤
E(0)E(s) +
E(t) |∇φ|2 dx dt,
2λ0
2
s
ω̂
Z T
Z
Z
Z T
∂h
∂φ
∂φ
k
dx dt ≤ αkhkW 1,∞
E(t) |∇φ|2 dx dt,
E(t)
α
s
ω̂
s
ω̂ ∂xi ∂xi ∂xk
Z
κ+1
κ Z T
E(0)E(s),
E 0 (t)
|∇ψ|2 − |ψ|2 dx dt ≤
4 s
4
Ω
Z
Z T
αkhk 1,∞
W
0
E (t)
φt (h · ∇φ) dx dt ≤
E(0)E(s),
α
2
s
Ω
Z
Z
Z T
α2 khk2
T
W 1,∞ kλk∞
E(t) λ(x)φt (h · ∇φ) dx dt ≤
E(0)E(s) + 2
E 2 (t)dt,
α
6
s
ω̂
s
Z
κα Z T
κα
E(t)
|∇ψ|2 dx dt ≤
E(0)E(s),
2 s
4β
Ω
Z
Z T
α Z T
a2 c21
E(t) φ|ψ|2 dx dt ≤
E(0)E(s) + 2
E 2 (t)dt,
2 s
32β
ω
s
Z
1 Z T
1
0
2
E (t)
|ψ| dx dt ≤ E(0)E(s),
4 s
4
Ω
Z
Z T
1 Z T
4
c1
2
E(t) φt |ψ| dx dt ≤
E(0)E(s) + 2
E 2 (t)dt,
2 s
8β
ω
s
14
M. N. POULOU, N. M. STAVRAKAKIS
EJDE-2012/179
and finally
Z
α
T
Z
E(t)
s
Ω
(F (x) · ∇ψ)(h · ∇φ) dx dt
Z T
a2 khk2W 1,∞ M 2
E(0)E(s) + 2
E 2 (t)dt,
≤
8β
s
Z T
Z
Z T
khk2
W 1,∞
E(0)E(s) + 2
E 2 (t)dt,
E(t)
φ(h · ∇φ) dx dt ≤
α
4λ
0
s
s
Ω
Z
Z
Z T
Z
1 Z T
1Z T
0
0
2
2
0
E (t) φ|ψ| dx dt ≤
E (t)
|φ| dx dt +
E (t)
|ψ|4 dx dt
2 s
4 s
ω
Ω
s
Ω
1 + c22
E(0)E(s).
≤
4
Taking into consideration the evaluations above and substituting them into (4.10),
we obtain
Z
Z
Z T
∂φ 2
αR T
E(t)
dΓdt ≤ R|Y | + 10R
E 2 (t)dt + RC1 E(s)
2 s
Γ(x0 ) ∂n
s
(4.11)
Z
Z
3αkhkW 1,∞ R T
2
E(t) |∇φ| dx dt,
+
2
s
ω̂
where
h κ + 3 + c2 + 2αkhk 1,∞
α2 c21 + 4α2 khk2W 1,∞ M 2
αkh|W 1,∞
W
2
C1 :=
+
+
4
32β
2λ0
i
2
2
2
khkW 1,∞
κα α kh|W 1,∞ kλk∞
+
+
+
E(0).
4λ0
4β
6
α
, we deduce
Combining (4.9) and (4.11) and choosing = 80R
Z T
1
R
RC1
1
E 2 (t)dt ≤ |χ| + |Y | + (C0 +
)E(s)
24 s
α
α
α
(4.12)
Z T
Z
3
2
+ kh|W 1,∞ R
E(t) |∇φ| dx dt,
2
s
ω̂
which completes the proof of the Lemma.
RT
R
To evaluate s E(t) ω̂ |∇φ|2 dx dt, we construct a function η ∈ W 1,∞ (Ω) such
that
0 ≤ η ≤ 1,
a.e. in Ω, with η = 1, a.e. in ω̂,
η = 0,
a.e. in Ω\ω,
(4.13)
(4.14)
2
|∇η|
∈ L∞ (ω).
η
Finally, we state and prove the main result of the present work.
(4.15)
Theorem 4.4. Let Assumptions 2.1 and 2.2 hold and β > 0. Then there exists
some positive constant C = C(E(0)) such that the following decay rate holds for
each solution (ψ, φ, φt ) of (1.1)- (1.4),
E(t) ≤
CE(0)
,
1+t
for all t ≥ 0.
(4.16)
EJDE-2012/179
UNIFORM DECAY
15
Proof. To prove the Theorem it is sufficient to prove that
Z T
E 2 (t)dt ≤ CE(S), for all 0 ≤ s < T < +∞,
s
for some positive constant C independent of T . Letting ξ = η in (4.6) we obtain
Z
Z T
Z
h
λ(x)φ iT
dx − α
αE(t) φη φt +
E(t) η|φt |2 dx dt
2
s
s
ω
ω
Z T
Z
Z T
Z
+α
E(t) φ(∇η · ∇φ) dx dt + α
E(t) η|∇φ|2 dx dt
s
ω
s
ω
(4.17)
Z T
Z
Z T
Z
α
0
2
2
E (t) ηλ(x)|φ| dx dt
+α
E(t) η|φ| dx dt +
2 s
ω
s
ω
Z
Z T
Z
Z T
E(t) ηφ∇ψ dx dt + α
E 0 (t) ηφt φ dx dt.
= −α
s
ω
ω
s
Evaluating the integrals above produces
Z T
Z
E(t) η|∇φ|2 dx dt
α
s
ω
Z
≤ |Z| + C2 E(s) + 2
s
T
α
E (t)dt +
2
2
Z
T
(4.18)
Z
E(t)
s
2
η|∇φ| dx dt,
ω
where
Z
λ(x)φ iT
φη φt +
dx ,
2
s
ω
|∇η|2
1
1
α2
4α
αi
+k
k∞ +
+
+
+
+
E(0).
η
2αλ0
αλ0
8β
3λ0
2
h
Z := αE(t)
C2 :=
h αkλk
∞
2
1
and taking into
Therefore combining (4.12) and (4.18), choosing = 288khk
∞R
consideration
Z T
Z
Z T
Z
E(t) η|∇φ|2 dx dt =
E(t) |∇φ|2 dx dt,
s
ω̂
s
ω̂
we obtain
Z T
1
1
R
RC1
E 2 (t)dt ≤ |χ|+ |Y |+3khk∞ R|Z|+(C0 +
+khk∞ RC2 )E(s). (4.19)
48 s
α
α
α
Note that the following estimate holds
R
1
|χ| + |Y | + 3khk∞ R|Z| ≤ C3 E(0)E(s),
α
α
where C3 = C3 (R, α, κ, δ, λ0 , c1 , khk∞ , kλk∞ ). Then
Z T
E 2 (t)dt ≤ CE(0)E(s),
s
where C = C(R, α, κ, δ, n, c1 , β, khk∞ , kλk∞ ) is independent of T . Then employing
Lemma 2.5 we deduce the desired decay rate.
Acknowledgments. This work was financially supported by a grant No 65/1790
from the Basic Research Committee of the National Technical University of Athens,
Greece.
16
M. N. POULOU, N. M. STAVRAKAKIS
EJDE-2012/179
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Marilena N. Poulou
Department of Mathematics, National Technical University, Zografou Campus 157 80,
Athens, Hellas, Greece
E-mail address: mpoulou@math.ntua.gr
Nikolaos M. Stavrakakis
Department of Mathematics, National Technical University, Zografou Campus 157 80,
Athens, Hellas, Greece
E-mail address: nikolas@central.ntua.gr