Int. J. Dynamical Systems and Differential Equations, Vol. 2, Nos. 1/2, 2009
Rates of decay for structural damped models
with decreasing in time coefficients
Xiaojun Lu
Department of Mathematics,
Zhejiang University,
310027 Hangzhou, PR China
E-mail: [email protected]
Michael Reissig*
Faculty for Mathematics and Computer Science,
TU Bergakademie Freiberg,
Prüferstr.9, 09596 Freiberg, Germany
E-mail: [email protected]
*Corresponding author
Abstract: The goal of this paper is to study the Cauchy problem for
a family of structural damped models interpolation between classical
damped wave and wave model with viscoelastic dissipation. We are
interested to observe the parabolic effect in high order energy estimates,
that is, decay rates depend on the order of the energy. The main tools are
related to WKB-analysis. One has to apply elliptic as well as hyperbolic
WKB-analysis in different parts of the extended phase space.
Keywords: decay rates; parabolic behaviour; time-dependent dissipation;
WKB analysis.
Reference to this paper should be made as follows: Lu, X. and Reissig, M.
(2009) ‘Rates of decay for structural damped models with decreasing in
time coefficients’, Int. J. Dynamical Systems and Differential Equations,
Vol. 2, Nos. 1/2, pp.21–55.
Biographical notes: Xiaojun Lu is a PhD student at Advanced Institute
of Mathematics in Zhejiang University. His research interest lies in
evolution equations, semi-linear damped wave equations, viscoelastic
damped systems and exact controllability problems.
Michael Reissig is a Professor at Institute of Applied Analysis
of TU Bergakademie Freiberg. He received Dr.rer.nat 1987 at
Martin-Luther Universität Halle-Wittenberg, Dr.sc. and Dr.habil. 1991 at
TU Bergakademie Freiberg. His research interests lies in wave equations,
partial differential equations of hyperbolic, elliptic or p-evolution type
with low regular coefficients and models of thermo-elasticity.
Copyright © 2009 Inderscience Enterprises Ltd.
21
22
X. Lu and M. Reissig
1 Introduction
In this paper we are interested in decay rates of the energy of solutions to the
Cauchy problem
utt − u + b(t)(−)σ ut = 0, σ ∈ (0, 1], b(t) = µ(1 + t)−δ , µ > 0,
δ ∈ [0, 1], u(0, x) = u0 (x), ut (0, x) = u1 (x).
(1.1)
The damping term is a special time-dependent case of a family of damping
operators which are introduced in Chen and Russell (1982). Theoretical arguments
and empirical studies motivated them to consider such damping operators
describing strong or structural damping effects. Lp − Lq decay estimates are studied
in the case σ = 0 (classical damped wave model with time-dependent dissipation)
in Wirth (2006, 2007). In these papers the cases of non-effective or effective
dissipation are discussed. Lp − Lq decay estimates for the case b(t) ≡ 1 and σ = 1
which corresponds to a viscoelastic model in Shibata (2000) or to a wave model
with internal damping in Guenther and Lee (1996) are studied in Shibata (2000).
The case σ ∈ (0, 1] of a mixed problem for a bounded domain in Rn has been
studied in Chen and Trigiani (1989, 1990). Here properties of the solutions are
obtained from the properties of the corresponding semigroup to the abstract elastic
differential equation with structural damping
utt + Au + Aσ ut = 0,
(1.2)
where A is a self-adjoint operator on a Hilbert space X, strictly positive, with
dense domain D(A). Corresponding problems are considered in Lions (1988).
It is shown that the value σ = 1/2 is a critical one in the sense that the properties
of the corresponding semigroup change from σ ∈ (0, 1/2) to σ ∈ [1/2, 1]. The case
σ ≥ 1 has been studied in Mugnolo (2008), which points out that the value σ = 1
is critical in the sense that one has an analytic semigroup under quite weak
assumptions for A. This observation results from the dominance of Aσ ut , σ > 1,
in comparison with Au. Local (in time) existence of weak solutions to quasi-linear
problems has been proved.
Recently, there are new results for the time-dependent case, for instance, in
Batty et al. (2008) the Cauchy problem
utt + A(t)u + B(t)ut = f (t), u(0) = ut (0) = 0
has been studied. In some special cases the property of Lp -maximal regularity of
the solution has already been proved.
In the present paper we consider the Cauchy problem for the special elastic
operator A = − and for the structural damping Aσ = (−)σ for σ ∈ (0, 1].
But we include in the dissipation term Aσ ut a decreasing time-dependent coefficient
b = b(t) which covers the constant case and model cases of strictly decreasing
coefficients. We are interested in the influence of the structural dissipation
b(t)(−)σ ut on energy decay rates, namely, on L2 − L2 estimates. To explain our
main goal let us recall a result from Shibata (2000) (formulated in correspondence
to our goal) for the solutions to the Cauchy problem
utt − u − ut = 0, u(0, x) = u0 (x), ut (0, x) = u1 (x).
(1.3)
Rates of decay for structural damped models
23
Proposition 1.1: The solution u = u(t, x) to (1.3) satisfies the following estimates:
∇u(t, ·)2L2 ≤ C1 (1 + t)−1 u0 2H 1 + u1 2L2 ,
∇β u(t, ·)2L2 ≤ Cβ (1 + t)−|β| u0 2H |β| + (1 + t)−(|β|−1) u1 2H |β|−2 if |β| ≥ 2,
∇β ut (t, ·)2L2 ≤ Cβ (1 + t)−(|β|+1) u0 2H |β| + (1 + t)−|β| u1 2H |β|
if |β| ≥ 0.
(1.4)
Although, the classical elastic and the kinetic energy have a decay if and only if
u1 ≡ 0, the model shows a parabolic effect, namely, higher order energies have a
better decay, therefore we need of course more regularity of the data.
Our main goal is to study under which conditions do we have a parabolic effect
for the solutions to the model (1.1).
Our considerations are based upon the following well-posedness result for (1.1)
in the scale of Sobolev spaces. Here we are interested in the existence of energy
solutions (s ≥ 1 in the following result).
Proposition 1.2: The Cauchy problem (1.1) is well-posed, that is, to every data
u0 ∈ H s , s ≥ 1, and u1 ∈ H s−1 there exists a unique solution u ∈ L∞ ([0, ∞), H s )
with ut ∈ L∞ ([0, ∞), H s−1 ). The solution depends continuously on the data.
For the energy of higher order
E |β| (u)(t) := u(t, ·)2H |β|+1 + ut (t, ·)2H |β|
with |β| ≥ 0 we have the estimate
E |β| (u)(t) ≤ C(t)(u0 H |β|+1 + u1 H |β| ).
(1.5)
Proof: Due to our assumptions we consider in the phase space the Cauchy
problem
vtt + |ξ|2 v + b(t)|ξ|2σ vt = 0, v(0, ξ) = v0 (ξ), vt (0, ξ) = v1 (ξ)
with v0 ∈ L2,s , v1 ∈ L2,s−1 . Here we use L2,s := F (H s ), that is, the Fourier
image of H s . Firstly, we restrict ourselves to the set of large frequencies
{|ξ| ≥ M }. We define the energy 2E(v)(t, ξ) = vt (t, ξ)2 + |ξ|2 v(t, ξ)2 . Straightforward calculations give dt E(v)(t, ξ) ≤ 0, hence, E(v)(t, ξ) ≤ E(v)(0, ξ). For
small frequencies {|ξ| ≤ M } we define the energy 2Ẽ(v)(t, ξ) = vt (t, ξ)2 + v(t, ξ)2 .
Straight-forward calculations give dt Ẽ(v)(t, ξ) ≤ C Ẽ(v)(t, ξ), hence, Ẽ(v)(t, ξ) ≤
C(t)Ẽ(v)(0, ξ). Both estimates together yield the unique existence and continuous
dependence on the data of the solution v satisfying v ∈ L∞ ([0, ∞), L2,s ) and
vt ∈ L∞ ([0, ∞), L2,s−1 ). This gives the desired statement about the well-posedness.
In the final analysis, the estimate of E |β| (u)(t) follows from the differentiation of
E |β| (u)(t).
24
X. Lu and M. Reissig
Remark 1.1: More careful considerations yield v ∈ C([0, ∞), L2,s ) and vt ∈
C([0, ∞), L2,s−1 ).
Having such a well-posedness result without loss of regularity we are able to study
decay estimates, which will be done in the following sections.
Section 2 is devoted to the constant case b(t) ≡ µ. The case σ ∈ (0, 12 ] and
b(t) = µ(1 + t)−δ , δ ∈ (0, 1], µ > 0, is discussed in Section 3. We will explain in
detail how hyperbolic or elliptic WKB analysis comes in. In Section 4 we study the
case σ ∈ ( 12 , 1] and b(t) = µ(1 + t)−δ , δ ∈ (0, 1], µ > 0. Some concluding remarks
complete the paper.
2 Time-independent dissipation
Let us consider the special case of (1.1)
utt − u + µ(−)σ ut = 0, u(0, x) = u0 (x),
ut (0, x) = u1 (x), µ > 0, σ ∈ (0, 1].
(2.1)
Theorem 2.1: The solution u = u(t, x) to (2.1) satisfies the following estimates:
In the case σ ∈ (0, 12 ] :
|β|
∇β u(t, ·)2L2 (1 + t)− 1−σ u0 2H |β| + (1 + t)−
for all |β| 1,
|β|−2σ
1−σ
u1 2H |β|−1
|β|+2−2σ
∇β ut (t, ·)2L2 (1 + t)− 1−σ u0 2H |β|+1 + (1 + t)max
for all |β| 0.
−
|β|+2−4σ
|β|
,− σ
1−σ
u1 2H |β|
(2.2)
In the case σ ∈ ( 12 , 1]:
|β|
|β|−1
σ
|β|
|β|−1
σ
∇β u(t, ·)2L2 (1 + t)− σ u0 2H |β| + (1 + t)−
if |β| = 1,
∇β u(t, ·)2L2 (1 + t)− σ u0 2H |β| + (1 + t)−
if |β| 2,
|β|+1
u1 2L2
u1 2H |β|−2σ
∇β ut (t, ·)2L2 (1 + t)− σ u0 2H |β|+2−2σ + (1 + t)−
for all |β| 0.
|β|
σ
u1 2H |β|
(2.3)
Remark 2.1: The statements for σ = 1 coincide with those from Proposition 1.1.
Proof: After partial Fourier transformation we have
vtt + |ξ|2 v + µ|ξ|2σ vt = 0, v(0, ξ) = v0 (ξ), vt (0, ξ) = v1 (ξ).
Let
λ1 (ξ) =
−µ|ξ|2σ +
−µ|ξ|2σ −
µ2 |ξ|4σ − 4|ξ|2
, λ2 (ξ) =
2
µ2 |ξ|4σ − 4|ξ|2
.
2
25
Rates of decay for structural damped models
We have the following explicit representation of the solution:
exp(λ1 t) − exp(λ2 t)
λ1 exp(λ2 t) − λ2 exp(λ1 t)
v0 (ξ) +
v1 (ξ) if λ1 =
λ2 ,
λ1 − λ2
λ1 − λ2
v(t, ξ) = exp(λ1 t)v0 (ξ) + t exp(λ1 t)(v1 (ξ) − λ1 v0 (ξ))
if λ1 = λ2 .
v(t, ξ) =
In the further calculations we use the following properties of λ1 and λ2 :
Lemma 2.2: It holds
µ2 |ξ|4σ − 4|ξ|2 , λ1 λ2 = |ξ|2 ,
1
λ1 + λ2 = −µ|ξ|2σ , λ1 − λ2 =
2
λ1 (ξ) = λ2 (ξ) = −µ|ξ|2σ if µ|ξ|2σ − 2|ξ| 0,
3
− µ2 |ξ|2−2σ λ1 (ξ) − µ1 |ξ|2−2σ , −µ|ξ|2σ λ2 (ξ) − µ2 |ξ|2σ if µ|ξ|2σ − 2|ξ| > 0.
1. Case: σ = 1/2
We only study the case µ < 2. We distinguish between small frequencies
{0 < |ξ| 1} and large frequencies {|ξ| 1}.
For small frequencies we conclude
µ|ξ|
t (|ξ||β| |v0 (ξ)| + |ξ||β|−1 |v1 (ξ)|) if |β| 1,
|ξ| |v(t, ξ)| exp −
2
µ|ξ|
|β|
t (|ξ||β|+1 |v0 (ξ)| + |ξ||β| |v1 (ξ)|) if |β| 0.
|ξ| |vt (t, ξ)| exp −
2
|β|
For |β| 1 we arrive at
u0 2L2 + u1 2L2
for 0 < t 1,
|ξ| |v(t, ξ)| dξ 0<|ξ|1
(1+t)−2|β| u0 2L2 + (1 + t)−2(|β|−1) u1 2L2 for t 1.
2β
2
For large frequencies we conclude
µ|ξ|
t (|ξ||β| |v0 (ξ)| + |ξ||β|−1 |v1 (ξ)|) if |β| 1,
|ξ||β| |v(t, ξ)| exp −
2
µ|ξ|
t (|ξ||β|+1 |v0 (ξ)| + |ξ||β| |v1 (ξ)|) if |β| 0.
|ξ||β| |vt (t, ξ)| exp −
2
So, for |β| 1 we arrive at
|ξ|1
|ξ|2β |v(t, ξ)|2 dξ 
u0 2H |β| + u1 2H max{|β|−1,0}
for 0 < t 1,
(1 + t)−2|β| u 2 + (1+ t)−2(|β|−1) u 2 for t 1.
0 L2
1 L2
Both estimates together yield the first inequality from (2.2) for µ < 2. Analogously
we prove the second one for µ < 2. The cases µ = 2 and µ > 2 are treated in the
same way.
26
X. Lu and M. Reissig
To study the other cases σ = 1/2 we introduce the constant C(µ, σ) :=
2 1/(2σ−1)
and divide the phase space into the following four regions:
µ
Z1 := {|ξ| ∈ (0, 12 C(µ, σ)]}, Z2 := {|ξ| ∈ [ 12 C(µ, σ), C(µ, σ)]}, Z3 := {|ξ| ∈ [C(µ, σ),
2C(µ, σ)]}, and Z4 := {|ξ| ∈ [2C(µ, σ), ∞)}. We will use the representation of
solution
v(t, ξ) =
exp(λ1 t) − exp(λ2 t)
λ1 exp(λ2 t) − λ2 exp(λ1 t)
v0 (ξ) +
v1 (ξ), |ξ| = C(µ, σ).
λ1 − λ 2
λ1 − λ2
2. Case: σ ∈ (0, 1/2)
Z1 : Here we use for |β| 1
|β|
1
2
− |ξ|2−2σ λ1 (ξ) − |ξ|2−2σ , |ξ|2|β| exp(−C|ξ|2−2σ t) t− 1−σ
µ
µ
|λ2 (ξ)| ∼ |ξ|2σ , |ξ|2(|β|−1) exp(−C|ξ|2σ t) t−
|β|−1
σ
for t > 0,
for t > 0,
where C is used as a universal constant. Thus we may conclude for |β| ≥ 1

v0 2 2 + v1 2 2
for t ∈ (0, 1),
L
L
2|β|
2
|ξ| |v(t, ξ)| dξ |β|
|β|−2σ
(1 + t)− 1−σ v 2 + (1 + t)− 1−σ v 2 for t ∈ [1, ∞).
Z1
0 L2
1 L2
In the same way we get for |β| 0
|ξ|2|β| |vt (t, ξ)|2 dξ Z1

v 2 + v1 2L2 for t ∈ (0, 1),


 0 L2
(1 + t)−



|β|+2−2σ
1−σ
|β|+2−4σ |β| v0 2L2 + (1 + t)max − 1−σ ,− σ v1 2L2
for t ∈ [1, ∞).
Z2 : Here we use the above inequality for λ1 and λ2 λ1 . There is no surprise to
get exponential decay in t by using L2 regularity of the data. We obtain
|ξ|2|β| |v(t, ξ)|2 dξ exp(−Ct) v0 2L2 + v1 2L2 for |β| ≥ 1,
Z2
|ξ|2|β| |vt (t, ξ)|2 dξ exp(−Ct) v0 2L2 + v1 2L2 for |β| ≥ 0.
Z2
2σ
Z3 : Here we have λ1 = λ2 = − µ|ξ|2 . There is the same behaviour as in Z2 .
Z4 : Now the additional regularity of the data comes in. We are interested in
decay estimates, thus for small t we should combine |ξ||β| with v0 , v1 , respectively.
We obtain
|ξ|2|β| |v(t, ξ)|2 dξ exp(−Ct) v0 2H |β| + v1 2H |β|−1 for |β| ≥ 1,
Z4
|ξ|2|β| |vt (t, ξ)|2 dξ exp(−Ct) v0 2H |β|+1 + v1 2H |β| for |β| ≥ 0.
Z4
27
Rates of decay for structural damped models
Taking into consideration all the estimates we arrive at the estimates for
∇β u(t, ·)L2 and for ∇β ut (t, ·)L2 in the case σ ∈ (0, 1/2).
3. Case: σ ∈ (1/2, 1]
Z1 : Here we use for |β| 1
|λ1 (ξ)| ∼ |ξ|, λ1 (ξ) = −µ|ξ|2σ , |ξ|2|β| exp(−C|ξ|2σ t) t−
|β|
σ
|λ2 (ξ)| ∼ |ξ|, λ2 (ξ) = −µ|ξ|2σ , |ξ|2(|β|−1) exp(−C|ξ|2σ t) t
for t > 0,
|β|−1
− σ
for t > 0.
Thus we conclude
|ξ|2|β| |v(t, ξ)|2 dξ Z1

v0 2 2 + v1 2 2
L
L
(1 + t)
|β|
− σ
v0 2L2 + (1 + t)
for t ∈ (0, 1),
|β|−1
− σ
v1 2L2 for t ∈ [1, ∞).
In the same way we get for |β| ≥ 0
|ξ|2|β| |vt (t, ξ)|2 dξ Z1

v0 2 2 + v1 2 2
L
L
(1 + t)
|β|+1
− σ
for t ∈ (0, 1),
|β|
− σ
v0 2L2 + (1 + t)
v1 2L2 for t ∈ [1, ∞).
Z2 : We get the same estimates as in the case σ ∈ (0, 1/2).
Z3 : Here we use
2
1
− |ξ|2−2σ λ1 (ξ) − |ξ|2−2σ .
µ
µ
We get the same estimates as in the case σ ∈ (0, 1/2).
Z4 : Now the additional regularity of the data comes in. We are interested in
decay estimates, thus for small t we should combine |ξ||β| with v0 , v1 , respectively.
We obtain
if |β| 2σ,
|ξ|2|β| |v(t, ξ)|2 dξ exp(−Ct) v0 2H |β| + v1 2H |β|−2σ
Z4
if |β| < 2σ,
|ξ|2|β| |v(t, ξ)|2 dξ exp(−Ct) v0 2H |β| + v1 2L2
Z4
|ξ|2|β| |vt (t, ξ)|2 dξ exp(−Ct) v0 2H |β|+2−2σ + v1 2H |β| if |β| ≥ 0.
Z4
Taking into consideration all the estimates we arrive at the estimates for
∇β u(t, ·)L2 and for ∇β ut (t, ·)L2 in the case σ ∈ (1/2, 1].
Remark 2.2: Theorem 2.1 tells us that we have the parabolic effect for σ ∈ (0, 1].
In general the elastic and kinetic energy have no decay (only if u1 ≡ 0), but higher
order energies have a decay. Moreover, the decay comes from the small frequencies
of Z1 .
28
X. Lu and M. Reissig
3 Time dependent strictly decreasing dissipation – σ ∈ (0, 1/2]
3.1 The case σ ∈ (0, 1/2): main results
In this section we study for σ ∈ (0, 1/2) the Cauchy problem
utt − u + b(t)(−)σ ut = 0, u(0, x) = u0 (x), ut (0, x) = u1 (x)
(3.1)
with b(t) = µ(1 + t)−δ , µ > 0, and δ ∈ (0, 1]. We will prove the following results.
Theorem 3.1: If δ ∈ (0, 1 − 2σ), then the following a-priori estimates hold:
1+δ
∇β u(t, ·)2L2 (1 + t)−|β| 1−σ u0 2H |β|
1+δ
2σ
max −(|β|−1) 1−δ
,− |β|− 1−δ
σ
1−σ
u1 2H |β|−1 , for |β| ≥ 1,
+ (1 + t)
1+δ
∇β ut (t, ·)2L2 (1 + t)−(|β|+1) 1−σ u0 2H |β|+1
1−δ
1+δ
2σ
+ (1 + t)max −|β| σ ,− |β|+1− 1−δ 1−σ u1 2H |β| ,
for |β| ≥ 0.
Theorem 3.2: If δ ∈ [1 − 2σ, 1], then the following a-priori estimates hold:
δ ∈ [1 − 2σ, 1) :
∇β u(t, ·)2L2 (1 + t)−
|β|(1−δ)
σ
+ (1 + t)−
∇β ut (t, ·)2L2 (1 + t)
u0 2H |β|
(|β|−1)(1−δ)
σ
(|β|+1)(1−δ)
−
σ
+ (1 + t)−
|β|(1−δ)
σ
u1 2H |β|−1 for |β| ≥ 1,
u0 2H |β|+1
u1 2H |β|
for |β| ≥ 0.
δ = 1:
∇β u(t, ·)2L2 (log(e + t))−
|β|
σ
u0 2H |β|
+ (log(e + t))−
∇β ut (t, ·)2L2 (log(e + t))−
|β|−1
σ
|β|+1
σ
u0 2H |β|+1
|β|
− σ
+ (log(e + t))
u1 2H |β|−1 for |β| ≥ 1,
u1 2H |β|
for |β| ≥ 0.
Remark 3.1: If we compare the estimates for δ = 1 − 2σ from Theorem 3.2 with
those ones from Theorem 3.1 if we formally set δ = 1 − 2σ, then both estimates
coincide.
3.1.1 The proof to Theorem 3.1
After partial Fourier transformation v(t, ξ) = Fx→ξ (u(t, x))(t, ξ) we obtain from
(3.1) the Cauchy problem
vtt + |ξ|2 v + b(t)|ξ|2σ vt = 0, v(0, ξ) = v0 (ξ), vt (0, ξ) = v1 (ξ)
with b(t) = µ(1 + t)−δ , µ > 0, and δ ∈ (0, 1 − 2σ).
The proof is divided into several steps.
(3.2)
Rates of decay for structural damped models
29
3.1.1.1 Division of the extended phase space into zones
We divide the extended phase space {(t, ξ) ∈ [0, ∞) × Rn } into the following zones
t
with Λ(t) = 1 + 0 b(s)ds, where ε is small and N is in general large:
1
the hyperbolic zone Zhyp (ε) = (t, ξ) : b(t)|ξ|2σ−1 ≤ 1 − ε ,
2
1
the reduced zone Zred (ε) = (t, ξ) : 1 − ε ≤ b(t)|ξ|2σ−1 ≤ 1 + ε ,
2
the elliptic zone Zell (ε, N )
1
2σ−1
2σ
≥ 1 + ε, and Λ(t)|ξ| ≥ N ,
= (t, ξ) : b(t)|ξ|
2
the pseudo-differential zone Zpd (ε, N )
1
= (t, ξ) : b(t)|ξ|2σ−1 ≥ 1 + ε, and Λ(t)|ξ|2σ ≤ N .
2
Moreover, we introduce separating lines. By tk = tk (|ξ|), k = 0, 1, 2, we denote the
separating lines between the pseudo-differential zone and the elliptic zone (k = 0),
between the elliptic zone and the reduced zone (k = 1) and between the reduced
zone and the hyperbolic zone (k = 2).
We define by the aid of the solution to (3.2) the energies |ξ||β| |v(t, ξ)| for |β| ≥ 1
and |ξ||β| |Dt v(t, ξ)| for |β| ≥ 0. Our goal is to derive in every zone estimates for
these energies.
3.1.1.2 Considerations in the hyperbolic zone
Lemma 3.3: The following estimates hold for all t ∈ [t2 (|ξ|), ∞):
|ξ|2σ t
|ξ| |v(t, ξ)| exp −
b(τ )dτ
2
t2 (|ξ|)
× |ξ||β| |v(t2 (|ξ|), ξ)| + |ξ||β|−1 |Dt v(t2 (|ξ|), ξ)| for |β| ≥ 1,
|ξ|2σ t
|ξ||β| |Dt v(t, ξ)| exp −
b(τ )dτ
2
t2 (|ξ|)
|β|+1
× |ξ|
|v(t2 (|ξ|), ξ)| + |ξ||β| |Dt v(t2 (|ξ|), ξ)| for |β| ≥ 0.
|β|
2σ t
Proof: Applying the transformation v(t, ξ) = exp − |ξ|2 0 b(s)ds w(t, ξ) leads to
the following Cauchy problem:

wtt + |ξ|2 w − 1 b2 (t)|ξ|4σ w − b (t) |ξ|2σ w = 0,
4
2
w(0, ξ) = w0 (ξ) = v0 (ξ), wt (0, ξ) = w1 (ξ) = 1 b(0)|ξ|2σ v0 (ξ) + v1 (ξ).
2
We define the micro-energy
T
W (t, ξ) = (p(t, ξ)w, Dt w) , p(t, ξ) :=
|ξ|2 −
µ2
(1 + t)−2δ |ξ|4σ , p(t, ξ) ∼ |ξ|
4
30
X. Lu and M. Reissig
in this zone. Here we consider p(t, ξ)2 w := |ξ|2 w − 14 b2 (t)|ξ|4σ w as a term of the
principal part and m(t, ξ)w := − b 2(t) |ξ|2σ w as a term of lower order. Thus, we have
to apply tools from hyperbolic WKB analysis. Transformation to a system of first
order in Dt leads to


Dt p(t, ξ)
0

 p(t, ξ)
0
p(t, ξ)


Dt W −
W −
 W = 0.

p(t, ξ)
0
 m(t, ξ)
0
p(t, ξ)
Using W = M W0 , M = 11 −1
1 , then after the first step of diagonalisation
we obtain
p(t, ξ)
0
W0
Dt W0 −
0
−p(t, ξ)


Dt p(t, ξ) m(t, ξ)
Dt p(t, ξ) m(t, ξ)
+
−
−
p(t, ξ)
p(t, ξ)
p(t, ξ) 
1
 p(t, ξ)

− 
 W0 = 0
2  Dt p(t, ξ) m(t, ξ) Dt p(t, ξ) m(t, ξ) 
+
−
−
p(t, ξ)
p(t, ξ)
p(t, ξ)
p(t, ξ)
with
−∂t q(t, ξ)
m(t, ξ)
1
=
, q(t, ξ) := b(t)|ξ|2σ .
2
2
p(t, ξ)
2
|ξ| − q(t, ξ)
In the following we use that
s
t
−∂r q(r, ξ)
1
dr 2
2
|ξ|
|ξ| − q(r, ξ)
t
−dq(r, ξ) Cε ,
s
is uniformly bounded with respect to (s, ξ), (t, ξ) ∈ Zhyp (ε). This and p(t, ξ) ∼ |ξ|
in the hyperbolic zone allow to stop the diagonalisation procedure after the first
step.
Thus we may conclude
|ξ|w(t, ξ) ≤ C |ξ|w(t2 (|ξ|), ξ) for t ≥ t2 (|ξ|),
|W (t, ξ)| ≤ C|W (t2 (|ξ|), ξ)|, Dt w(t, ξ)
Dt w(t2 (|ξ|), ξ) respectively, with a constant C which is independent of t ∈ [t2 (|ξ|), ∞).
Consequently, we derived for |β| ≥ 1 the a-priori estimates
|ξ|2σ t
b(τ )dτ |w(t, ξ)|
|ξ||β| |v(t, ξ)| = |ξ||β| exp −
2
0
2σ t
|ξ|
|ξ||β|−1 exp −
b(τ )dτ (|ξ||w(t2 (|ξ|), ξ)| + |Dt w(t2 (|ξ|), ξ)|)
2
0
2σ t
|ξ|
|β|−1
|ξ|
exp −
b(τ )dτ (|ξ||v(t2 (|ξ|), ξ)| + |Dt v(t2 (|ξ|), ξ)|).
2 t2 (|ξ|)
31
Rates of decay for structural damped models
In the same way we conclude for |β| ≥ 0 the a-priori estimates
|ξ|
|β|
|β|
|Dt v(t, ξ)| |ξ|
|ξ|2σ t
exp −
b(τ )dτ (|ξ||v(t2 (|ξ|), ξ)| + |Dt v(t2 (|ξ|), ξ)|).
2 t2 (|ξ|)
All estimates of the lemma are proved.
3.1.1.3 Considerations in the reduced zone
Lemma 3.4: The following estimates hold for all t ∈ [t1 (|ξ|), t2 (|ξ|)]:
|ξ|2σ t
b(τ )dτ
|ξ||β| |v(t, ξ)| exp −
4
t1 (|ξ|)
|β|
× |ξ| |v(t1 (|ξ|), ξ)| + |ξ||β|−1 |Dt v(t1 (|ξ|), ξ)| for |β| ≥ 1,
|ξ|2σ t
|β|
b(τ )dτ
|ξ| |Dt v(t, ξ)| exp −
4
t1 (|ξ|)
× |ξ||β|+1 |v(t1 (|ξ|), ξ)| + |ξ||β| |Dt v(t1 (|ξ|), ξ)| for |β| ≥ 0.
Proof: In the reduced zone we have |ξ| ≤ p1 and (1 + t)−δ |ξ|2σ−1 ∼ 1.
We study directly the equation vtt + |ξ|2 v + µ(1 + t)−δ |ξ|2σ vt = 0. The
δ
dissipation coefficient behaves as µ(1 + t)−δ |ξ|2σ ∼ (1 + t)− 1−2σ , that is, as
(1 + t)−a , a ∈ (0, 1).
Now let us devote to
vtt + |ξ|2 v + g(t, ξ)vt = 0, g(t, ξ) = µ(1 + t)−δ |ξ|2σ .
The special structure of g = g(t, ξ) with
|∂tk g(t, ξ)| ≤ ck (1 + t)−k g(t, ξ), g(t, ξ) ∼ (1 + t)−a , a =
δ
,
1 − 2σ
allows us to apply the treatment of effective dissipation from Wirth (2007).
We only have to understand in which part of the extended phase space (related
to the effective dissipation g in the language of Wirth (2007)) our reduced zone
Zred (ε) is located. Setting γ(t, ξ) = 12 g(t, ξ) the condition that (t, ξ) ∈ Zred (ε) is
transformed to
1
1
γ(t, ξ) ≤ |ξ| ≤
γ(t, ξ).
1+ε
1−ε
Therefore in the language of Wirth (2007) the point (t, ξ) belongs to
Zred
ε
1−ε
if
ξγ(t,ξ) ≤
ε
γ(t, ξ) , where ξ2γ(t,ξ) = ||ξ|2 − γ(t, ξ)2 |.
1−ε
It turns out that we can use the procedure for the reduced zone from Wirth (2007).
32
X. Lu and M. Reissig
The transformation
t
v(t, ξ) = exp −
γ(τ, ξ)dτ w(t, ξ)
0
transforms the above equation into
1
1
2
wtt + |ξ| − g(t, ξ) − gt (t, ξ) w = 0.
4
2
2
We introduce the micro-energy W :=
T
ε
.
1−ε γ(t, ξ)w, Dt w
Then the equation
Dt2 w − (|ξ|2 − γ(t, ξ)2 − γt (t, ξ))w = 0
is transformed into the following system of first order:

Dt γ(t, ξ)
γ(t, ξ)


Dt W (t, ξ) =  2
 |ξ| − γ(t, ξ)2 − γt (t, ξ)
ε
1−ε γ(t, ξ)

ε
γ(t, ξ)
1−ε

 W (t, ξ), W (s, ξ) is given.

0
To estimate the entries of the matrix we use
•
ε
γ(t, ξ) ∼ (1 + t)−a in Zred ( 1−ε
) with a ∈ (0, 1),
•
ξγ(t,ξ) εγ(t, ξ),
•
−γt (t, ξ) =
•
consequently, εγ(t, ξ) +
δ
1+t γ(t, ξ),
δ
ε
1−ε (1+t)
≤ 2εγ(t, ξ) for t ≥ T (δ, σ, ε).
ε by
Thus, the modulus of all entries of the matrix can be estimated in Zred 1−ε
2εγ(t, ξ) for t ≥ T (δ, σ, ε). Then we conclude immediately the following statement
by taking into account the compactness of the set {(t, ξ) ∈ [t1 (|ξ|), T ] × {|ξ| ≤ p1 }}:
Lemma 3.5: The fundamental solution E(t, s, ξ) to the system

ε
γ(t,
ξ)


1−ε


Dt W (t, ξ) =  2
 W (t, ξ)
 |ξ| − γ(t, ξ)2 − γt (t, ξ)

0
ε
1−ε γ(t, ξ)

Dt γ(t, ξ)
γ(t, ξ)
can be estimated by
E(t, s, ξ) exp
t
2εγ(τ, ξ)dτ
s
for all (s, ξ), (t, ξ) ∈ Zred (ε).
Rates of decay for structural damped models
33
From the backward transformation and the equivalence γ(t, ξ) ∼ |ξ| in Zred (ε) we
conclude the next a-priori estimates for all t ∈ [t1 (|ξ|), t2 (|ξ|)]:
t
|β|
2ε − 1 γ(τ, ξ)dτ (|ξ||β| |v(t1 (|ξ|), ξ)|
|ξ| |v(t, ξ)| exp
t1 (|ξ|)
|β|−1
+ |ξ|
|ξ||β| |Dt v(t, ξ)| exp
|Dt v(t1 (|ξ|), ξ)|)
t
for |β| ≥ 1,
(2ε − 1)γ(τ, ξ)dτ
t1 (|ξ|)
× (|ξ||β|+1 |v(t1 (|ξ|), ξ)| + |ξ||β| |Dt v(t1 (|ξ|), ξ)|) for |β| ≥ 0.
The last estimates imply the statements of the lemma.
3.1.1.4 Considerations in the elliptic zone
Lemma 3.6: The following estimates hold for all t ∈ [t0 (|ξ|), t1 (|ξ|)]:
t
dτ
|ξ||β| |v(t, ξ)| exp −C|ξ|2−2σ
t0 (|ξ|) b(τ )
|β|
2σ
× |ξ| |v(t0 (|ξ|), ξ)| + |ξ||β|− 1−δ |Dt v(t0 (|ξ|), ξ)| for |β| ≥ 1,
t
dτ
|ξ||β| |Dt v(t, ξ)| exp −C|ξ|2−2σ
t0 (|ξ|) b(τ )
2−δ
1
|ξ||β|+2−2σ |v(t0 (|ξ|), ξ)| + |ξ||β|+2−2σ 1−δ |Dt v(t0 (|ξ|), ξ)|
×
b(t)
t
+ exp −|ξ|2σ
b(τ )dτ |ξ||β| |Dt v(t0 (|ξ|), ξ)| for |β| ≥ 0.
t0 (|ξ|)
Proof: The proof is divided into several steps.
Introducing the micro-energy V (t, ξ) := (|ξ|v, Dt v)T we obtain from (3.2) the
system of first order
0
|ξ|
V = 0.
(3.3)
Dt V − AV := Dt V −
|ξ| ib(t)|ξ|2σ
Tools of the approach
We define the following classes of symbols related to (3.2), the structure of b = b(t)
and Zell (ε, N ):
S{m1 , m2 , m3 } = a(t, ξ) ∈ C ∞ (Zell (ε, N )) : |Dtk Dξα a(t, ξ)|
m3 +k
b(t)
≤ Ck,α |ξ|m1 −|α| b(t)m2
Λ(t)
for all multi-indices α and non-negative integers k .
This definition is reasonable for δ < 1 which fits to our case. The considerations
are based on the following properties of the symbol classes:
34
X. Lu and M. Reissig
•
S{m1 , m2 , m3 } ⊂ S{m1 + 2σk, m2 + k, m3 − k} for k ≥ 0;
•
if a ∈ S{m1 , m2 , m3 } and b ∈ S{k1 , k2 , k3 }, then
ab ∈ S{m1 + k1 , m2 + k2 , m3 + k3 };
•
if a ∈ S{m1 , m2 , m3 }, then Dtk a ∈ S{m1 , m2 , m3 + k}, and
Dξα a ∈ S{m1 − |α|, m2 , m3 };
t
if a ∈ S{−2σ, −1, 2}, then tξ |a(τ, |ξ|)|dτ ≤ C for all (t, ξ) ∈ Zell (ε, N )
(here we use the additional assumption Λ(t)|ξ|2σ ≥ N ).
•
Our goal is to estimate the fundamental solution E = E(t, s, ξ) = (Ekl (t, s, ξ))2k,l=1
to the system (3.3) in two steps.
Step 1:
A straight-forward estimate for the fundamental solution
Lemma 3.7: The fundamental solution E satisfies for all (s, ξ), (t, ξ) ∈ Zell (ε, N )
the following estimate:


1
1
t 2−2σ 2σ−1
b(s)|ξ|


|E11 (t, s, ξ)| |E12 (t, s, ξ)|
|ξ|
,
dτ 
exp −C


b(τ )
|E21 (t, s, ξ)| |E22 (t, s, ξ)|
b(t)
s
2σ−1
b(t)|ξ|
b(s)
where the constant C is independent of (s, ξ), (t, ξ) ∈ Zell (ε, N ).
Proof: The characteristic roots of the matrix A from (3.3) are
ib(t)|ξ|2σ + (−1)k−1 i b2 (t)|ξ|4σ − 4|ξ|2
, k = 1, 2.
λk =
2
The corresponding matrix M = M (t, ξ) of eigenvectors is
M (t, ξ) :=
1
1
.
λ1 (t, ξ)|ξ|−1 λ2 (t, ξ)|ξ|−1
Moreover,
b2 (t)|ξ|4σ − 4|ξ|2
| det M (t, ξ)| =
= b2 (t)|ξ|4σ−2 − 4 ≥ 4(1 + ε)2 − 4.
|ξ|
This fact indicates that M (t, ξ) is invertible in Zell (ε, N ) and its inverse is
M
−1
(t, ξ) = i
b2 (t)|ξ|4σ−2 − 4
λ2 (t, ξ)|ξ|−1 −1
.
−λ1 (t, ξ)|ξ|−1 1
From the definition of Zell (ε, N ) we conclude M −1 (t, ξ) ≤ Cε here.
35
Rates of decay for structural damped models
Let
V (t, ξ) = M (t, ξ)V1 (t, ξ),
Straight-forward calculations imply
then
Dt V1 = M −1 AM V1 − M −1 Dt M V1 .

 2
i b (t)|ξ|4σ −4|ξ|2 +ib(t)|ξ|2σ
0


2

D := M −1 AM =


−i b2 (t)|ξ|4σ −4|ξ|2 +ib(t)|ξ|2σ
0
2
and
−1
−1
−Dt λ1 (t, ξ) −Dt λ2 (t, ξ)
Dt λ2 (t, ξ)
Dt λ1 (t, ξ)
B := M Dt M = (det M (t, ξ)|ξ|)

b (t)|ξ|2σ
b(t)b (t)|ξ|4σ
+
 i b2 (t)|ξ|4σ − 4|ξ|2
i(b2 (t)|ξ|4σ − 4|ξ|2 )




1

= 
2σ
2
b(t)b (t)|ξ|4σ
− b (t)|ξ|
−

2
2
4σ
2
i(b (t)|ξ|4σ − 4|ξ|2 )
 i b (t)|ξ| − 4|ξ|



b (t)|ξ|2σ
i b2 (t)|ξ|4σ − 4|ξ|2 


4σ

b(t)b (t)|ξ|

−
i(b2 (t)|ξ|4σ − 4|ξ|2 ) 

.

b (t)|ξ|2σ

− 
2
4σ
2
i b (t)|ξ| − 4|ξ| 


b(t)b (t)|ξ|4σ
+
i(b2 (t)|ξ|4σ − 4|ξ|2 )
Using the symbol classes we have M −1 AM ∈ S{2σ, 1, 0} and M −1 Dt M ∈
S{0, 0, 1}. To carry out the second step of diagonalisation we follow the procedure
of the asymptotic theory of ordinary differential equations. Namely, we look
for a matrix N1 (t, ξ) having the representation N1 (t, ξ) := I + N (1) (t, ξ), where
N (0) := I, B (0) := B, F (0) := diag B (0) ,
(0)
(1)
Nqr
:=
Bqr
(1)
, q = r, Nqq
:= 0, τk , k = 1, 2, are the characteristic roots,
τq − τr
B (1) := (Dt − D + B)(I + N (1) ) − (I + N (1) )(Dt − D + F (0) ).
According to the properties of symbols we have N (1) ∈ S{−2σ, −1, 1} and
F (0) ∈ S{0, 0, 1}. For B (1) we obtain the relation
B (1) = B + [N (1) , D] − F (0) + Dt N (1) + BN (1) − N (1) F (0) .
The construction principle implies that the sum of the first three terms vanishes,
hence B (1) ∈ S{−2σ, −1, 2}. Finally, let us define
R1 = N1−1 ((Dt − D + B)(I + N (1) ) − (I + N (1) )(Dt − D + F (0) )).
But this means R1 = N1−1 B (1) ∈ S{−2σ, −1, 2}. Due to the fourth property of the
t
symbol hierarchies we conclude tξ |R1 (s, ξ)|ds ≤ C with a uniform constant C
for all (t, ξ) ∈ Zell (ε, N ). From the construction we have N (1) ∈ S{−2σ, −1, 1}.
Due to the definition of symbols and the definition of elliptic zone this means
(1)
|Nqr | ≤ CN1 . Consequently, for N large enough N1 − I < 12 in Zell (ε, N ) implies
36
X. Lu and M. Reissig
the invertibility of N1 . Introducing V2 := N1−1 V1 , then we get the following system
after the second step of diagonalisation:
0 = (Dt − D + B)N1 V2 = N1 (Dt − D + F (0) + R1 )V2 , where R1 ∈ S{−2σ, −1, 2}.
Thus it remains to study the system

 1
1
b2 (t)|ξ|4σ − 4|ξ|2 − b(t)|ξ|2σ
0
−


2
0 = ∂t V 2 −  2
V2
1
1 2
0
b (t)|ξ|4σ −4|ξ|2 − b(t)|ξ|2σ
2
2

b (t)|ξ|2σ
0
 2

 b (t)|ξ|4σ − 4|ξ|2



4σ


1
b(t)b (t)|ξ|
V2 + iR1 V2 .
+ 
+ 2


2 b (t)|ξ|4σ − 4|ξ|2



b (t)|ξ|2σ
b(t)b (t)|ξ|4σ 

0
−
+ 2
b2 (t)|ξ|4σ − 4|ξ|2 b (t)|ξ|4σ − 4|ξ|2
We are interested in the fundamental solution E2 to this system. First we estimate
E1 = E1 (t, s, ξ) as the fundamental solution of the diagonal part of this system,
that is,
t

b (τ )|ξ|2σ
1

(11)

= exp
− 1+ 2
E1 (t, s, ξ)


2
b (τ )|ξ|4σ − 4|ξ|2

s





2σ
2
4σ
2

dτ ,
b
(τ
)|ξ|
−
4|ξ|
+
b(τ
)|ξ|




t 1
b (τ )|ξ|2σ
(22)

1+ 2
= exp
E1 (t, s, ξ)


b (τ )|ξ|4σ − 4|ξ|2

s 2





2σ
2
4σ
2

dτ ,
b (τ )|ξ| − 4|ξ| − b(τ )|ξ|





E1 (t, s, ξ)(12) = E1 (t, s, ξ)(21) = 0.
Lemma 3.8: We have in the elliptic zone Zell (ε, N ) the following estimate:
t 2−2σ |ξ|
dτ
E1 (t, s, ξ) exp −C(ε, N )
b(τ )
s
with a positive constant C(ε, N ) which is independent of (s, ξ), (t, ξ) ∈ Zell (ε, N ).
Proof: The estimate for E1 will be determined by the estimate of E1 (t, s, ξ)(22) .
Using
1 2
|ξ|2−2σ
,
b (t)|ξ|4σ − 4|ξ|2 − b(t)|ξ|2σ ≤ −
2
b(τ )
|b (t)|
b2 (t)|ξ|4σ − 4|ξ|2 ∼ b2 (t)|ξ|4σ ,
≤ CN in Zell (ε, N ),
b2 (t)|ξ|2σ
the constant CN is small if N is large, the statement follows immediately.
37
Rates of decay for structural damped models
The fundamental solution E2 = E2 (t, s, ξ) satisfies
(Dt − D(t, ξ) + F (0) (t, ξ) + R1 (t, ξ))E2 = 0, E2 (s, s, ξ) = I.
Thus we have
t
∂t exp −i
(D(τ, ξ) − F (0) (τ, ξ))dτ E2 (t, s, ξ)
s
t
= −i exp −i
(D(τ, ξ) − F (0) (τ, ξ))dτ R1 (t, ξ)E2 (t, s, ξ),
s
and, consequently,
t
(0)
E2 (t, s, ξ) = exp i
(D(τ, ξ) − F (τ, ξ))dτ E2 (s, s, ξ)
s
t
t
exp i
(D(τ, ξ) − F (0) (τ, ξ))dτ R1 (θ, ξ)E2 (θ, s, ξ)dθ.
−i
s
θ
The statement of Lemma 3.8 motivates to introduce the weight
b (t)|ξ|2σ
1
( b2 (t)|ξ|4σ − 4|ξ|2 − b(t)|ξ|2σ ).
w(t, ξ) =
1+ 2
4σ
2
2
b (t)|ξ| − 4|ξ|
Let us define
t
Q(t, s, ξ) := exp −
w(τ, ξ)dτ E2 (t, s, ξ).
s
Then we get
t
(0)
(iD(τ, ξ) − iF (τ, ξ) − w(τ, ξ)I)dτ
Q(t, s, ξ) = exp
s
t
t
(0)
−
exp
(iD(τ, ξ) − iF (τ, ξ) − w(τ, ξ)I)dτ R1 (θ, ξ)Q(θ, s, ξ)dθ.
s
θ
Furthermore,
t
H(t, s, ξ) = exp
(iD(τ, ξ) − iF (0) (τ, ξ) − w(τ, ξ)I)dτ
s
t
b (τ )|ξ|2σ
2
4σ
2
b (τ )|ξ| − 4|ξ| +
dτ , 1 .
= diag exp −
b2 (τ )|ξ|4σ − 4|ξ|2
s
Hence, the matrix H is uniformly bounded for (s, ξ), (t, ξ) ∈ Zell (ε, N ). Taking
account of R1 ∈ S{−2σ, −1, 2} the matrix Q = Q(t, s, ξ) which is given by the
matrizant representation
Q(t, s, ξ) = H(t, s, ξ) +
···
s
∞
k=1
tk−1
(−i)k
t
H(t, t1 , ξ)R1 (t1 , ξ)
s
t1
H(t1 , t2 , ξ)R1 (t2 , ξ)
s
H(tk−1 , tk , ξ)R1 (tk , ξ)dtk · · · dt2 dt1
38
X. Lu and M. Reissig
is uniformly bounded in Zell (ε, N ). From the last considerations we may conclude
E2 (t, s, ξ) exp −C
s
t
|ξ|2−2σ
dτ
b(τ )
for all (s, ξ), (t, ξ) ∈ Zell (ε, N ).
representation
The
1 1
1 1
backward
transformation
gives
the
E(t, s, ξ) = M (t, ξ)N1 (t, ξ)E2 (t, s, ξ)N1−1 (s, ξ)M −1 (s, ξ),
but now we are only interested in s ≤ t. Due to Lemma 3.8 and the uniform
boundedness of Q, N1 and N1−1 the statement of Lemma
3.7 follows if we take
into consideration the estimate of the norm |M (t, ξ)| 11 11 |M −1 (s, ξ)|. Using
2−2σ
|λ1 (t, ξ)| ∼ b(t)|ξ|2σ , |λ2 (t, ξ)| ∼ |ξ|b(t) and | det M (t, ξ)| ∼ b(t)|ξ|2σ−1 , then the
definition of Zell (ε, N ), the fact that b is decreasing and s ≤ t yield
1
|M (t, ξ)|
1
1
|M (t, ξ)|
1
1
|M (t, ξ)|
1
1
|M (t, ξ)|
1
(11)
1 −1
(s,
ξ)|
|M
1 (12)
1 −1
|M (s, ξ)|
1 (21)
1 −1
|M (s, ξ)|
1 (22)
1 −1
|M
(s,
ξ)|
1 1,
1
,
b(s)|ξ|2σ−1
b(t)|ξ|2σ−1 ,
b(t)
.
b(s)
These estimates give the desired statement.
Remark 3.2: Taking account the dissipative character of our model we cannot be
satisfied with the estimate for E21 from Lemma 3.7. For this reason we need a
refined estimate which we present in the next step.
Step 2:
A refined estimate for the fundamental solution
Lemma 3.9: The fundamental solution E satisfies for all (s, ξ), (t, ξ) ∈ Zell (ε, N )
the following estimate:
|E11 (t, s, ξ)| |E12 (t, s, ξ)|
|E21 (t, s, ξ)| |E22 (t, s, ξ)|


1
1
t 2−2σ 2σ−1


b(s)|ξ|
|ξ|

exp −C
dτ 


b(τ )
1
1
s
b(t)|ξ|2σ−1 b(t)b(s)|ξ|4σ−2
t
0 0
2σ
+ exp −|ξ|
b(τ )dτ
,
0 1
s
where the constant C is independent of (s, ξ), (t, ξ) ∈ Zell (ε, N ).
39
Rates of decay for structural damped models
Remark 3.3: If we formally set δ = 0, then we see that the estimates coincide with
those ones from Theorem 2.1 for the case of constant coefficients.
Proof: If Φk (t, s, ξ), k = 1, 2, solves the equation Φtt + b(t)|ξ|2σ Φt + |ξ|2 Φ = 0
with initial values Φk (s, s, ξ) = δ1k , ∂t Φk (s, s, ξ) = δ2k , we have
|ξ|v(t, ξ)
Dt v(t, ξ)



 |ξ|v(s, ξ)

=
 Dt Φ1 (t, s, ξ)
 D v(s, ξ) .
t
iDt Φ2 (t, s, ξ)
|ξ|
Φ1 (t, s, ξ)
i|ξ|Φ2 (t, s, ξ)
Hence, it follows from Lemma 3.7
|ξ|2−2σ
dτ ,
|Φ1 (t, s, ξ)| exp −C
b(τ )
s
t 2−2σ |ξ|
1
|Φ2 (t, s, ξ)| dτ ,
exp
−C
2σ
b(s)|ξ|
b(τ )
s
t 2−2σ |ξ|
dτ ,
|∂t Φ1 (t, s, ξ)| b(t)|ξ|2σ exp −C
b(τ )
s
t 2−2σ |ξ|
b(t)
|∂t Φ2 (t, s, ξ)| exp −C
dτ .
b(s)
b(τ )
s
t
Let
Ψk (t, s, ξ) = ∂t Φk (t, s, ξ), k = 1, 2,
then
Ψk (s, s, ξ) = δk2 . Standard calculations lead to
∂t Ψk + b(t)|ξ|2σ Ψk = −|ξ|2 Φk ,
t
t
Ψ1 (t, s, ξ) = −|ξ|2
Φ1 (τ, s, ξ) exp −|ξ|2σ
b(θ)dθ dτ,
s
τ
t
t
t
2σ
2
2σ
b(θ)dθ − |ξ|
Φ2 (τ, s, ξ) exp − |ξ|
b(θ)dθ dτ.
Ψ2 (t, s, ξ) = exp − |ξ|
s
s
τ
Using the estimates for |Φ1 (t, s, ξ)| and |Φ2 (t, s, ξ)|, we have
τ 2−2σ
t
t
|ξ|
dθ − |ξ|2σ
exp −C
b(θ)dθ dτ,
|Ψ1 (t, s, ξ)| ≤ C1 |ξ|2
b(θ)
s
s
τ
t
|Ψ2 (t, s, ξ)| ≤ exp −|ξ|2σ
b(θ)dθ
s
t
τ 2−2σ
|ξ|2−2σ t
|ξ|
2σ
dθ − |ξ|
+ C1
exp −C
b(θ)dθ dτ.
b(s)
b(θ)
s
s
τ
If we are able to derive the desired estimate for |Ψ1 (t, s, ξ)|, then we conclude
immediately the desired estimate for |Ψ2 (t, s, ξ)|. Applying partial integration we get
t
τ 2−2σ |ξ|2−2σ
|ξ|
|Ψ1 (t, s, ξ)| ≤ C1 |ξ| exp −C
dθ
dθ
exp −C
b(θ)
b(θ)
s
s
t
t
b(θ)dθ dτ
× exp −|ξ|2σ
2
τ
t
40
X. Lu and M. Reissig
t 2−2σ t
τ 2−2σ |ξ|
1
|ξ|
dθ
exp −C
dθ
= C1 |ξ|2−2σ exp −C
b(θ)
b(τ
)
b(θ)
s
s
t
t
2σ
× ∂τ exp −|ξ|
b(θ)dθ dτ
τ
τ 2−2σ t 2−2σ 1
|ξ|
|ξ|
2−2σ
dθ
exp −C
dθ
= C1 |ξ|
exp −C
b(θ)
b(τ )
b(θ)
s
t
t t
t
t
× exp −|ξ|2σ
b(θ)dθ −
exp −|ξ|2σ
b(θ)dθ
τ
s
τ
s
τ 2−2σ 1
|ξ|
× ∂τ
exp −C
dθ
dτ .
b(τ )
b(θ)
t
Using with a universal constant C < 1 the estimate
t 2−2σ t
|ξ|
dθ ≤ exp C|ξ|2σ
b(θ)dθ
exp C
b(θ)
τ
τ
gives
s 2−2σ t 2−2σ 1
1
|ξ|
|ξ|
dθ
+
exp −C
dθ
|Ψ1 (t, s, ξ)| ≤ C1 |ξ|2−2σ exp −C
b(θ)
b(t) b(s)
b(θ)
s
t
t
t
t
× exp −|ξ|2σ
b(θ)dθ +
exp −|ξ|2σ
b(θ)dθ
s
s
τ
τ 2−2σ |b (τ )| |ξ|2−2σ
|ξ|
dθ
+
× exp −C
dτ
b(θ)
b2 (τ )
b2 (τ )
t
t 2−2σ t
1
1
|ξ|
2−2σ
2σ
dθ
+
exp −C|ξ|
exp −C
b(θ)dθ
≤ C1 |ξ|
b(θ)
b(t) b(s)
s
s
t
t
|b (τ )| |ξ|2−2σ
+ 2
+
exp −C|ξ|2σ
b(θ)dθ
dτ
b2 (τ )
b (τ )
s
τ
t 2−2σ t
t
1
|ξ|
dθ
+
≤ C1 |ξ|2−2σ exp −C
exp −C|ξ|2σ
b(θ)dθ
b(θ)
b(t)
s
s
τ
2−2σ
|b (τ )| |ξ|
×
+ 2
dτ .
b2 (τ )
b (τ )
Here we used that b = b(t) is decreasing and s ≤ t. Now we consider the part
t
t
b (τ )
|ξ|2−2σ
+ 2
dτ.
exp −C|ξ|2σ
b(θ)dθ
− 2
Θ(t, s, ξ) :=
b (τ )
b (τ )
s
τ
Using again partial integration
t
t
Θ(t, s, ξ) = −C −1
∂τ exp −C|ξ|2σ
b(θ)dθ
−
b (τ )
|ξ|2−4σ
dτ
+
b3 (τ )|ξ|2σ
b3 (τ )
s
τ
t
t
b (τ )
|ξ|2−4σ −1
2σ
b(θ)dθ
− 3
+ 3
= −C exp −C|ξ|
b (τ )|ξ|2σ
b (τ ) s
τ
t
t
|ξ|2−4σ
b (τ )
−1
2σ
dτ.
+C
exp −C|ξ|
b(θ)dθ ∂τ − 3
+ 3
b (τ )|ξ|2σ
b (τ )
s
τ
41
Rates of decay for structural damped models
We study the first part and second part, respectively. Notice that
−
C
C
b (t)
≤
,
≤
b3 (t)|ξ|2σ
Λ(t)|ξ|2σ b(t)
N b(t)
the first part can be estimated by
s
t
C
b(t) .
1
C
,
≤
b3 (t)|ξ|4σ−2
b(t)
As for the second part, namely, for
|ξ|2σ t
b (τ )
3(b (τ ))2
3b (τ )
dτ,
exp −
b(θ)dθ
− 3
+
−
2
b (τ )|ξ|2σ
b4 (τ )|ξ|2σ
b4 (τ )|ξ|4σ−2
τ
since
b (τ )
b (τ )
C b (τ )
b (τ )
=
≤
−
,
b3 (τ )|ξ|2σ
b (τ )b(τ ) b2 (τ )|ξ|2σ
N b2 (τ )
≤−
(b (τ ))2
b4 (τ )|ξ|2σ
b (τ )
C b (τ )
b (τ )
,
−
,
≤
−ε
N b2 (τ )
b4 (τ )|ξ|4σ−2
b2 (τ )
so we can move it to the left-hand side, and get the desired estimates for |Ψ1 (t, s, ξ)|
and |E21 (t, s, ξ)|, respectively:
t 2−2σ |ξ|
|ξ|2−2σ
|Ψ1 (t, s, ξ)| exp −C
dθ ,
b(t)
b(θ)
s
t 2−2σ |ξ|
1
dθ .
|E21 (t, s, ξ)| exp
−C
b(t)|ξ|2σ−1
b(θ)
s
The estimates for |Ψ2 (t, s, ξ)| and |E22 (t, s, ξ)| follow immediately, they are
|Ψ2 (t, s, ξ)| = |E22 (t, s, ξ)| exp −|ξ|
2σ
t
b(θ)dθ
s
t 2−2σ |ξ|
1
dθ .
exp −C
+
b(t)b(s)|ξ|4σ−2
b(θ)
s
Thus all desired estimates are proved.
Remark 3.4: We are able to derive a refined estimate for the fundamental solution
because we use in the proof to Lemma 3.9 only the estimates for E11 and E12 from
Lemma 3.7. Both estimates are optimal with our analytical tools.
Conclusion
Taking into consideration the estimates from Lemma 3.9 and the behaviour of
b(t0 (|ξ|)) we arrive at the estimates which we wanted to prove.
42
X. Lu and M. Reissig
3.1.1.5 Considerations in the pseudo-differential zone
Lemma 3.10: The following estimates hold for all t ∈ [0, t0 (|ξ|)]:
1−δ
1−δ
|ξ||β| |v(t, ξ)| (1 + t)−|β| 2σ |v(0, ξ)| + (1 + t)1−|β| 2σ |Dt v(0, ξ)| for |β| ≥ 1,
1−δ
|β|
|ξ| |Dt v(t, ξ)| (1 + t)1−(|β|+2) 2σ |v(0, ξ)|
t
+ exp −|ξ|2σ
b(τ )dτ |ξ||β| |Dt v(0, ξ)|
for |β| ≥ 0.
0
Proof: Let us introduce the micro-energy
V =
T
N
(1 + t)
1−δ
2σ
v, Dt v
.
Then we get the system

1−δ
−1
− 1−δ
2σ
N (1 + t)

−i 2σ (1 + t)
 V (t, ξ),
Dt V (t, ξ) = 
 |ξ|2

1−δ
2σ
(1 + t) 2σ
ib(t)|ξ|
N

and for its fundamental solution


1−δ
−1
− 1−δ
2σ
(1
+
t)
iN
(1
+
t)
 2σ

 E(t, s, ξ),
∂t E(t, s, ξ) = 
 |ξ|2

1−δ
2σ
(1 + t) 2σ
i
−b(t)|ξ|
N
1 0
E(s, s, ξ) =
for all s, t ∈ [0, t0 (|ξ|)].
0 1
Setting s = 0 we get for t ∈ [0, t0 (|ξ|)] the following representation of solutions:
E11 (t, 0, ξ) =
1
(1 + t)
1−δ
2σ
1
E21 (t, 0, ξ) = 2
λ (t, ξ)
E12 (t, 0, ξ) =
0
iN
(1 + t)
+i
1−δ
2σ
N
(1 + t)
t
1−δ
2σ
t
E21 (τ, 0, ξ)dτ,
0
i|ξ|2 (1 + τ )
λ (τ, ξ)
N
t
E22 (τ, 0, ξ)dτ,
2
0
i|ξ|2
1
+
E22 (t, 0, ξ) = 2
λ (t, ξ) N λ2 (t, ξ)
t
(1 + τ )
1−δ
2σ
1−δ
2σ
E11 (τ, 0, ξ)dτ,
λ2 (τ, ξ)E12 (τ, 0, ξ)dτ,
0
t
where λ2 (t, ξ) = exp(|ξ|2σ 0 b(s)ds).
Now let us estimate the modulus of |Ekl (t, 0, ξ)|, k, l = 1, 2.
43
Rates of decay for structural damped models
|E11 (t, 0, ξ)| and |E21 (t, 0, ξ)|
First case:
We have
(1 + t)
1−δ
2σ
E11 (t, 0, ξ) = 1 − |ξ|2
t
0
1
λ2 (τ, ξ)
τ
λ2 (s, ξ)(1 + s)
1−δ
2σ
E11 (s, 0, ξ)ds dτ,
0
hence,
|(1 + t)
1−δ
2σ
E11 (t, 0, ξ)| ≤ 1 + |ξ|2
t
0
0
τ
1−δ
λ2 (s, ξ)
|(1 + s) 2σ E11 (s, 0, ξ)|ds dτ.
λ2 (τ, ξ)
Using the monotonicity of λ2 (t, ξ) in t and the definition of the pseudo-differential
zone we arrive at
|E11 (t, 0, ξ)| ≤
C
(1 + t)
1−δ
2σ
.
Then we conclude immediately
|E21 (t, 0, ξ)| ≤ CN |ξ|2 (1 + t) ≤ CN
1
(1 + t)
1−δ
σ −1
.
Second case: |E12 (t, 0, ξ)| and |E22 (t, 0, ξ)|
From the representations for E12 and E22 we get
|ξ|2
1
− 2
E22 (t, 0, ξ) = 2
λ (t, ξ) λ (t, ξ)
t
2
λ (τ, ξ)
0
τ
E22 (s, 0, ξ)ds dτ,
0
hence,
λ (t, ξ)E22 (t, 0, ξ) = 1 − |ξ|
2
t
τ
λ (τ, ξ)
E22 (s, 0, ξ)ds dτ
0
0
t τ 2
λ (τ, ξ) 2
λ (s, ξ)E22 (s, 0, ξ)ds dτ.
= 1 − |ξ|2
2
0 0 λ (s, ξ)
2
2
We have in the pseudo-differential zone λ−2 (s, ξ) ≤ C and λ2 (τ, ξ) ≤ C.
Consequently,
|λ2 (t, ξ)E22 (t, 0, ξ)| ≤ C(1 + |ξ|2 (1 + t)2 ) ≤ C 1 +
1
(1 + t)
1−δ
σ −2
≤C
C
for δ ∈ (0, 1 − 2σ). Thus we derived |E22 (t, 0, ξ)| ≤ λ2 (t,ξ)
≤ C. It remains to
estimate |E12 (t, 0, ξ)|. The representation for E12 (t, 0, ξ) gives immediately
|E12 (t, 0, ξ)| ≤
C
(1 + t)
1−δ
2σ −1
.
44
X. Lu and M. Reissig
As an application we can estimate the elastic energy in the pseudo-differential zone
as follows:
|ξ||v(t, ξ)| 1
(1 + t)
1
(1 + t)
1−δ
2σ
1−δ
2σ
|v(t, ξ)| |E11 (t, 0, ξ)||v(0, ξ)| + |E12 (t, 0, ξ)||Dt v(0, ξ)|
|v(0, ξ)| +
1
(1 + t)
1−δ
2σ −1
|Dt v(0, ξ)|
for δ ∈ (0, 1 − 2σ).
For the kinetic energy we obtain the following estimates:
|Dt v(t, ξ)| |E21 (t, 0, ξ)||v(0, ξ)| + |E22 (t, 0, ξ)||Dt v(0, ξ)|
t
1
2σ
|v(0,
ξ)|
+
exp
−|ξ|
b(τ
)dτ
|Dt v(0, ξ)|
1−δ
(1 + t) σ −1
0
for δ ∈ (0, 1 − 2σ).
By using the definition of the pseudo-differential zone we conclude for the
elastic energy and the kinetic energy of higher order the estimates from the lemma.
3.1.1.6 Gluing procedure
Now we have to glue the estimates from the Lemmas 3.3, 3.4, 3.6 and 3.10.
If we consider the set Mε := {ξ : |ξ| ≥ ε}, then it is clear that an exponential
type decay for the higher order energies follows from Lemma 3.3 under the usual
regularity assumption for the data from the Cauchy problem for the wave equation.
Thus the interesting case is to glue all estimates for small frequencies, let us say
for {ξ : |ξ| ≤ p}, where p is sufficiently small. In this case all the zones restricted to
small frequencies are unbounded.
1. Case:
t ≤ t0 (|ξ|)
In this Case we apply Lemma 3.10.
2. Case:
t ∈ [t0 (|ξ|), t1 (|ξ|)]
Now we have to glue the estimates from Lemmas 3.10 and 3.6.
Corollary 3.11: The following estimates hold for all t ∈ [t0 (|ξ|), t1 (|ξ|)]:
t
dτ
|ξ||β| |v(t, ξ)| exp −C|ξ|2−2σ
0 b(τ )
2σ
|β|
|ξ| |v(0, ξ)| + |ξ||β|− 1−δ |Dt v(0, ξ)| , |β| ≥ 1,
t
1
dτ
|β|
2−2σ
|ξ| |Dt v(t, ξ)| exp −C|ξ|
b(τ
)
b(t)
0
2σ
|β|+2−2σ
× |ξ|
|v(0, ξ)| + |ξ||β|+2−2σ− 1−δ |Dt v(0, ξ)|
t
2σ
b(τ )dτ |ξ||β|+2− 1−δ |v(0, ξ)|
+ exp −|ξ|2σ
0
|β|
+ |ξ| |Dt v(0, ξ)| , |β| ≥ 0.
45
Rates of decay for structural damped models
Proof: In the pseudo-differential zone the terms with the phase functions have no
meaning. This follows from
|ξ|
2σ
t0 (|ξ|)
1+
b(τ )dτ
= N, |ξ|
2−2σ
t0 (|ξ|)
1+
0
0
∼ |ξ|
2−2σ
1
dτ
b(τ )
(1 + t0 (|ξ|))
∼ |ξ|
1+δ
4σ
2−
1−δ
∼ |ξ| 1−δ .
1+δ
2−2σ
−2σ
|ξ|
1+δ
1−δ
2−2σ−2σ
∼ |ξ|
4σ
> 0 for δ ∈ (0, 1 − 2σ) and |ξ| ≤ p in Zpd (ε, N )
Taking into consideration 2 − 1−δ
we have the uniform boundedness. Consequently, the integral in the phases in the
estimates from Lemma 3.6 can be extended from [t0 (|ξ|), t] to [0, t]. Let us begin to
estimate |ξ||β| |v(t, ξ)|. If we choose real constants r, q ∈ [1, |β|], then the statements
of Lemmas 3.6 and 3.10 imply
|β|
|ξ|
|v(t, ξ)| exp −C|ξ|
2−2σ
0
t
dτ
b(τ )
2σ
× |ξ||β|−r |ξ|r |v(t0 (|ξ|), ξ)| + |ξ||β|−q |ξ|q− 1−δ |Dt v(t0 (|ξ|), ξ)|)
t
dτ
exp −C|ξ|2−2σ
0 b(τ )
1−δ
1−δ
|β|−r
(1 + t0 (|ξ|))−r 2σ |v(0, ξ)| + (1 + t0 (|ξ|))1−r 2σ |Dt v(0, ξ)|
× |ξ|
t
dτ
2−2σ
+ exp −C|ξ|
0 b(τ )
1−δ
2σ
2σ
|β|−q
(1 + t0 (|ξ|))1−(q− 1−δ +2) 2σ |v(0, ξ)| + |ξ|q− 1−δ |Dt v(0, ξ)| .
× |ξ|
Taking account the definition of t0 (|ξ|) and 2(1 − δ) − 4σ > 0 we conclude
immediately
dτ |β|
|ξ| |v(0, ξ)|
0 b(τ )
2σ
+ |ξ||β|− 1−δ |Dt v(0, ξ)| , |β| ≥ 1.
|ξ||β| |v(t, ξ)| exp −C|ξ|2−2σ
t
In the same way we obtain
|ξ||β| |Dt v(t, ξ)| exp −C|ξ|2−2σ
0
t
1
dτ
b(τ ) b(t)
2σ
× |ξ||β|+2−2σ |v(0, ξ)| + |ξ||β|+2−2σ− 1−δ |Dt v(0, ξ)|
t
2σ
2σ
b(τ )dτ |ξ||β|+2− 1−δ |v(0, ξ)|
+ exp −|ξ|
0
|β|
+ |ξ|
This completes the proof.
|Dt v(0, ξ)| , |β| ≥ 0.
46
X. Lu and M. Reissig
t ∈ [t1 (|ξ|), ∞)
3. Case:
From Lemmas 3.3 and 3.4 we obtain the following statement:
Corollary 3.12: The following estimates hold for all t ∈ [t1 (|ξ|), ∞):
|ξ|2σ t
b(τ )dτ
|ξ| |v(t, ξ)| exp −
4
t1 (|ξ|)
× |ξ||β| |v(t1 (|ξ|), ξ)| + |ξ||β|−1 |Dt v(t1 (|ξ|), ξ)| for |β| ≥ 1,
|ξ|2σ t
b(τ )dτ
|ξ||β| |Dt v(t, ξ)| exp −
4
t1 (|ξ|)
|β|+1
× |ξ|
|v(t1 (|ξ|), ξ)| + |ξ||β| |Dt v(t1 (|ξ|), ξ)| for |β| ≥ 0.
|β|
Finally, we have to glue the estimates from Corollaries 3.11 and 3.12.
Corollary 3.13: The following estimates hold for all t ∈ [t1 (|ξ|), ∞):
|ξ||β| |v(t, ξ)| exp −C|ξ|2−2σ
|ξ|2σ t
b(τ )dτ
exp −
4
0
t1 (|ξ|)
|β|
2σ
|β|− 1−δ
× |ξ| |v(0, ξ)| + |ξ|
|Dt v(0, ξ)| ,
2σ t
2σ
|ξ|
+ exp −
b(τ )dτ |ξ||β|+1− 1−δ |v(0, ξ)|
4
0
+ |ξ||β|−1 |Dt v(0, ξ)| , |β| ≥ 1,
t1 (|ξ|)
dτ
|ξ|2σ t
|β|
2−2σ
|ξ| |Dt v(t, ξ)| exp −C|ξ|
b(τ )dτ
exp −
b(τ )
4
0
t1 (|ξ|)
2σ
× |ξ||β|+1 |v(0, ξ)| + |ξ||β|+1− 1−δ |Dt v(0, ξ)| ,
2σ
|ξ|2σ t
b(τ )dτ |ξ||β|+2− 1−δ |v(0, ξ)|
+ exp −
4
0
|β|
+ |ξ| |Dt v(0, ξ)| , |β| ≥ 0.
t1 (|ξ|)
dτ
b(τ )
3.1.1.7 Energy estimates
Using (1.5) we can restrict ourselves to t ≥ 1. From Lemma 3.10 we conclude the
following statement:
Corollary 3.14: The following estimates hold for all t ∈ [1, t0 (|ξ|)]:
1−δ
|ξ||β| |v(t, ξ)| (1 + t)−|β| 2σ |v(0, ξ)| + (1 + t)1−|β|
for |β| ≥ 1,
|ξ||β| |Dt v(t, ξ)| (1 + t)1−(|β|+2)
1−δ
2σ
1−δ
2σ
|Dt v(0, ξ)|
|v(0, ξ)| + (1 + t)−|β|
for |β| ≥ 0.
From Corollary 3.11 we obtain the following result:
1−δ
2σ
|Dt v(0, ξ)|
47
Rates of decay for structural damped models
Corollary 3.15: The following estimates hold for all t ∈ [t0 (|ξ|), t1 (|ξ|)]:
1+δ
1+δ
2σ
|ξ||β| |v(t, ξ)| (1 + t)−|β| 2−2σ |v(0, ξ)| + (1 + t)− |β|− 1−δ 2−2σ |Dt v(0, ξ)|
for |β| ≥ 1,
1+δ
|ξ||β| |Dt v(t, ξ)| (1 + t)δ−(|β|+2−2σ) 2−2σ |v(0, ξ)|
2σ
1+δ
+ (1 + t)δ−(|β|+2−2σ− 1−δ ) 2−2σ |Dt v(0, ξ)|
2σ
+ (1 + t)−(|β|+2− 1−δ )
−|β| 1−δ
2σ
+ (1 + t)
1−δ
2σ
|v(0, ξ)|
|Dt v(0, ξ)| for |β| ≥ 0.
To derive the corresponding energy estimates from Corollary 3.13 we have to
estimate the term
t1 (|ξ|)
dτ
|ξ|2σ t
Sr (t, |ξ|) := |ξ|r exp −C|ξ|2−2σ
b(τ )dτ .
exp −
b(τ )
4
0
t1 (|ξ|)
Lemma 3.16: If the constant C is sufficiently small, then
Sr (t, |ξ|) maxn |ξ| exp −C|ξ|
r
ξ∈R
2−2σ
0
t
dτ
b(τ )
(1 + t)−
r(1+δ)
2−2σ
for r ≥ 0.
Proof: Here we follow ideas from Wirth (2007). To estimate the term Sr (t, |ξ|) it is
important, that the first partial derivative ∂|ξ| Sr (t, |ξ|) is negative for |ξ| ≤ εr . This
follows from
1 2σ
r
1
2−2σ
+
|ξ| b(t1 (|ξ|)) − C|ξ|
∂|ξ| Sr (t, |ξ|) < Sr (t, |ξ|)
d|ξ| t1 (|ξ|)
|ξ|
4
b(t1 (|ξ|))
and from
|ξ|2σ b(t1 (|ξ|)) ∼ |ξ|, |ξ|2−2σ
1−2σ
1
∼ |ξ|, d|ξ| t1 (|ξ|) ∼ −|ξ|−1− δ .
b(t1 (|ξ|))
Here we use that C is sufficiently small, that δ ∈ (0, 1 − 2σ) and that to a given
r ≥ 0 it is sufficient to study small frequencies with |ξ| ≤ εr . For |ξ| ≥ εr we have
an exponential type decay from the hyperbolic zone. Let us now fix t > 0, then the
˜ For t = t1 (|ξ|)
˜ the
˜ satisfying t = t1 (|ξ|).
above term takes its maximum for the |ξ|
second integral vanishes in Sr (t, |ξ|). Consequently,
t1 (|ξ̃|)
dτ
˜ |ξ|)
˜ = |ξ|
˜ r exp −C|ξ|
˜ 2−2σ
Sr (t, |ξ|) ≤ Sr (t1 (|ξ|),
b(τ )
0
t
r(1+δ)
dτ
(1 + t)− 2−2σ .
≤ maxn |ξ|r exp −C|ξ|2−2σ
ξ∈R
0 b(τ )
In this way the lemma is proved.
Corollary 3.13 and Lemma 3.16 yield the following result.
48
X. Lu and M. Reissig
Corollary 3.17: The following estimates hold for all t ∈ [t1 (|ξ|), ∞):
1+δ
1+δ
2σ
|ξ||β| |v(t, ξ)| (1 + t)−|β| 2−2σ |v(0, ξ)| + (1 + t)− |β|− 1−δ 2−2σ |Dt v(0, ξ)|
1−δ
2σ
+ (1 + t)− |β|+1− 1−δ 2σ |v(0, ξ)|
+ (1 + t)−(|β|−1)
1−δ
2σ
|Dt v(0, ξ)| for |β| ≥ 1,
1+δ
2σ
|ξ||β| |Dt v(t, ξ)| (1 + t)
|v(0, ξ)| + (1 + t)− |β|+1− 1−δ 2−2σ |Dt v(0, ξ)|
1−δ
2σ
+ (1 + t)− |β|+2− 1−δ 2σ |v(0, ξ)|
1+δ
−(|β|+1) 2−2σ
+ (1 + t)−|β|
1−δ
2σ
|Dt v(0, ξ)|
for |β| ≥ 0.
From the Corollaries 3.14, 3.15 and 3.17 we obtain the statements of Theorem 3.1.
3.1.2 The proof to Theorem 3.2
In this case we have only the hyperbolic zone Zhyp (ε), the reduced zone Zred (ε)
and the pseudo-differential zone Zpd (ε, N ). We will explain later why the elliptic
zone disappears. If δ ∈ [1 − 2σ, 1] we are able to extend the estimate from Zhyp (ε)
to Zred (ε). For this reason, after repeating the approach for the hyperbolic zone
from Section 3.1.1, we obtain for t ≥ t1 (|ξ|), where t1 = t1 (|ξ|) is the separating line
between the pseudo-differential zone and the reduced zone, the a-priori estimates
|ξ|2σ t
b(τ )dτ
|ξ||β| |v(t, ξ)| exp −
4
t1 (|ξ|)
|β|
× |ξ| |v(t1 (|ξ|), ξ)| + |ξ||β|−1 |Dt v(t1 (|ξ|), ξ)| for |β| ≥ 1,
|ξ|2σ t
b(τ )dτ
|ξ||β| |Dt v(t, ξ)| exp −
4
t1 (|ξ|)
|β|+1
× |ξ|
|v(t1 (|ξ|), ξ)| + |ξ||β| |Dt v(t1 (|ξ|), ξ)| for |β| ≥ 0.
Now let us discuss the behaviour of solutions in the pseudo-differential zone
Zpd (ε, N ) = {(t, ξ) : b(t)|ξ|2σ−1 ≥ 2(1 + ε) and Λ(t)|ξ|2σ ≤ N }.
1. Case:
δ ∈ [1 − 2σ, 1), σ ∈ (0, 1/2)
The separating line between the reduced zone and the elliptic region Rell (ε) =
{(t, ξ) : (1 + t)−δ |ξ|2σ−1 ≥ µ2 (1 + ε)} is implicitly defined by
(1 + t1 (|ξ|))−δ |ξ|2σ−1 =
2
1
(1 + ε), hence, t1 (|ξ|) ∼
1−2σ .
µ
|ξ| δ
The separating line t0 = t0 (|ξ|) between the pseudo-differential zone and the
exterior of pseudo-differential zone is defined by
(1 + t0 (|ξ|))1−δ |ξ|2σ =
2(1 − δ)
1
N, hence, t0 (|ξ|) ∼
2σ .
µ
|ξ| 1−δ
If t ≥ C 1 and δ ∈ [1 − 2σ, 1), then Zpd (N ) ⊃ Rell (ε).
49
Rates of decay for structural damped models
2. Case:
δ = 1, σ ∈ (0, 1/2)
For the first separating line we have t1 (|ξ|) ∼
we have
1
|ξ|1−2σ ,
for the second separating line
N
t0 (|ξ|) ∼ e µ|ξ|2σ .
If t ≥ C 1 and δ = 1, then Zpd (N ) ⊃ Rell (ε).
For t ∈ [0, t1 (|ξ|)] we obtain the same estimates as in the reduced zone and in the
hyperbolic zone, or in other words, we are able to continue the a-priori estimates
from the reduced and hyperbolic zone into the pseudo-differential zone. The gluing
procedure is trivial in this case. Finally, from
1 t
|β|
2σ
b(τ )|ξ| dτ
|ξ| |v(t, ξ)| exp −
2 0
× |ξ||β| |v(0, ξ)| + |ξ||β|−1 |Dt v(0, ξ)| for |β| ≥ 1,
1 t
b(τ )|ξ|2σ dτ
|ξ||β| |Dt v(t, ξ)| exp −
2 0
|β|+1
× |ξ|
|v(0, ξ)| + |ξ||β| |Dt v(0, ξ)| for |β| ≥ 0
we obtain immediately the estimates from Theorem 3.2.
3.2 The case σ =
1
2
The case σ = 1/2 and δ ∈ (0, 1] represents a special case. The separating line t2
between the hyperbolic zone and the reduced zone is defined by b(t2 ) = 2(1 − ε).
From Proposition 1.2 we conclude that it is sufficient to study the behaviour of
solutions in the hyperbolic zone.
Theorem 3.18: The solution u = u(t, x) to
1
utt − ∆u + b(t)(−∆) 2 ut = 0, u(0, x) = u0 (x), ut (0, x) = u1 (x)
with b(t) = µ(1 + t)−δ , µ > 0 and δ ∈ (0, 1] satisfies the following estimates:
δ ∈ (0, 1):
∇β u(t, ·)2L2 (1 + t)−2|β|(1−δ) u0 2H |β|
+ (1 + t)−2(|β|−1)(1−δ) u1 2H |β|−1 for |β| ≥ 1,
∇β ut (t, ·)2L2 (1 + t)−2(|β|+1)(1−δ) u0 2H |β|+1
+ (1 + t)−2|β|(1−δ) u1 2H |β|
for |β| ≥ 0.
δ = 1:
∇β u(t, ·)2L2 (log(e + t))−2|β| u0 2H |β|
+ (log(e + t))−2(|β|−1) u1 2H |β|−1 for |β| ≥ 1,
∇β ut (t, ·)2L2 (log(e + t))−2(|β|+1) u0 2H |β|+1
+ (log(e + t))−2|β| u1 2H |β|
for |β| ≥ 0.
50
X. Lu and M. Reissig
Proof: For t t2 the statement of Proposition 1.2 implies the above ones.
For t t2 the statements follow from Lemma 3.3.
Remark 3.5: If we set formally σ = 1/2 in the estimates from Theorem 3.2, then
we get the estimates from Theorem 3.18.
4 Time dependent strictly decreasing dissipation – σ∈ (1/2, 1]
In this section we study for σ ∈ (1/2, 1] the Cauchy problem
utt − u + b(t)(−)σ ut = 0, u(0, x) = u0 (x), ut (0, x) = u1 (x)
(4.1)
with b(t) = µ(1 + t)−δ , µ > 0 and δ ∈ (0, 1].
Theorem 4.1: The solution u = u(t, x) to (4.1) satisfies the following estimates:
δ ∈ (0, 1):
∇β u(t, ·)2L2 (1 + t)−
|β|(1−δ)
σ
+ (1 + t)−
∇β ut (t, ·)2L2 (1 + t)−
u0 2H |β|
(|β|−1)(1−δ)
σ
(|β|+1)(1−δ)
σ
|β|(1−δ)
−
σ
+ (1 + t)
u1 2H |β|−1 for |β| ≥ 1,
u0 2H |β|+1
u1 2H |β|
for |β| ≥ 0.
δ = 1:
∇β u(t, ·)2L2 (log(e + t))−
|β|
σ
u0 2H |β|
+ (log(e + t))−
∇β ut (t, ·)2L2 (log(e + t))
|β|−1
σ
|β|+1
− σ
+ (log(e + t))−
u1 2H |β|−1 for |β| ≥ 1,
u0 2H |β|+1
|β|
σ
u1 2H |β|
for |β| ≥ 0.
Proof: We will briefly sketch the changes to the proofs to Theorems 3.1 and 3.2.
We divide the extended phase space into the same zones as in Section 3.1.1.1.
But there is a big difference in the geometry of these zones. If we consider the
most interesting part {(t, ξ) : |ξ| ≤ p0 }, where p0 is sufficiently small, then this part
is completely contained in the hyperbolic zone, in other words, if a point (t, ξ)
belongs to one of the other three zones we have |ξ| ≥ p0 with a suitable positive p0 .
Moreover, the pseudo-differential zone degenerates to a compact set of the extended
phase space. For this reason we have only to study the behaviour of solutions in
the hyperbolic zone, the reduced zone and the elliptic zone. All these observations
hint to the following:
•
the decay is determined by the behaviour of solutions in the hyperbolic zone,
•
in the other zones we expect a faster decay, in the case δ ∈ (0, 1) even an
exponential type decay.
The separating curves t1 = t1 (|ξ|) and t2 = t2 (|ξ|) are defined as in Section 3.1.1.1.
Rates of decay for structural damped models
51
Considerations in Zhyp (ε)
Here we can follow the proof to Lemma 3.3 and obtain the following result:
Lemma 4.2: The following estimates hold for all t ∈ [t2 (|ξ|), ∞):
|ξ|2σ t
|β|
|ξ| |v(t, ξ)| exp −
b(τ )dτ
2
t2 (|ξ|)
× (|ξ||β| |v(t2 (|ξ|), ξ)| + |ξ||β|−1 |Dt v(t2 (|ξ|), ξ)|), |β| 1,
|ξ|2σ t
|ξ||β| |Dt v(t, ξ)| exp −
b(τ )dτ
2
t2 (|ξ|)
× (|ξ||β|+1 |v(t2 (|ξ|), ξ)| + |ξ||β| |Dt v(t2 (|ξ|), ξ)|), |β| 0.
Considerations in Zred (ε)
Here we can repeat the approach from Section 3.1.1.3, the difference is that
δ
now g(t, ξ) ∼ (1 + t)a , a = 2σ−1
. Then we obtain a corresponding statement to
Lemma 3.4 which after gluing with the statement from Lemma 4.2 reads as follows
(cf. with Corollary 3.12):
Lemma 4.3: The following estimates hold for all t ∈ [t1 (|ξ|), ∞):
|ξ|2σ t
|ξ||β| |v(t, ξ)| exp −
b(τ )dτ
4
t1 (|ξ|)
× (|ξ||β| |v(t1 (|ξ|), ξ)| + |ξ||β|−1 |Dt v(t1 (|ξ|), ξ)|), |β| 1,
|ξ|2σ t
|β|
|ξ| |Dt v(t, ξ)| exp −
b(τ )dτ
4
t1 (|ξ|)
× (|ξ||β|+1 |v(t1 (|ξ|), ξ)| + |ξ||β| |Dt v(t1 (|ξ|), ξ)|), |β| 0.
Considerations in Zell (ε, N )
Here we have to follow the proofs of Lemmas 3.7–3.9.
1. Case:
δ ∈ (0, 1)
Here we can follow the proofs. Taking account that the last term in the estimates
from Lemma 3.9 produces a better decay than the other terms we conclude
immediately.
Lemma 4.4: If δ ∈ (0, 1), then we have the following estimates for t ∈ [0, t1 (|ξ|)]:
t 2−2σ |ξ|
dτ (|ξ||β| |v(0, ξ)|
|ξ||β| |v(t, ξ)| exp −C
b(τ )
0
+ |ξ||β|−1 |Dt v(0, ξ)|), |β| 1,
t 2−2σ |ξ|
dτ (|ξ||β|+1 |v(0, ξ)|
|ξ||β| |Dt v(t, ξ)| exp −C
b(τ
)
0
+ |ξ||β| |Dt v(0, ξ)|), |β| 0.
52
X. Lu and M. Reissig
2. Case: δ = 1
Here we present the modifications to the case δ ∈ (0, 1) to obtain corresponding
estimates to Lemmas 3.7 and 3.8. In this case it is meaningless to define symbol
classes as in the considerations for δ ∈ (0, 1), since with this definition we cannot
expect any hierarchy of symbols in the second step of diagonalisation.
Let us define with a sufficiently large K the following elliptic zone, which is of
great significance for our discussion: Zell (K, N ) = {(t, ξ) : Λ(t)|ξ| ≥ N, |ξ| ≥ K}.
As in the proof to Lemma 3.7 we apply the first step of diagonalisation. The
antidiagonal part of the matrix B can be estimated by antidiag B ≤ C(µ)
1+t . Define
as usual N (1) , then we get in the case δ = 1 for all (t, ξ) ∈ Zell (K, N ) the estimate
N (1) (t, ξ) ≤
C(µ)
C(µ)
≤ 2σ .
2σ
|ξ|
K
Obviously, N1 is invertible if K is large enough. After the second step of
diagonalisation, we obtain for all (t, ξ) ∈ Zell (K, N ) the estimate
R1 (t, ξ) ≤
C(µ)
.
|ξ|2σ (1 + t)
Notice the fact that in Zell (K, N ) it holds
|b (t)|
≤ C(K, µ),
b2 (t)|ξ|2σ
where C(K, µ) is small if K is large, we conclude immediately a corresponding
statement to Lemma 3.8.
Lemma 4.5: We have in the elliptic zone Zell (K, N ) the following estimate:
E1 (t, s, ξ) exp −C(N )
s
t
|ξ|2−2σ
dτ ,
b(τ )
with a positive constant C(N ) which is independent of (s, ξ), (t, ξ) ∈ Zell (K, N )
for K ≥ K0 and K0 is sufficiently large.
Then we can follow the considerations after the proof to Lemma 3.8. We get that
the matrix H is uniformly bounded for (s, ξ), (t, ξ) ∈ Zell (K, N ). Taking account of
R1 (t, ξ) ≤
C(µ)
,
+ t)
|ξ|2σ (1
the matrix Q = Q(t, s, ξ) can be estimated by
C(µ)
Q(t, s, ξ) ≤ exp
|ξ|2σ
s
t
1
dτ .
1+τ
Rates of decay for structural damped models
53
Consequently, we conclude for E2 = E2 (t, s, ξ) the estimate
t
t
1
dτ
(1 + τ )dτ + C(µ)|ξ|−2σ
E2 (t, s, ξ) exp −C(µ)|ξ|2−2σ
s
s 1+τ
t
1
exp −C(µ)|ξ|2−2σ
dτ
b(τ
)
s
for all (s, ξ), (t, ξ) ∈ Zell (K, N ), where the constant C(µ) is independent of the
choice of K if K ≥ K0 and K0 is chosen sufficiently large. Then we can follow the
other considerations of the proofs to Lemmas 3.7 and 3.8 and conclude the final
estimates in the elliptic zone.
Lemma 4.6: If δ = 1, then we have the following estimates for t ∈ [0, t1 (|ξ|)]:
t 2−2σ |ξ|
dτ (|ξ||β| |v(0, ξ)|
|ξ||β| |v(t, ξ)| exp −C(µ)
b(τ )
0
+ |ξ||β|−1 |Dt v(0, ξ)|), |β| 1,
t 2−2σ |ξ|
dτ (|ξ||β|+1 |v(0, ξ)|
|ξ||β| |Dt v(t, ξ)| exp −C(µ)
b(τ
)
0
+ |ξ||β| |Dt v(0, ξ)|), |β| 0.
Gluing procedure
Lemma 4.7: If δ ∈ (0, 1), then to a given positive constant C we can find a
sufficiently small positive constant C1 such that
t1 (|ξ|)
t
dτ
2−2σ
2σ
exp −C|ξ|
b(τ )dτ
exp −C1 |ξ|
b(τ )
0
t1 (|ξ|)
t
1
exp − C1 |ξ|2σ
b(τ )dτ .
2
0
Proof: This follows by calculating the integrals and taking account of |ξ|2σ
(1 + t1 (|ξ|))1−δ ∼ |ξ|2−2σ (1 + t1 (|ξ|))1−δ .
Lemma 4.8: If δ = 1, then to a given positive constant K we can find a positive
constant CK such that for all {|ξ| ≥ K} it holds
t1 (|ξ|)
t
dτ
2−2σ
2σ
exp −C1 |ξ|
b(τ )dτ
exp −C(µ)|ξ|
b(τ )
0
t1 (|ξ|)
≤ CK (1 + t)−
C1
2
K 2σ
.
Proof: This follows by estimating
t1 (|ξ|)
t
dτ
b(τ )dτ
exp −C1 |ξ|2σ
exp −C|ξ|2−2σ
b(τ )
0
t1 (|ξ|)
t1 (|ξ|)
t
dτ
2−2σ
2σ
≤ exp −CK
exp −C1 K
b(τ )dτ
b(τ )
0
t1 (|ξ|)
and calculating the integrals on the right-hand side.
54
X. Lu and M. Reissig
Energy estimates
If δ ∈ (0, 1), then Lemma 4.7 tells us, that for large frequencies we have an
exponential type decay. For this reason we can restrict ourselves to small
frequencies and apply Lemma 4.3. For small frequencies we are able to continue
the integrals from (t1 (|ξ|), t) to (0, t). From Lemma 4.3 we obtain the statements
of Theorem 4.1.
If δ = 1, then Lemma 4.8 tells us, that for large frequencies we have an potential
decay with a rate larger than a given positive constant. For this reason we can
restrict ourselves to small frequencies and apply Lemma 4.3. For small frequencies
we are able to continue the integrals from (t1 (|ξ|), t) to (0, t). From Lemma 4.3 we
obtain the statements of Theorem 4.1.
Remark 4.1: If we set formally σ = 1/2 in the estimates from Theorem 4.1, then
we get the estimates from Theorem 3.18.
5 Concluding remarks and open problems
Remark 5.1: To exclude special effects coming from the structure of b = b(t) we
considered in this paper only the model case b(t) = µ(1 + t)−δ , µ > 0, δ ∈ (0, 1]
in (1.1). In forthcoming papers we will study model cases with a strictly increasing
time dependent coefficient and later structural dissipations with a general timedependent coefficient b.
Remark 5.2: In this paper we were interested in energy estimates, that is, in
L2 − L2 estimates for solutions to (1.1) and answered the question for the parabolic
effect. In a forthcoming project we are interested in Lp − Lq , in particular, in
L1 − L∞ estimates.
Remark 5.3: In a forthcoming project we will extend our studies to damped
models of the type
utt + (−)γ u + b(t)(−)σ ut = 0, σ, γ > 0,
u(0, x) = u0 (x), ut (0, x) = u1 (x).
For the special case σ = γ/2 we refer to Fang et al. (2009), where interesting
motivations to study this case are presented.
Acknowledgements
The discussions to this paper began during a research stay of the first author from
October 2006 to September 2007 at the TU Bergakademie Freiberg. The first author
would like to thank the Department of Mathematics and Computer Science for the
warm hospitality and assistance. Both authors thank Professor Fang Daoyuan for
the useful discussions. The first author is partly supported by NSFC 10871175.
Rates of decay for structural damped models
55
References
Batty, Ch.J.K., Chill, R. and Srivastava, S. (2008) ‘Maximal regularity for second order
non-autonomous Cauchy problems’, Stud. Math., Vol. 189, No. 3, pp.205–223.
Chen, G. and Russell, D.L. (1982) ‘A mathematical model for linear elastic systems with
structural damping’, Quarterly Journal of Applied Mathematics, Vol. 39, pp.433–454.
Chen, Sh. and Trigiani, R. (1989) ‘Proof of extensions of two conjectures on structural
damping for elastic systems’, Pacific Journal of Mathematics, Vol. 136, No. 1,
pp.15–55.
Chen, Sh. and Trigiani, R. (1990) ‘Gevrey class semigroups arising from elastic systems with
gentle dissipation: the case 0 < α < 12 ’, Proceedings of AMS, Providence, Vol. 110,
No. 2, pp.401–415.
Fang, D., Lu, X. and Reissig, M. (2009) ‘High-order energy decay for structural damped
systems in the electromagnetical field’, 11 A4, accepted for publication in Chinese
Annals of Mathematics, Series B.
Guenther, R.B. and Lee, J.W. (1996) Partial Differential Equations of Mathematical
Physics and Integral Equations, Dover Publications, New York.
Lions, J.L. (1988) Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués,
Masson, Paris.
Matsumura, A. (1976) ‘On the asymptotic behaviour of solutions of semi-linear wave
equations’, Publ. RIMS Kyoto Univ., Vol. 12, pp.169–189.
Mugnolo, D. (2008) ‘A variational approach to strongly damped wave models’,
in Amann, H., Arendt, W., Hieber, M., Neubrander, F., Nicaise, S. and von Below, J.
(Eds.): Functional Analysis and Evolution Equations: The Günter Lumer Volume,
Birkhäuser, pp.503–514.
Shibata, Y. (2000) ‘On the rate of decay of solutions to linear viscoelastic equation’,
Math. Meth. Appl. Sci., Vol. 23, pp.203–226.
Wirth, J. (2006) ‘Wave equations with time-dependent dissipation I. Non-effective
dissipation’, J. Differential Equations, Vol. 222, pp.487–514.
Wirth, J. (2007) ‘Wave equations with time-dependent dissipation II. Effective dissipation’,
J. Differential Equations, Vol. 232, pp.74–103.