Universität Konstanz
Wave equations with time-dependent coefficients
Johannes Emmerling
Konstanzer Schriften in Mathematik und Informatik Nr. 225, Februar 2007
ISSN 1430-3558
© Fachbereich Mathematik und Statistik
© Fachbereich Informatik und Informationswissenschaft Universität Konstanz
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URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-23501
JOHANNES EMMERLING
Abstract. We study the damped wave equation with time-dependent coef- ficientsutt(t, x)−a(t)2∆u(t, x)−b
′
b(t)ut(t, x) = 0 inRn and prove energy estimates for a new class of coefficients.
1. Introduction
In this paper we prove energy estimates for the damped wave equation with time- dependent coefficients, i.e.
(1.1) utt(t, x)−a(t)2∆u(t, x)−bb′(t)ut(t, x) = 0
fort∈[0,∞) andx∈Rn, wherea∈C([0,∞),R) andb∈C1([0,∞),R) are strictly positive functions. This is a simple transformation of
utt(t, x)−a(t)2∆u(t, x) +β(t)ut(t, x) = 0 with b(t) := exp(−Rt
0β(τ)dτ).
If uis a solution of (1.1) such thatut(t,·)∈L2(Rn) and∇u(t,·)∈ L2(Rn)n
for every t≥0, then we call
(1.2) Eu(t) :=
Z
Rn
ut(t, x)2+a(t)2|∇u(t, x)|2dx
theenergy ofuat timet≥0. Here,| · |denotes the Euclidean norm in Rn. We deal with solutionsuof (1.1) such that
u∈C0 [0,∞), H2(Rn)
∩C1 [0,∞), H1(Rn)
∩C2 [0,∞), L2(Rn) . These ushall be calledadmissible solutions of (1.1).
Supposeaand−bb′ are constant and positive, then Eu(t)≤Eu(0)
for each admissible solutionuof (1.1). As can be seen from the Fourier representa- tion, the estimate can only be improved by restrictions on the solutions. Whereas in thisL2-L2-estimate the “damping term”−bb′ seems to have no effect, its influence can be clearly seen in the Lp-Lq-estimate:
If bb′ = 0, then for every 2 ≤q ≤ ∞, 1p + 1q = 1, Np > n 1− 2q
there exists a constant C >0 such that
ut(t,·),∇u(t,·)
Lq ≤C(1 +t)−n−12 (1−2q)
ut(0,·),∇u(0,·) HNp,p
for every t ≥0, see [7]. But if −bb′ = 1, then for every 2 ≤q ≤ ∞, p1+ 1q = 1, Np > n 1−2q
there exists a constantC >0 such that
ut(t,·),∇u(t,·)
Lq ≤C(1 +t)−n2(1−2q)
ut(0,·),∇u(0,·) HNp,p
for every t ≥ 0, c.f. [6]. Here, the damping term improves the rate of decay, see also [2]. First results for wave equations with time-dependent damping term have
1
been proved by Matsumura [3] and Uesaka [13]: For every µ ≥ 0 there exists a constant C >0 such that for each admissible solutionuof
utt(t, x) + ∆u(t, x) +µ(1 +t)−1ut(t, x) = 0,
whose initial valuesu(0,·) andut(0,·) have compact support, the following inequal- ity holds:
Eu(t)≤C(1 +t)−min{µ,2} Eu(0) +ku(0,·)k2L2
.
Note that both energy and damping term decay with time and that the estimate needs more than the initial energy to give an upper bound. Recent works on the wave equation with time-dependent damping term are due to Reissig ([9],[10], [11], [12]) and Wirth ([12],[15]), for a good survey see [9]. In [15], the result above with µ ∈ [0,1) is generalized to a broader class of coefficients. This result will be discussed in section 5, subsequent to Corollary 5.2. In that work, estimates for larger, so called “effective” damping terms can be found, which we will not consider here.
A time-dependent propagation speedacan cause many difficulties. This is shown in [11] by means of the equation
utt(t, x)−(2 + sint)∆u(t, x) = 0.
For an admissible solutionuof this equation it follows immediately that Eu(t)≤CectEu(0)
with constants c, C > 0, see for example the following Lemma 2.1. Reissig and Yagdjian show that this energy estimate cannot be substantially improved, even if Lp-Lq estimates are considered. Thus oscillating coefficients have a deteriorat- ing effect on energy estimates. In [10], this effect is examined more closely for propagation speeds of the form
a(t)2= 2 + sin (log(t+ 30))α
where α >0. Compare example 4 in section 6, where a function ais given, which just does not damage the energy estimate. Equations with two oscillating coeffi- cients can be dealt with, too, see [8]. There, the equation
utt(t, x)−ϕ(t)2ω(t)2∆u(t, x) +αϕ(t)ω(t)ut(t, x) = 0
is considered, where αis a positive constant andϕ,ωare smooth, strictly positive functions,ϕincreasing andω oscillating.
In Theorem (5.1) of this paper we extend the theorem from [15] mentioned above, showing that
Eu(t)≤C a(t)b(t) Eu(0) +ku(0,·)k2L2
under considerably weaker assumptions on a and b. In particular, the case µ ∈ [1,∞) is included and oscillating damping terms are possible, even with decaying energy, see Example 2 in section 6. However, Theorem (5.1) does not cover a∈ L1([0,∞)). This is different for Theorem 5.3, which gives energy estimates for those functions under a smallness condition for |a−b|. Here, the estimate
Eu(t)≤C a(t)b(t)Eu(0)
is possible, where only the initial energy is needed. Again, oscillating coefficients are possible, see Example 5 in section 6.
The methods developed for the proof of these estimates are also applicable to the viscoelastic equation
utt(t, x)−b(t)∆ut(t, x)−a(t)∆u(t, x) = 0, t∈[0,∞), x∈Rn, c.f. [1]. This will be subject of a forthcoming paper.
Our paper is organized as follows: In Section 2 we will prove elementary estimates for the energy which will serve both as motivation and as benchmark for the final estimates. In Section 3 we will derive approximate solutions for equation (1.1).
These approximate solutions will be turned into Fourier space estimates in Sec- tion 4 and glued together to form energy estimates in Section 5. We conclude with examples in Section 6.
2. Preliminaries
Now we turn to estimates which can be obtained directly form equation (1.1).
Lemma 2.1.Let a, b∈C1([0,∞),R)be strictly positive functions.
(a) Fort≥0 letM(t) := maxa′
a(t),bb′(t) andm(t) := mina′
a(t),bb′(t) . Then (2.1) exp
2
Z t 0
m(τ)dτ
Eu(0)≤Eu(t)≤exp
2 Z t
0
M(τ)dτ
Eu(0) for each admissible solution uof (1.1) and everyt≥0.
(b) Suppose aa′ − bb′ is absolutely integrable on[0,∞), then there exists a constant C >0 such that
(2.2) Eu(t)≤Ca(t)b(t)Eu(0).
for each admissible solution uof (1.1) and everyt≥0.
Proof. Supposeuis an admissible solution of (1.1). Partial integration and substi- tuting the differential equation leads to
E′u(t) = 2 Z
Rn b′
b(t)ut(t, x)2+aa′(t)a(t)2|∇u(t, x)|2dx.
(a) Then we have
2m(t)Eu(t)≤Eu′(t)≤2M(t)Eu(t),
for every t≥0, and with a suitable differential inequality, e. g. [14], Theorem XI, we obtain (a).
(b) We write
E′u(t)≤
a′
a(t) +bb′(t)
Eu(t) +
a′
a(t)−bb′(t) Eu(t).
Now we can apply the same differential inequality or Gronwall’s Lemma to obtain Eu(t)≤exp
Z t 0
a′
a(τ) +bb′(τ) +
a′
a(τ)−bb′(τ) dτ
Eu(0)
≤Ca(t)b(t)Eu(0).
From the estimates of Lemma 2.1 we can draw several conclusions:
First, iff ∈C1([0,∞),R) is a strictly positive function, then there always exists an equation of the form (1.1), where the energy of each admissible solution behaves like f. In particular, exponential or polynomial stability can be obtained by choosing suitable coefficients.
Suppose aa′(t)≤ bb′(t) or bb′(t)≤ aa′(t) for everyt≥0, then by Lemma 2.1 (a) the inequalities
a(t)2Eu(0)≤Eu(t)≤b(t)2Eu(0) or b(t)2Eu(0)≤Eu(t)≤a(t)2Eu(0)
hold for each admissible solutionuof (1.1) and for everyt≥0.
If aa′ − bb′ is monotone and if there are contantsc, C >0 such that ca(t)≤b(t)≤Ca(t)
for every t≥0, then there exists a constantC′>0 such that Eu(t)≤C′a(t)b(t)Eu(0)
for each admissible solution u of (1.1) and for every t ≥ 0. Suppose a = b or
a′
a − bb′ = 0, thenM =mand we can determine the energy exactly:
Eu(t) = a(t)b(t) a(0)b(0)Eu(0).
Hence we can understand Lemma 2.1 as a stability result: As long as aand bare close together, for example if aa′−bb′ is absolutely integrable or aa′ −bb′ is monotone and there exist constantsc, C >0 such that
ca(t)≤b(t)≤Ca(t),
then the energy estimate of the case a=b is preserved. The question is how far this region of stability can be expanded. At least a little bit, as the example of Matsumura shows: Here we have a = 1 and b(t) = (1 +t)−µ, hence there is no constant c >0 such thatca(t)≤b(t) for everyt≥0. And since
a′
a(t)−bb′(t) =µ(1 +t)−1
the function aa′ − bb′ is not absolutely integrable. Nevertheless, the energy of each admissible solution decays likea·b.
Now we can begin with the proof of energy estimates. To this, end we transform the equation: Suppose uis an admissible solution of equation (1.1) and ˆu(t,·) is the Fourier transform ofu(t,·) fort≥0. Then ˆu(·, ξ)∈C2([0,∞]),C) satisfies for every ξ∈Rn the ordinary differential equation
(2.3) uˆtt(t, ξ) +a(t)2|ξ|2u(t, ξ)ˆ −bb′(t)ˆut(t, ξ) = 0.
In this equation, only |ξ| appears as a parameter, thus theξ-dependence of ˆu(t, ξ) is only due to the initial values ˆu(0, ξ) and ˆut(0, ξ). Therefore, we determine a fundamental system of the equation
(2.4) v′′(t, λ) +a(t)2λ2v(t, λ)−bb′(t)v′(t, λ) = 0 for every λ≥0.
3. Approximate Solutions
In order to get good energy estimates, we should know the solutions of (2.4) as com- pletely as possible. But even homogeneous linear differential equations of second order like equation (2.4) cannot be solved explicitly in general. Therefore we use
approximate solutions of the equation, and these can be derived from the following model equations:
y′′(t, λ) +λ2y(t, λ) =f(t, λ)y(t, λ), (3.1)
y′′(t, λ) =f(t, λ)y(t, λ), (3.2)
whereλ >0 andf(·, λ) is a continuous function. Iff = 0, then the equations have explicit solutions, which can serve as approximations of the general case. But when we replace the true solutions by approximations, we make an error, and this error has to be estimated.
For equations of type (3.1) this has been done by F. W. J. Olver, see [5]. The ap- proximate solutions covered in the next theorem are also known as WKB-solutions.
Theorem 3.1 (Olver).Suppose α ≤ γ ≤ β, λ > 0 and f(·, λ) ∈ C([α, β],C).
Then the differential equation (3.1) has a fundamental system y+(·, λ), y−(·, λ)∈ C2([α, β],C)such that
y±(t, λ) =e±iλt 1 +r±(t, λ) , where
r±(t, λ) ,
iλ1r′±(t, λ)±r±(t, λ) ≤exp
F(t, λ) λ
−1, and
(3.3) F(t, λ) :=
Z t
γ |f(τ, λ)|dτ .
Moreover, the interval [α, β] and the value of γ may be infinite provided the inte- gral (3.3) exists.
Now we prove a lemma for equations of type (3.2). To simplify notation, we write R := R∪ {−∞,∞} and we agree that intervals [α, β] with α, β ∈ R include nei- ther ∞ nor −∞. For functions f: [α,∞] → C we mean by f(∞) = c that the limit limt→∞f(t) exists and is equal to c. In the same way we define f(−∞) for f: (−∞, β]→C.
In order to get error bounds, we use the following variation-of-constants formula:
Lemma 3.2 (Variation of constants).Let α, β ∈ R, γ ∈ [α, β] and f, g, h ∈ C([α, β],C). Then there exists a functionx∈C2([α, β],C)solving
x′′+f x′+gx=h
with initial values x(γ) = x′(γ) = 0. If x1, x2 ∈ C2([α, β],C) is a fundamental system of solutions of the corresponding homogeneous equation, then
x(t) = Z t
γ
x1(τ)x2(t)−x1(t)x2(τ) x1(τ)x′2(τ)−x′1(τ)x2(τ)h(τ)dτ
for everyt∈[α, β]. The interval[α, β]and the value ofγ may be infinite, provided that the associated integral exists.
The next step is a Lemma of Gronwall type, which will turn the representation of the previous Lemma into error bounds. It allows for inequalities of the form
x(t)≤ Z t
0
k(t, τ) x(τ) +h(τ) dτ,
which contain an integral kernel (t, τ)7→k(t, τ) instead of a functionτ7→g(τ).
Lemma 3.3. Let α, β ∈ R, let k ∈ C [α, β]2,R
be differentiable with respect to the first variable and let x, h ∈ C [α, β],R
. Suppose k(t, τ), ∂1k(t, τ) ≥ 0 for every α ≤ τ ≤ t ≤ β and x(t), h(t) ≥ 0 for every t ∈ [α, β]. Assume also that K(t) :=Rt
αk(t, τ)dτ converges. If x(t)≤
Z t α
k(t, τ) x(τ) +h(τ) dτ for every t∈[α, β], then
x(t)≤ exp(K(t))−1 sup
τ∈[α,t]
h(τ) for every t∈[α, β].
Proof. Let [γ, δ] be a finite subinterval of [α, β]. Then P:C([γ, δ],R)−→C([γ, δ],R) defined by (P y)(t) =
Z t γ
k(t, τ)y(τ)dτ is a continuous, linear, positive operator with spectral radius <1 (see [16], p. 38).
Since
x(t)≤(P h)(t) + (P x)(t), the Abstract Gronwall Lemma ([16], p. 81) implies
x(t)≤ (id−P)−1P h (t) =
∞
X
n=0
(Pn+1h)(t).
UsingH(t) = maxτ∈[γ,t]h(τ), we obtain (P h)(t) =
Z t γ
k(t, τ)h(τ)dτ ≤ Z t
γ
k(t, τ)H(t)dτ ≤K(t)H(t)
(P2h)(t)≤ Z t
γ
k(t, τ)K(τ)H(τ)dτ ≤ Z t
γ
k(t, τ)K(τ)dτ
H(t).
We estimate the integral
I(t) :=
Z t γ
k(t, τ)K(τ)dτ by differentiation:
I′(t) =k(t, t)K(t) + Z t
γ
∂1k(t, τ)K(τ)dτ
≤
k(t, t) + Z t
γ
∂1k(t, τ)dτ
K(t) =K′(t)K(t).
This estimate is possible because ∂1k(t, τ) is always nonnegative and K is mono- tonically increasing. We conclude that
(P2h)(t)≤ Z t
γ
K′(τ)K(τ)dτ
H(t) = 12K(t)2H(t).
By induction, we get
(Pnh)(t)≤ n!1K(t)nH(t), hence
x(t)≤ exp(K(t))−1 H(t)
for every t∈[γ, δ].Sinceγ, δ∈[α, β] were arbitrary, we can extend this inequality
to everyt∈[α, β].
Now we can state the approximation lemma for equations of type (3.2).
Lemma 3.4. Let α ∈ R, β ∈ R, and f ∈ C([α, β),C). Then the differential equation
y′′(t) =f(t)y(t)
has a fundamental system y1, y2∈C2([α, β),C)such that y1(t) = 1 +r1(t), y2(t) = (t−α) 1 +r2(t)
, where
|r1(t)|,|r2(t)| ≤exp F(t)
−1,
|r′1(t)|,|r2′(t)| ≤F′(t) exp F(t) and
F(t) :=
Z t α
(t−τ)|f(τ)|dτ.
Proof. We can deal with both solutions at the same time. For that purpose, let ϕ: [α, β] → R and suppose ϕ = 1 or ϕ(t) = t−α for everyt ∈ [α, β]. Now we determine a functionr∈C2([α, β],C) such that
y(t) :=ϕ(t) +r(t)
defines a solution of the differential equation. Such a functionrsatisfies the equa- tion
(3.4) r′′(t) =y′′(t)−ϕ′′(t) =f(t) ϕ(t) +r(t) .
By Lemma 3.2, there exists a solution r∈C2([α, β],C) of this equation such that
|r(t)| ≤ Z t
α
(t−τ)|f(τ)| ϕ(τ) +|r(τ)| dτ.
Now we can apply Lemma 3.3, and sinceϕis monotonically increasing, we get
|r(t)| ≤ϕ(t) exp F(t)
−1 . By integrating equation (3.4), we obtain
r′(t) = Z t
α
f(τ) ϕ(τ) +r(τ) dτ.
This leads to the estimate
|r′(t)| ≤ Z t
α |f(τ)| ϕ(τ) +|r(τ)| dτ ≤
Z t
α |f(τ)|ϕ(τ) exp F(τ) dτ
≤ϕ(t) exp F(t) Z t
α |f(τ)|dτ =ϕ(t) exp F(t) F′(t).
Put q(t) := ϕ(t)r(t) fort≥αandq(α) = 0. It follows that
|q(t)|,|q′(t)| ≤ exp F(t)
−1 for t≥α, hence q∈C2([α, β],C). Thusy(t) =ϕ(t) 1 +q(t)
is a solution of the original equation. The solutionsy1and y2 belonging toϕ(t) = 1 andϕ(t) =t−α satisfy y1(α) = 1, y2(α) = 0 and y1′(α) = 0, y′2(α) = 1, thus y1, y2 is even a
fundamental system.
In the following we give approximate solutions of equation (2.4), sorted by validity for increasingλ. We frequently use the abbreviations
A(t) :=
Z t 0
a(τ)dτ, B(t) :=
Z t 0
b(τ)dτ.
Ifλis small, then solutions of (2.4) behave like solutions of the equation w′′(t, λ) = bb′(t)w′(t, λ).
A fundamental system is given by the functions 1 andB, which we take as a model for the following approximate solutions.
Lemma 3.5. Letλ≥0, leta∈C0([0,∞),R),b∈C1([0,∞),R)such thata(t)≥0 and b(t)>0 for everyt∈ [0,∞). Then equation (2.4) has a fundamental system v1(·, λ), v2(·, λ)∈C2([0,∞),C)such that
v1(t, λ) = 1 +ρ1(t, λ), v2(t, λ) =B(t) 1 +ρ2(t, λ) , and
|ρ1(t, λ)|,|ρ2(t, λ)| ≤exp λ2R(t)
−1,
|ρ′1(t, λ)|,|ρ′2(t, λ)| ≤λ2R′(t) exp λ2R(t) , where
R(t) :=
Z t 0
B(t)−B(τ)
b(τ) a(τ)2dτ.
Proof. Since b is strictly positive, B is strictly increasing. Thus s = B(t) or σ=B(τ) define a transformation of variables. Suppose w(s, λ) = v(t, λ), then w satisfies the differential equation
wss(s, λ) +a(t)2λ2
b(t)2 w(s, λ) = 0.
Put
f(s, λ) :=−a(t)2λ2 b(t)2 . According to Lemma 3.4, the differential equation
wss(s, λ) =f(s, λ)w(s, λ)
has a fundamental system
w1(·, λ), w2(·, λ)∈C2 B([0,∞)),C such that
w1(s, λ) = 1 +r1(s, λ), w2(s, λ) =s 1 +r2(s, λ) , where
|r1(s, λ)|,|r2(s, λ)| ≤exp F(s, λ)
−1,
|(r1)s(s, λ)|,|(r2)s(s, λ)| ≤Fs(s, λ) exp F(s, λ) and
F(s, λ) = Z s
0
(s−σ)|f(σ, λ)|dσ.
Put R(t) =λ−2F(s, λ). It follows that R(t) =
Z s 0
(s−σ)a(τ)2 b(τ)2 dσ=
Z t 0
B(t)−B(τ)a(τ)2 b(τ) dτ,
and R′(t) = λ−2b(t)Fs(s, λ). Thus the original differential equation possesses a fundamental systemv1(·, λ),v2(·, λ)∈C2 [0,∞),C
such that v1(t, λ) = 1 +ρ1(t, λ), v2(t, λ) =B(t) 1 +ρ2(s, λ)
, where
ρ1(t, λ) =r1(s, λ), ρ2(t, λ) =r2(s, λ), ρ′1(t, λ) =b(t)(r1)s(s, λ), ρ′2(t, λ) =b(t)(r2)s(s, λ).
We obtain the inequalities
|ρ1(t, λ)|,|ρ2(t, λ)| ≤exp λ2R(t)
−1,
|ρ′1(t, λ)|,|ρ′2(t, λ)| ≤λ2R′(t) exp λ2R(t)
.
The notation R(t) = Rt 0
B(t)−B(τ)
b(τ) a(τ)2dτ will be used again in the next section.
In order to find good estimates for R(t), we use the following condition:
(R)
Letε <1 andc >0 such that b(t)
b(τ) ≤ca(t) a(τ)
1 +A(t)ε
1 +A(τ)ε
for every 0≤τ ≤t.
This condition implies R′(t) =b(t)
Z t 0
a(τ)2
b(τ) dτ =a(t) Z t
0
b(t) b(τ)
a(τ)2 a(t) dτ
≤c a(t) 1 +A(t)εZ t 0
a(τ) 1 +A(τ)−ε
dτ ≤ c
ε+ 1a(t) 1 +A(t) , thus
R(t)≤ c
2(ε+ 1) 1 +A(t)2
, bearing in mind that−ε >−1.
Approximate solutions as in the previous lemma can be found for every second order ordinary differential equation with a parameterλ. This implies, though, that they
do not respect the special form of the equation. This is different for the following approximation:
We assume thataandbare almost equal. Then we can use solutions of the equation w′′(t, λ) +b(t)2λ2w(t, λ)−bb′(t)w′(t, λ) = 0
as approximations. This equation has the same structure as (2.4) and can be solved explicitly. When we regroup (2.4) in this way, the term b(t)2−a(t)2
λ2v(t, λ) remains. Like the remainder term in the previous lemma, this term is quadratic in λ.
Lemma 3.6. Let λ > 0 andα≥0. Suppose a∈C0([0,∞],R), b∈C1([0,∞),R) are functions such that a(t)≥0 andb(t)>0 for everyt∈[0,∞). Then the differ- ential equation (2.4) has the fundamental system v+(·, λ), v−(·, λ)∈C2([0,∞),C) such that
v±(t, λ) = exp ±iλB(t)
1 +ρ±(t, λ) where
ρ±(t, λ) ,
iλb(t)1 ρ′±(t, λ)±ρ±(t, λ)
≤exp λ|Q(t)−Q(α)|
−1 and
Q(t) :=
Z t 0
|a(τ)2−b(τ)2| b(τ) dτ.
Proof. Since b is strictly positive, B is strictly increasing. Thus s = B(t) or σ=B(τ) define a transformation of variables. Suppose w(s, λ) = v(t, λ), then w satisfies the differential equation
wss(s, λ) +a(t)2λ2
b(t)2 w(s, λ) = 0.
Put
f(s, λ) :=λ2−a(t)2λ2 b(t)2 , then we can write
wss(s, λ) +λ2w(s, λ) =f(s, λ)w(s, λ)
and apply Theorem 3.1. Hence this equation has a fundamental system w+(·, λ), w−(·, λ)∈C2(B([0,∞)),C)
such that
w±(s, λ) = exp(±iλs) 1 +r±(s, λ) , where
r±(s, λ) ,
iλ1r±′ (s, λ)±r±(s, λ)
≤exp(λ−1F(s, λ))−1 and
F(s, λ) =
Z s
B(α)|f(σ, λ)|dσ . ExpressingF in terms oftgives
F(s, λ) =λ2
Z s B(α)
|b(τ)2−a(τ)2| b(τ)2 dσ
=λ2
Z t α
|b(τ)2−a(τ)2|
b(τ) dτ
.
Thus the original equation has a fundamental systemv+(·, λ),v−(·, λ)∈C2 [0,∞),C such that
v±(t, λ) = exp ±iλB(t)
1 +ρ±(t, λ) , where
ρ±(t, λ) =r±(s, λ), ρ′±(t, λ) =b(t)(r±)s(s, λ).
Using the estimates for r, we obtain ρ±(t, λ)
,
iλb(t)1 ρ±(t, λ)±ρ±(t, λ)
≤exp λ|Q(t)−Q(α)|
−1 where
Q(t) :=
Z t 0
|a(τ)2−b(τ)2| b(τ) dτ.
The notation Qwill be used again in the next section. We can estimate Qeasily using the following assumption:
(Q)
There exist a nonnegative, monotone function ϕQ ∈C1([0,∞),R) and constantsc1, c2>0 such that
|a(t)−b(t)| ≤c1|ϕ′Q(t)|, a(t), b(t)≥c2|ϕ′Q(t)| for everyt≥0.
Using ϕQ is advantageous, because we do not have to assume that a or b are monotone. Under condition (Q), we obtain the following estimates
|a(t)2−b(t)2| ≤c1|ϕ′Q(t)| a(t) +b(t)
≤c1|ϕ′Q(t)| b(t) +c1|ϕ′Q(t)|+b(t)
≤c3|ϕ′Q(t)|b(t).
SinceϕQ is monotone,ϕ′Q does not change sign, hence (3.5) Q(t)−Q(α)≤c3
Z t
α |ϕ′Q(τ)|dτ =c3|ϕQ(t)−ϕQ(α)|. Moreover, ab and ab are bounded, for
a(t) b(t)
≤ b(t) +c1|ϕ′Q(t)|
b(t) ≤1 + c1
c2
,
b(t) a(t)
≤ a(t) +c1|ϕ′Q(t)|
a(t) ≤1 +c1
c2
. Thus there exist constantsc, C >0 such that
(3.6) c a(t)b(t)≤a(t)2≤C a(t)b(t), c a(t)b(t)≤b(t)2≤C a(t)b(t).
For large λ, we can apply Theorem 3.1 after a suitable transformation. For this purpose, the following function will prove to be useful:
(3.7) ψ:= 1
2 1 a
a′ a − b′
b
. We will use this function frequently later on.
Lemma 3.7. Let λ >0 andα≥0, let a∈ C2([0,∞),R), b∈C2([0,∞),R) such that a(t), b(t) >0 for every t ∈ [0,∞). Then there exists a fundamental system v+(·, λ), v−(·, λ)∈C2([0,∞),C)for the differential equation (2.4) such that
v±(t, λ) =q
b
a(t) exp ±iλA(t)
1 +ρ±(t, λ) and
ρ±(t, λ) ,
iλa(t)1 ρ′±(t, λ)±ρ±(t, λ)
≤exp λ−1|S(t)−S(α)|
−1, where
S(t) :=
Z t
0 |ψ′(τ) +a(τ)ψ(τ)2|dτ.
The error term S will reappear in the next section.
Proof. In order to use Theorem 3.1, we have to transform equation (2.4). For this purpose, put
ϕ:=
q b a . Since ψ = 121a aa′ − bb′
, it follows thatϕ′ =−aψϕ. Since a is strictly positive, A is strictly increasing. Thus s = A(t) and σ = A(τ) define a transformation of variables. Define w(·, λ) ∈ C2 A([α,∞)),C
by w(s, λ) := v(t, λ)ϕ(t)−1. We obtain the following equations
v(t, λ) =ϕ(t)w(s, λ),
v′(t, λ) =−a(t)ψ(t)ϕ(t)w(s, λ) +a(t)ϕ(t)ws(s, λ), v′′(t, λ) = −a′(t)ψ(t)−a(t)ψ′(t) +a(t)2ψ(t)2
ϕ(t)w(s, λ) + −2a(t)2ψ(t) +a′(t)
ϕ(t)ws(s, λ) +a(t)2ϕ(t)wss(s, λ).
Substitutingv in the differential equation leads to 0 =a(t)2wss(s, λ) +
−2a(t)2ψ(t) +a′(t)−bb′(t) ws(sλ)
−
a′(t)ψ(t) +a(t)ψ′(t)−a(t)2ψ(t)2−bb′(t)a(t)ψ(t)−a(t)2λ2 w(s, λ)
=a(t)2wss(s, λ) +a(t)2λ2w(s, λ)
−a(t)2
a(t)−1aa′(t)ψ(t) +a(t)−1ψ′(t)−ψ(t)2−a(t)−1bb′(t)ψ(t) w(s, λ)
=a(t)2wss(s, λ) +a(t)2λ2w(s, λ)−a(t)2h
ψ(t)2+a(t)−1ψ′(t)i w(s, λ).
Fors∈A([α,∞)) putf(s) := [ψ(t)2+a(t)−1ψ′(t)]. With this notation, it follows that
(3.8) wss(s, λ) +λ2w(s, λ) =f(s)w(s).
By Theorem 3.1 there exists a fundamental system
w+(·, λ), w−(·, λ)∈C2 A([0,∞)),C for this equation such that
w±(s, λ) = exp(±iλs) 1 +r±(s, λ) , where
r±(s, λ) ,
iλa(t)1 r±′ (s, λ)±r±(s, λ) ≤exp
F(s) λ
−1,
and
F(s) =
Z s
A(α)|f(σ)|dσ .
It follows that the original equation has a fundamental system v+(·, λ), v−(·, λ)∈C2
0,∞),C such that
v±(t, λ) =p
ϕ(t) exp ±iλA(t)
1 +ρ±(t, λ) , where
ρ±(t, λ) =r±(s, λ), ρ′±(t, λ) =a(t)(r±)s(s, λ).
Substitutingr± we obtain the estimates ρ±(t, λ)
,
ia(t)λ1 ρ±(t, λ)±ρ±(t, λ)
≤exp λ−1|S(t)−S(α)|
−1 where
S(t) :=
Z s
0 |f(σ)|dσ= Z t
0
a(t)|f(σ)|dτ = Z t
0 |ψ′(τ) +a(τ)ψ(τ)2|dτ.
The following assumption will help in estimatingS:
(S)
There exists a nonnegative, monotone function ϕS ∈ C1([0,∞),R) and a constant c > 0 such that
|ψ(t)| ≤c ϕS(t),
|ψ′(t) +a(t)ψ(t)2| ≤c|ϕ′S(t)| for every t≥0.
The function ϕS helps to avoid conditions on the monotonicity of more important functions, in this case we avoid special assumptions on ψ. We assume (S). Since ϕ′S does not change sign,
(3.9) S(t)−S(α)≤c|ϕS(t)−ϕS(α)| for every t≥α≥0.
In the following, we assume thata, b∈C2([0,∞),R) are strictly positive.
4. Fourier space estimates
Now we use the approximate fundamental systems of equation (2.4) to prove bounds for the energy density
eu(t, ξ) :=|uˆt(t, ξ)|2+a(t)2|ξ|2|u(t, ξ)ˆ |2.
of admissible solutionsuof (1.1). To this end we use the Euclidean matrix norm (aij)
:=
v u u t
n
X
i,j=1
|aij|2,
which is compatible with the Euclidean norm for vectors. Let λ > 0. Suppose v1(·, λ), v2(·, λ)∈C2([0,∞),C) is a fundamental system of equation (2.4) and
(4.1) V(·, λ) :=
v1(·, λ) v2(·, λ) v′1(·, λ) v′2(·, λ)
the associated fundamental matrix. For an admissible solution uof equation (1.1) and for every ξ∈Rn put
U(·, ξ) :=
u(ˆ ·, ξ) ˆ ut(·, ξ)
. For every s, t∈[0,∞) andξ∈Rn\ {0}we can write
U(t, ξ) =V(t,|ξ|)V(s,|ξ|)−1U(s, ξ).
For every M(·, λ) : [0,∞) →GL(2,C) ,N(·, λ) : [0,∞)→GL(2,C) and for every t, s∈[0,∞) we define
FMN(t, s, λ) :=
M(t, λ)V(t, λ)V(s, λ)−1N(s, λ)−1
2.
This definition is independent of the choice of the fundamental matrix V(·, λ):
If X, Y are fundamental matrices of the same linear differential equation, then X(t)X(s)−1 = Y(t)Y(s)−1. Suppose L(·, λ) : [0,∞) → GL(2,C) is another map- ping. Since the Euclidian matrix norm is submultiplicative, it follows that
FLN(t, r, λ)≤FLM(t, s, λ)FMN(s, r, λ) for every t≥s≥r. In the following, let
A(t, λ) := diag a(t)λ,1
, B(t, λ) := diag a(t)hλi,1 where hλi=√
1 +λ2. We observe that
|A(t,|ξ|)U(t, ξ)|2=eu(t, ξ), |B(t,|ξ|)U(t, ξ)|2=eu(t, ξ) +a(t)2|u(t, ξ)ˆ |2, for every ξ∈Rn\ {0}. Hence
eu(t, ξ) =
A(t,|ξ|)V(t,|ξ|)V(0,|ξ|)−1A(0,|ξ|)−1A(0,|ξ|)U(0, ξ)
2 ≤FAA(t,0,|ξ|)eu(0, ξ), and
eu(t, ξ)≤FAB(t,0,|ξ|) eu(0, ξ) +a(0)2|u(0, ξ)ˆ |2 . If we are able to proof an estimate of the form
FAA(t,0, λ)≤d(t) or FAB(t,0, λ)≤d(t) for every t≥0 andλ >0, then
Eu(t) = Z
Rn
eu(t, ξ)dξ≤ Z
Rn
FAA(t,0,|ξ|)eu(0, ξ)dξ≤d(t)Eu(0) or
Eu(t) = Z
Rn
eu(t, ξ)dξ≤ Z
Rn
FAB(t,0,|ξ|) eu(0, ξ) +a(t)2|u(0, ξ)ˆ |2 dξ
≤d(t) Eu(0) +a(0)2ku(0,·)k2L2
, respectively. We will now prove such estimates.
Lemma 4.1. Letc >0. Then there exists a constant C >0, depending only onc, such that
FAB(t,0, λ)≤C(a(0)−2+b(0)−2) a(t)2(1 +B(t))2λ2+b(t)2 for every t≥0 andλ >0satisfying λ2R(t)≤c andλR′(t)≤ca(t).
Proof. Letv1, v2be the fundamental system as in Lemma 3.5, and lett≥0,λ >0 such that λ2R(t)≤c andλR′(t)≤c. Then
|v1(t, λ)| ≤ec, |v2(t, λ)| ≤ecB(t),
|v′1(t, λ)| ≤cecλa(t), |v′2(t, λ)| ≤ecb(t) +cecλ a(t)B(t).
It follows that
|A(t, λ)V(t, λ)|2≤c21 2a(t)2λ2+a(t)2λ2B(t)2+ (b(t) +a(t)B(t)λ)2
≤c2 a(t)2 1 +B(t)2
λ2+b(t)2 .
Note that the constants c1, c2 depend only onc. The initial values are v1(0, λ) = 1, v2(0, λ) = 0,
v′1(0, λ) = 0, v′2(0, λ) =b(0),
hence|V(0, λ)−1B(0, λ)−1|2=a(0)−2hλi−2+b(0)−2≤a(0)−2+b(0)−2. Combining
this with the last inequality completes the proof.
Corollary 4.2.Let c >0. If condition (R) is satisfied, then there exists a constant C >0 such that
FAB(t,0, λ)≤C a(t)2 1 +B(t)2
λ2+b(t)2 for every t ≥0 and every λ >0 satisfying 1 +A(t)
λ≤c. The constantC >0 depends only on c,a(0),b(0)and the constant appearing in (R).
Proof. Let t ≥ 0 and λ > 0 such that λ ≤ c(1 +t)−1+ℓ. By the remark on condition (R), see p. 9, it follows that
λ2R(t)≤c′λ2 1 +A(t)2
≤c′c, λR′(t)≤c′λ a(t) 1 +A(t)
≤c′ca(t).
Lemma 4.1 completes the proof.
Lemma 4.3. Letc >0. Then there exists a constant C >0, depending only onc, such that
FAA(t, α, λ)≤C a(t)2+b(t)2
a(α)−2+b(α)−2 for every α≥0,t≥αand every λ >0 satisfying λ Q(t)−Q(α)
≤c.
Proof. Let v+, v− be the fundamental system as in Lemma 3.6 and V(t, λ) the associated fundamental matrix as in (4.1). Suppose t ≥α ≥0, λ > 0 such that λ Q(t)−Q(α)
≤c. Then
|a(t)λv±(t, λ)| ≤a(t)λec,
|v±′ (t, λ)|=
exp ±iλB(t)
±iλb(t) 1 +ρ±(t, λ)
+ρ′±(t, λ)
≤b(t)λec, hence
|A(t, λ)V(t, λ)|2≤2e2c a(t)2+b(t)2 λ2. The initial values are
v±(α, λ) =a(α)λexp ±iλB(α)
, v′±(α, λ) =±ib(α)λexp ±iλB(α) .
We compute the determinant
det V(α, λ)−1A(α, λ)−1
=−2ia(α)b(α)λ2, and since|M−1|=|det|MM|| for every matrixM ∈C2×2, we have
|V(α, λ)−1A(α, λ)−1|2≤2e2c a(α)2+b(α)2 λ2 4a(α)2b(α)2λ4 = e2c
2 a(α)−2+b(α)−2 λ−2.
This completes the proof.
Corollary 4.4. Assume condition (Q) and suppose c >0.
(a) IfϕQ is increasing, then there exists a constantC >0 such that FAA(t,0, λ)≤Ca(t)b(t)
a(0)b(0) for everyt≥0 and everyλ >0 such thatλϕQ(t)≤c.
(b) If ϕQ is decreasing, then there exists a constant C >0 such that FAA(t, α, λ)≤C a(t)b(t)
a(α)b(α)
for everyα≥0, for every t≥αandλ >0 such thatλϕQ(α)≤c.
The constant C >0 depends only on c and on the constants in (Q).
Proof. By inequality (3.5), there exists a constantc′>0 such that Q(t)−Q(α)≤c′|ϕQ(t)−ϕQ(α)|
for every α≥0 and everyt≥α. SupposeϕQ is increasing,t≥0 andλ >0 such that λϕQ(t)≤c. Then
λ Q(t)−Q(0)
≤c′λϕQ(t)≤c′c.
By Lemma 4.3,
FAA(t,0, λ)≤C a(t)2+b(t)2
a(0)−2+b(0)−2
≤C′a(t)b(t) a(0)b(0).
In the last step we used that under assumption (Q) there arek, K >0 such that k a(t)b(t)≤a(t)2≤K a(t)b(t), k a(t)b(t)≤b(t)2≤K a(t)b(t),
see inequality (3.6). IfϕQis decreasing,t≥α≥0 andλ >0 such thatλϕQ(α)≤c, then
λ Q(t)−Q(α)
≤c′λϕQ(α)≤c′c, and we obtain
FAA(t, α, λ)≤C a(t)b(t) a(α)b(α)
by Lemma 4.3.
Lemma 4.5. Let c >0. Then there exists a constantC >0 depending only on c, such that
FAA(t, α, λ)≤C a(t)b(t) a(α)b(α), FAB(t, α, λ)≤C λ2
hλi2
a(t)b(t) a(α)b(α)
for every α≥0,t≥αand everyλ >0such thatS(t)−S(α)≤cλand|ψ(t)| ≤cλ.
Proof. Let v+, v− be the fundamental system as in Lemma 3.7 and let V(t, λ) be the associated fundamental matrix as in (4.1). Suppose t≥α≥0 andλ >0 such that S(t)−S(α)≤cλ and|ψ(t)| ≤cλ. Then
|a(t)λv±(t, λ)| ≤a(t)λq
b
a(t)ec≤c1
pa(t)b(t)λ and
v′±(t, λ) =±iλp
a(t)b(t) exp ±iλA(t)
1 +ρ±(t, λ)∓iλ1ρ′±(t, λ)
+q
b a
′
(t) exp ±iλA(t)
1 +ρ±(t, λ) . Since
qb a
′
(t)
=a(t)|ψ(t)| qb
a(t)≤cλp
a(t)b(t), we obtain
|v′±(t, λ)| ≤p
a(t)b(t)λec+cp
a(t)b(t)λec≤c1
pa(t)b(t)λ, where c1:= (1 +c)ec. Hence
|A(t, λ)V(t, λ)|2≤4c1a(t)b(t)λ2. The initial values are
v±(α, λ) =q
b
a(α)λexp ±iA(α) , v′±(α, λ) =
±ip
a(α)b(α)λ+q
b a
′
(α)
exp ±iA(α) , thus
det V(α, λ)−1A(α, λ)−1
=−2ia(α)b(α)λ2. Since|M−1|= |det|M|M| for every matrixM ∈C2×2, it follows that
|V(α, λ)−1A(α, λ)−1|2≤4c1
a(α)b(α)λ2 4a(α)2b(α)2λ4 =c1
1 a(α)b(α)λ2 and
|V(α, λ)−1B(α, λ)−1|2≤4c1
a(α)b(α)λ2
4a(α)2b(α)2hλi2λ2 =c1
1
a(α)b(α)hλi2.
Corollary 4.6. Assume (S) and let c >0.
(a) IfϕS is increasing, then there exists a constant C >0 such that FAA(t,0, λ)≤Ca(t)b(t)
a(0)b(0), FAB(t,0, λ)≤C λ2 hλi2
a(t)b(t) a(0)b(0) for everyt≥0 and everyλ >0 such thatϕS(t)≤c λ.
(b) If ϕS is decreasing, then there exists a constant C >0 such that FAA(t, α, λ)≤C a(t)b(t)
a(α)b(α), FAB(t, α, λ)≤C λ2 hλi2
a(t)b(t) a(α)b(α) for everyt≥α≥0 and every λ >0 mitϕS(α)≤c λ.
The constant C >0 depends only on c and on the constant appearing in (S).
Proof. Let ϕS be increasing, let t ≥ 0 and λ > 0 such that ϕS(t) ≤ c λ. By condition (S) and inequality (3.9), there exists a constant c′ such that
S(t)−S(0)≤c′ϕS(t)≤cc′λ and
|ψ(t)| ≤c′ϕS(t)≤cc′λ.
Lemma 4.5 implies (a).
IfϕS is increasing,t≥α≥0 andλ >0 such thatϕS(α)≤c λ, then S(t)−S(α)≤c′ϕS(α)≤cc′λ
and
|ψ(t)| ≤c′ϕS(t)≤c′ϕS(α)≤cc′λ.
As above, (b) is an immediate implication of Lemma 4.5.
5. Energy inequalities
Now we glue together the estimates of the previous section in order to get energy estimates. As before, we consider only admissible solutions of equation
(1.1) utt(t, x)−a(t)2∆u(t, x)−bb′(t)ut(t, x) = 0, i.e. solutions u∈C0 [0,∞), H2(Rn)
∩C1 [0,∞), H1(Rn)
∩C2 [0,∞), L2(Rn) . Under the conditions of the following theorem, the existence of nonvanishing ad- missible solutions is guaranteed, see Theorem 5.4 below.
To state the main theorem, we only need the following abbreviations introduced in the previous sections:
A(t) :=
Z t 0
a(τ)dτ, B(t) :=
Z t 0
b(τ)dτ and ψ:= 1 2
1 a
a′ a − b′
b
.
Theorem 5.1. Leta, b∈C2([0,∞),R)be strictly positive, and supposea /∈L1([0,∞)).
Let ε <1 andc1, c2, c3>0 such that b(t) b(τ) ≤c1
a(t) a(τ)
1 +A(t)ε
1 +A(τ)ε, (5.1)
|ψ(t)| ≤c2 1 +A(t)−1
, (5.2)
|ψ′(t)| ≤c3a(t) 1 +A(t)−2
(5.3)
for every 0≤τ ≤t. LetM: [0,∞)→[0,∞) be a function such that M(t)
M(τ) ≥ a(t)b(t)
a(τ)b(τ), M(t)≥a(t)2 1 +B(t)2
1 +A(t)2 + b(t)2 (5.4)
(a)
λ
t (b)
λ t
Figure 1. Sketch to the proof of Theorem 5.1 and to the proof of Theorem 5.3 showing (a) the curve defined by ϕS(t) = λ and (b) the curve defined byϕQ(t) = 1λ.
for every 0≤τ ≤t. Then there exists a constantC >0such that Eu(t)≤C M(t) Eu(0) +ku(0,·)k2L2
for each admissible solution of equation (1.1) and everyt≥0.
Proof. By (5.1), the functionsaandbsatisfy condition (R), and by (5.2) and (5.3), they satisfy condition (S) with
ϕS(t) := 1 +A(t)−1
.
The curve defined by ϕS(t) = λ is sketched in Figure 1 (a). Since a is strictly positive, A is strictly increasing, and since a is not integrable, A is unbounded.
Hence,ϕS is strictly decreasing and tends to 0.
1. Supposeλ≥ϕS(0). By Corollary 4.6, FAB(t,0, λ)≤c4a(t)b(t)
a(0)b(0) ≤c5M(t) for everyt≥0.
2. Supposeλ≤ϕS(0) and putα:=ϕ−1S λ). Here, the estimates fort≤αand for t≥αhave to be glued together.
(a) Let 0≤t≤α. Then
λ=ϕS(α) = 1 +A(α)−1
≤ 1 +A(t)−1
, and by Corollary 4.2 it follows that
FAB(t,0, λ)≤c6 a(t)2 1 +B(t)2
1 +A(t)−2
+b(t)2
≤c6M(t).
(b) Lett≥α. By Corollary 4.6 and sinceϕS(α) =λthe inequality FAA(t, α, λ)≤c7
a(t)b(t) a(α)b(α) holds. Fort≥αit follows that
FAB(t,0, λ)≤FAA(t, α, λ)FAB(α,0, λ)≤c8 a(t)b(t)
a(α)b(α)M(α)m≤c8M(t).
Now it has been proven that
FAB(t,0, λ)≤c9M(t) for every t≥0 and everyλ >0, and thus
Eu(t)≤ Z
Rn
FAB(t,0,|ξ|) eu(0, ξ) +|u(0, ξ)|2 dξ
≤CM(t) Eu(0) +ku(0,·)k2L2
.
for each admissible solution uof (1.1) and every t≥0.
As can be seen from the proof, (5.3) could be replaced by the slightly weaker, but more cumbersome condition
|ψ′(t) +aψ(t)2| ≤c a(t) 1 +A(t)−2
.
The conditions of Theorem 5.1 can be simplified when only solutions with energy behaving likea·bare considered:
Corollary 5.2. Leta, b∈C2([0,∞),R)be strictly positive and supposea /∈L1([0,∞)).
Let ε <1 andc1, . . . , c4>0 such that
|ψ(t)| ≤c1 1 +A(t)−1
, |ψ′(t)| ≤c2 a(t) 1 +A(t)−2
, b(t)
b(τ) ≤c3a(t) a(τ)
1 +A(t)ε
1 +A(τ)ε, b(t) a(t) ≥c4
1 +B(t)2
1 +A(t)2 + b(t)2 a(t)2
!
for every t≥τ ≥0. Then there exists a constantC >0such that Eu(t)≤C a(t)b(t) Eu(0) +ku(0,·)k2L2
for each admissible solution of equation (1.1) and for every t≥0.
In [15], Wirth proves an analogous result to Theorem 5.1 for a = 1, where the following conditions are used:
(A1) bb′(t)≤0 for everyt≥0, (A2) bb′′
(t)>0 for everyt≥0, (A3) bb′2
(t)≤c bb′′
(t) for everyt≥0, (A4)
b
′
b
(k)
(t)
≤c(1 +t)−1−k fork= 0,1 and everyt≥0.
(C1) lim supt→∞tb(t)<1.
Wirth shows the theorem under conditions (A1) – (A3) and (C1) or under conditions (A1), (A4), and (C1). Of these, Theorem 5.1 only needs condition (A4), which is equivalent to (A3) together with
b′ b
′
(t)> c(1 +t)−2
for everyt≥0. Condition (C1) is replaced by the much weaker condition (5.4). In particular, Theorem 5.1 needs no restrictions on the sign of the coefficient bb′. As the second example of section 6 shows, a decay of the energy is possible even if bb′ changes sign infinitely often
Theorem 5.1 can be reached via an alternative route: First, Theorem 5.1 has to be proved for the special casea= 1. For an arbitrary strictly positive function
a∈C2([0,∞),R)\L1([0,∞))
the change of variables s=A(t) leads to
wss(s, x)−∆w(s, x) + 2ψ(t)ws(s, x) = 0 fors∈A([0,∞)).
Since a is not integrable over [0,∞), it follows that A([0,∞)) = [0,∞), and the estimate for the special casea= 1 can be carried over to the general case.
This is not possible ifa∈L1([0,∞))[0,∞) andψis unbounded: Putβ(s) = 2ψ(t), then β is unbounded as well and hence has a singularity at the right boundary of A([0,∞)). A continuous extension of β on [0,∞) is then impossible. But those cases are covered be the following theorem:
Theorem 5.3. Let c >0. Suppose a, b∈C2([0,∞),R) are strictly positive func- tions such that conditions (Q) and (S) are satisfied. Assume that ϕQ or ϕS is strictly decreasing and tends to 0, and suppose
ϕQ(t)ϕS(t)≤c
for every t≥0. Then there exists a constantC >0 such that Eu(t)≤C a(t)b(t)Eu(0)
for each admissible solution of equation (1.1) and everyt≥0.
Proof. First suppose ϕQ is strictly decreasing and tends to 0. In Figure 1 (b) the curve defined by ϕQ(t) =λ1 is shown for the example ϕQ =e−12t.
1. Suppose λ1 ≥ϕQ(0). Then, by Corollary 4.4, FAA(t,0, λ)≤C′a(t)b(t)
a(0)b(0) for everyt≥0.
2. Suppose λ1 ≤ϕQ(0) and putα:=ϕ−1Q λ1). Here, the estimates fort≤αand for t≥αmust be glued together.
(a) For every 0≤t≤αthe inequality
(5.5) 1λϕS(t) =ϕQ(α)ϕS(t)≤ϕQ(t)ϕS(t)≤c
holds. IfϕS is increasing, this is exactly the inequality we need for Corol- lary 4.6 (a). SupposeϕSis decreasing. Then we can apply Corollary 4.6 (b), ifϕS(0)≤cλ, which is only a special case of (5.5). Thus, by Corollary 4.6,
FAA(t,0, λ)≤C′a(t)b(t) a(0)b(0) for every 0≤t≤α.
(b) Supposet≥α. By Corollary 4.4 and sinceλϕQ(α) = 1 it follows that FAA(t, α, λ)≤C′ a(t)b(t)
a(α)b(α). Thus
FAA(t,0, λ)≤FAA(t, α, λ)FAA(α,0, λ)≤C′a(t)b(t) a(0)b(0).
