DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS SERIES S
Volume 2, Number 3, September 2009
doi:10.3934/dcdss.2009.2.609
pp. 609–629
HIGHER ORDER ENERGY DECAY RATES FOR DAMPED
WAVE EQUATIONS WITH VARIABLE COEFFICIENTS
Petronela Radu
Department of Mathematics
University of Nebraska-Lincoln, NE 68588, USA
Grozdena Todorova and Borislav Yordanov
Department of Mathematics
University of Tennessee-Knoxville, TN 37996, USA
Abstract. Under appropriate assumptions the energy of wave equations with
damping and variable coefficients c(x)utt − div(b(x)∇u) + a(x)ut = h(x, t) has
been shown to decay. Determining the decay rate for the higher order energies
of the kth order spatial and time derivatives has been an open problem with
the exception of some sparse results obtained for k = 1, 2. We establish the
sharp gain in the decay rate for all higher order energies in terms of the first
energy, and also obtain the sharp gain of decay rates for the L2 norms of the
higher order spatial derivatives. The results concern weighted (in time) and
also pointwise (in time) energy decay estimates. We also obtain L∞ estimates
for the solution u in dimension n = 3. As an application we compute explicit
decay rates for all energies which involve the dimension n and the bounds for
the coefficients a(x) and b(x) in the case c(x) = 1 and h(x, t) = 0.
1. Introduction. We will study hyperbolic equations of the form
c(x)utt − div(b(x)∇u) + a(x)ut = h(x, t),
x ∈ Rn , t > 0,
(1)
where a, b, c, and h satisfy the assumptions listed below. This equation is generally
accepted as a model of wave propagation in a heterogeneous medium with friction
(given by a(x)ut ) and source terms h(x, t). The coefficient c(x) accounts for variable
mass density, while b(x) is responsible for temperature changes as the wave travels
in space (see the derivation in [6]). More surprisingly, (1) has been considered as a
model for heat propagation by Cattaneo [2], and independently by Vernotte [17]. In
this setting the constitutive equation for the flux q as given by
qt + a(x)q = b(x)∇u
(2)
replaces the classical Fourier’s law q = b(x)∇u. The Cattaneo-Vernotte equation
(2) is usually considered as appropriate to describe unsteady heat conduction. For
more on the applicability of this model see [3, 8, 9]. The finite speed of propagation
feature that the solutions enjoy under this formulation renders this system as a more
accurate way to study the real effects of heat diffusion. We will show later how these
aspects regarding the physical interpretation of (1) tie in with the mathematical
analysis of finding the decay rates, but we believe that this paper offers compelling
2000 Mathematics Subject Classification. Primary: 35L05, 35L15; Secondary: 37L15.
Key words and phrases. wave equations with variable coefficients, hyperbolic diffusion, linear
dissipation, decay rates, higher order energy.
609
610
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
evidence as to why the diffusion phenomenon is an essential feature for the damped
wave equations.
The literature is replete with results on energy estimates for hyperbolic systems
with linear damping, but they mainly concern the case when b(x) = c(x) = 1
and h(x, t) = 0. In [13] we considered wave equations with variable coefficients
in the case c(x) = 1, h(x, t) = 0 for which we obtained explicit decay rates that
depend on the dimension and the growth of the coefficients. We continue here the
analysis of the damped wave equations with variable coefficients a(x), b(x), c(x) and
nonhomogeneous source h(x, t). More precisely, the main goals of this work are:
1. Establish the exact gain in the decay rate for all higher order energies E(t; ∂tk u)
in terms of the first order energy E(t; u), where
Z
1
(cvt2 + b|∇v|2 ) dx.
E(t; v) =
2
2. Obtain the sharp gain of decay rates for the L2 norms of the higher order
spatial derivatives.
Regarding the first problem, Nakao [11] uses the decay of the damping to find
decay rates for the first and second order energies when c(x) = b(x) = 1, and
h(x, t) = 0. He shows that the energy has the following decay rate
E(t; u) :=
1
(||ut (t)||2 + ||∇u(t)||2 ) . (1 + t)−1 ,
2
while for the second order energy the decay rate improves by t−1 , so one has
1
(||utt (t)||2 + ||∇ut (t)||2 ) . (1 + t)−2 .
2
By using these estimates in the equation he consequently obtains
E(t; ut ) =
||∆u(t)||2 . (1 + t)−1 .
Note that these estimates show no improvement in the rate of decay for a higher
order spatial derivative, as the L2 norms of the gradient and the Laplacian have the
same decay rate (1 + t)−1/2 .
The idea of using the diffusion effect in order to show a faster decay for the higher
order energies is also present in [5] where the authors show that for the elasticity
system with linear damping ut one has
E(t; u) + E(t; ut ) + E(t; utt ) . (1 + t)−2 .
One expects, however, that the gain in the decay rate when moving from the
first energy to the second should be t−2 . This is motivated by the fact that the long
time behavior of the linearly damped wave equation with constant coefficients
utt − ∆u + aut = 0
(3)
resembles the behavior of the corresponding parabolic equation
− ∆u + aut = 0,
(4)
as it was suggested in [10], [12], [16]. By estimating the higher order energies for
the Gaussian as a solution of (4), one notices that E(t; ∂tk+1 u) decays faster than
E(t; ∂tk u) by a factor of t−2 . This suggests that a similar phenomena should occur
for (3). Indeed, our paper shows that this conjecture is correct, so in the absence
HIGHER ORDER ENERGY DECAY RATES
611
of other effects caused by inhomogeneities and source terms (a = b = c = 1, h = 0),
one basically has
E(t; ∂tk u) . E(t, u)(1 + t)−2k ,
k = 0, 1, 2, ...
Regarding the decay of the L2 norms of the spatial derivatives one notices a gain
of t−1/2 for any additional order of the derivatives, i.e.
||∂xk u||2L2 . ||u||2L2 (1 + t)−k ,
a phenomenon analogous with the one obtained for higher order spatial derivatives
of the Gaussian (this is shown here for some spatial derivatives up to the fourth
order, but the computation can be extended for arbitrarily high orders). In this
spirit, by matching the gain in decay t−2 of E(t; ∂tk+1 u) versus E(t; ∂tk u), and the
decay t−1/2 for L2 norms of spatial derivatives (in the constant coefficient case and
no sources), with the decays exhibited by solutions of diffusion problems we show
the optimality of our decay results. The presence of variable coefficients makes the
analysis more cumbersome so that the gain in the decay rates from one order to the
next is much more complicated as seen in Theorems 2.1, 2.2, 2.3. One of the most
important contributions of this paper is developing a methodology to determine
sharp decay rates for all higher order energies and L2 norms of higher order spatial
derivatives for a variety of hyperbolic systems with linear damping. The control of
spatial derivatives allows us to obtain L∞ estimates for the solution which are a
key tool in rendering different nonlinearities harmless in many equations.
The paper is organized as follows. In the next section we present the assumptions
which govern our results and state two of the main theorems regarding the decay of
the higher order energies involving time derivatives. The proofs of these theorems
together with some other weighted decay estimates involving the kinetic energy are
presented in Section 3. The weighted decay of the spatial derivatives is obtained
for orders k = 2 and k = 4 from which we deduce L∞ estimates of the solution in
dimension n = 3; these are contained in Sections 4 and 5. We apply these results
in the case c = 1, h = 0 in Section 6. In the Appendix one can find the theorem
and proof regarding the finite speed of propagation of solutions of (1) for variable
coefficients b and c.
2. Main results. Let a, b, and c be C 1 functions satisfying the conditions:
a0 (1 + |x|)−α ≤ a(x) ≤ a1 (1 + |x|)−α ,
b0 (1 + |x|)β ≤ b(x) ≤ b1 (1 + |x|)β ,
c0 (1 + |x|)−γ ≤ c(x) ≤ c1 (1 + |x|)−γ ,
(5)
where ai , bi , ci , i = 0, 1 are positive constants and the exponents α, β, γ satisfy:
0 < β + γ < 2,
2α + β − γ < 2.
(6)
We consider initial data
u|t=0 = u0 ∈ H k+1 (Rn ),
ut |t=0 = u1 ∈ H k (Rn ),
(7)
and a source h such that
h ∈ C k ([0, ∞), L2 (Rn )),
(8)
where k is a nonnegative integer. These three functions u0 , u1 , and h have compact
support:
u0 (x) = 0, u1 (x) = 0 for |x| > R,
(9)
612
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
h(x, t) = 0 for q(|x|) > t + q(R),
(10)
where q measures the increase in the support of u as given by Proposition 1 below.
Before stating our main theorems we state this finite speed of propagation result
which will be used throughout the paper. This property reflects the impact of the
variable coefficients b(x) and c(x) on the size of the support of the solution. In fact,
this finite speed of propagation type result is a corollary of a more general theorem,
both of which are stated and proven in the Appendix.
Proposition 1. Let u be a solution of (1) with data (u, ut )|t=0 = (u0 , u1 ), such
that (u0 , u1 ) ∈ H 1 (Rn ) × L2 (Rn ). Assume that u0 , u1 , and h have compact support
as given by (9) - (10). Let b and c satisfy (5) with 0 < β + γ < 2. Then there exists
q satisfying
s
0 < q ′ (|x|) ≤
c(x)
, x ∈ Rn
b(x)
and
q(|x|) ≥ q0 |x|1−(β+γ)/2 , x ∈ Rn ,
with q0 > 0 such that u(x, t) = 0 for
2/(2−β−γ)
t + q(R)
|x| >
.
q0
As a consequence of the above result, the support of the solution u at time t
satisfies
2
supp u(·, t) ⊆ {x ∈ Rn : |x| ≤ R + Ct 2−β−γ }
for some C > 0.
We are now in position to state some of our main results. The first two theorems
concern decay rates for all the higher order energies, as weighted and pointwise
(in time) estimates. The third theorem gives decay rates for the L2 norm of some
fourth order spatial derivatives in terms of the L2 norm of second order spatial
derivatives of the solution. Similarly, we can deduce decay rates for other higher
order spatial derivatives. Other results concerning decay of the L∞ norm of solutions
are presented in Section 5.
Theorem 2.1. Let θ > 0 and let a, b, c ∈ C 1 (Rn ) be coefficients which satisfy (5),
(6). Assume that the initial data and the inhomogeneity h are chosen such that
(7), (8), (9), (10) hold. Then for any k = 0, 1, 2... the solution u of equation (1)
satisfies the following weighted energy estimate:
" k
#
Z T
X
θ+2k
k
θ+ω
i
(1 + t)
E(t; ∂t u) dt . (1 + T0 )
E(T0 ; ∂t u)
(11)
T0
+
i=0
Z
T
(1 + t)θ E(t; u)dt +
T0
where ω is such that
Z
T
T0
Z
(1 + t)θ+1 a−1
k
X
[(1 + t)2i (∂ti h)2 ] dxdt,
i=0
2(α − γ)
max 0,
< ω < 1.
2−β−γ
(12)
A pointwise (in time) version of the above energy decay rate is also available as
the following theorem states:
HIGHER ORDER ENERGY DECAY RATES
613
Theorem 2.2. Under the same assumptions of Theorem 2.1, with the same parameters and notation, we have:
"
k
X
E(T ; ∂tk u) . (1 + T )−θ−2k−1 (1 + T0 )θ+ω [
E(T0 ; ∂ti u)] (13)
i=0
+
Z
T
θ
(1 + t) E(t; u)dt +
T0
Z
T
T0
Z
θ+1
(1 + t)
k
X
(1 + t)2i (∂ i h(x, t))2
t
i=0
a(x)
#
dxdt .
The decay of some of the spatial derivatives of the solution can be deduced by
using the above estimates and by
Theorem 2.3. Let a, b, c ∈ C 2 (Rn ) satisfy (5) and (6) and h ∈ C 2 ([0, ∞); H 2 (Rn )).
Define
div(b∇u)
Mu =
.
a
Then the following weighted estimates hold for the solution u of equation (1) with
initial conditions (7) which satisfy (9):
Z TZ
(i)
(1 + t)θ+1 a(M u)2 dxdt
T0
.
θ+1
(1 + T0 )
Z
1
X
i
[
E(T0 ; ∂t u)] +
Z
T
T
Z
T0
Z
(ii)
T0
.
Z
(1 + t)θ+1
1
X
(1 + t)2i (∂ i h)2
t
i=0
a
+
T
+
Z
T0
T0
T
(1 + t)θ E(t; u)dt
T0
i=0
T
dxdt,
(1 + t)θ+3−λ2 a(M 2 u)2 dxdt
Z
3
X
E(T0 ; ∂ti u)] +
(1 + T0 )θ+3 [
Z
(1 + t)θ E(t; u) dt
T0
i=0
+
T
2
X
(1 + t) (∂ti h)2
dxdt
a
i=0
"
2 #
Z
2
2
h
h
h
t
(1 + t)θ+3−λ2
+ (1 + t)2λ1 tt + a M
dxdt,
a
a
a
Z
(1 + t)θ+1
2i
where λ1 , λ2 are given by (32) and (33).
3. Weighted L2 estimates for time derivatives. The strategy behind determining the rate of decay for E(t; ∂tk u) in terms of the decay for E(t; u) is based on
some simple facts. First, since ∂tk u solves the equation (7) with the different inhomogeneity ∂tk h, the higher order energy E(t; ∂tk u) satisfies (almost) the same decay
estimate as E(t; u) with possibly different constants (the actual decay will depend
on ∂tk h). The question is whether these decay rates can be improved similarly to
the case a = b = c = 1 and h = 0 where the Fourier representation of u shows faster
decay rates of norms involving higher derivatives.
Let us recall the definition
Z
1
E(t; u) =
(cu2t + b|∇u|2 ) dx
2
614
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
and energy identity
d
E(t; u) +
dt
Z
au2t
dx =
Z
hut dx.
(14)
A simple consequence is the following.
Proposition 2. Assume that 0 ≤ T0 ≤ T and µ ≥ 0. Then
Z TZ
Z T
(1 + t)µ a(x)u2t dxdt . (1 + T0 )µ E(T0 ; u) +
(1 + t)µ−1 E(t; u)dt
T0
T0
+
Z
T
T0
Z
(1 + t)µ
2
h (x, t)
dxdt.
a(x)
Proof. The energy identity (14) implies
Z
Z
d
[(1 + t)µ E(t; u)] + (1 + t)µ au2t dx = (1 + t)µ hut dx + µ(1 + t)µ−1 E(t; u).
dt
Notice that |hut | ≤ au2t /2 + a−1 h2 /2. The result follows from integration on the
time interval [T0 , T ].
We will now proceed to estimate higher order norms and energies. Let v = ut to
simplify notations. Our next result is:
Proposition 3. Assume that the a, b, c and h satisfy
the o
assumptions of Theorem
n
< ω < 1 we have:
2.1. Then for every θ > 0 and ω such that max 0, 2(α−γ)
2−β−γ
Z
T
Z
T0
+
Z
(1 + t)θ E(t; v) dxdt . (1 + T0 )θ+ω [E(T0 ; v) + E(T0 ; u)]
T
θ−2
(1 + t)
T0
E(t; u)dt +
Z
T
T0
Z
(15)
(1 + t)θ−1 a−1 ((1 + t)2 h2t + h2 ) dxdt,
Proof. Differentiating (1) with respect to t,
c(x)vtt − div(b(x)∇v) + a(x)vt = ht (x, t),
x ∈ Rn , t > 0,
and
1 d
2 dt
Z
(cvt2 + b|∇v|2 ) dx +
Z
avt2 dx =
Z
ht vt dx.
(16)
In order to obtain integrals of E(t; v), we multiply the equation for v with 21 W (t)v
where W will be constructed later. Integration on Rn yields
Z Z
1 d
W a − Wt c 2
W cvt v +
v
dx − W cvt2 dx
2 dt
2
Z 1
Wtt c − Wt a 2
+
W cvt2 + W b|∇u|2 +
v
dx
(17)
2
2
Z
1
=
W ht v dx.
2
HIGHER ORDER ENERGY DECAY RATES
615
We add (16) and (17) and obtain
Z Z
1 d
W a − Wt c 2
2
2
cvt + b|∇v| + W cvt v +
v
dx + (a − W c)vt2 dx
2 dt
2
Z 1
Wtt c − Wt a 2
2
2
+
W cvt + W b|∇v| +
v
dx
(18)
2
2
Z 1
=
ht vt + W ht v dx.
2
Next we estimate the term ht vt with the Cauchy-Scwarz inequality and absorb avt2
on the LHS so we have
Z Z a
1 d
W a − Wt c 2
cvt2 + b|∇v|2 + W cvt v +
v
dx +
− W c vt2 dx
2 dt
2
2
Z 1
Wtt c − Wt a 2
+
W cvt2 + W b|∇v|2 +
v
dx
(19)
2
2
Z 1 2 1
≤
ht + W ht v dx.
2a
2
We will choose W such that
a(x)
,
(20)
(i) W (t) ≤
inf
x∈ supp u(·,t) 2c(x)
(ii) Wtt c − Wt a ≥ 0,
W a − Wt c 2
(iii) 0 ≤ cvt2 + b|∇v|2 + W cvt v +
v ≤ 2cvt2 + b|∇v|2 + C0 W av 2 ,
2
(iv) w1 (1 + t)−ω ≤ W (t) ≤ w2 (1 + t)−ω , 0 < ω < 1, 0 < w1 ≤ w2 ,
where C0 ≥ 0 is some constant. We will point out the role of these conditions in
our proof and at the end we will exhibit a weight function W which satisfies (20).
Next multiply identity (19) by (1 + t)ν , for an arbitrary ν > 0, and integrate on
[T0 , T ]. Use (20)(i) to obtain the following inequality:
Z
Z 1 T
W a − Wt c 2
ν d
2
2
(1 + t)
cvt + b|∇v| + W cvt v +
v
dxdt
2 T0
dt
2
Z Z
1 T
+
(1 + t)ν W cvt2 + W b|∇v|2 dxdt
2 T0
Z TZ
1 2 1
ν
≤
(1 + t)
h + W ht v dxdt.
2a t 2
T0
After performing an integration by parts (in time) in the first integral and by using
the left inequality in (20) (ii) it follows that:
Z Z
1 T
(1 + t)ν W cvt2 + W b|∇v|2 dxdt
2 T0
Z 1
W a − Wt c 2
ν
2
2
≤
(1 + T0 )
cvt + b|∇v| + W cvt v +
v
dx
2
2
t=T0
Z TZ
ν
W a − Wt c 2
+
(1 + t)ν−1 cvt2 + b|∇v|2 + W cvt v +
v
dxdt
2 T0
2
Z TZ
1 2 1
ν
+
(1 + t)
h + W ht v dxdt.
2a t 2
T0
616
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
Our assumptions (20) (ii) on the quadratic form yield a simpler estimate:
Z TZ
(1 + t)ν W cvt2 + W b|∇v|2 dxdt
T0
Z
≤ (1 + T0 )ν
cvt2 + b|∇v|2 + C0 W av 2 dx
t=T0
Z
+ν
+
Z
T
T0
T Z
Z
(1 + t)ν−1 cvt2 + b|∇v|2 + C0 W av 2 dxdt
(1 + t)ν
T0
h2t
+ W ht v
a
dxdt.
Choose T0 sufficiently large, such that
(1 + t)ν W > 2ν(1 + t)ν−1
for t > T0 ,
so by combining similar terms we obtain
Z TZ
(1 + t)ν W cvt2 + W b|∇v|2 dxdt
T0
Z
ν
2
2
2
≤ (1 + T0 )
cvt + b|∇v| + C0 W av dx
+νC0
Z
T
T0
+
T
Z
T0
Z
Z
(1 + t)ν−1 W av 2 dxdt +
Z
T
T0
Z
(21)
t=T0
(1 + t)ν
h2t
dxdt
a
(1 + t)ν W ht v dxdt.
We can estimate the last integral as follows:
Z TZ
(1 + t)ν W ht v dxdt
T0
≤
Z
T
T0
Z 1
1
ν−1
2
ν+1
−1 2
(1 + t)
W av + (1 + t)
W a ht dxdt.
2
2
Hence, (21) implies
Z TZ
(1 + t)ν W cvt2 + W b|∇v|2 dxdt
T0
Z
. (1 + T0 )ν
cvt2 + b|∇v|2 + W av 2 dx
+
Z
T
T0
Z
ν−1
(1 + t)
2
W av dxdt +
Z
T
T0
Z
(22)
t=T0
(1 + t)ν+1 W a−1 h2t dxdt,
where we used the fact that for large times
(1 + t)ν . (1 + t)ν+1 W (t)
since the weight function W satisfies (20) (iv):
w1 (1 + t)−ω ≤ W (t) ≤ w2 (1 + t)−ω ,
HIGHER ORDER ENERGY DECAY RATES
617
where 0 < ω < 1 and w2 ≥ w1 > 0. The inequalities in (20)(iv) and estimate (22)
yield:
Z TZ
(1 + t)ν−ω cvt2 + b|∇v|2 dxdt
T0
Z
ν
2
2
2
. (1 + T0 )
cvt + b|∇v| + W av dx
t=T0
+
Z
T
T0
Z
(1 + t)ν−ω−1 av 2 dxdt +
Z
T
T0
ν−ω−1
Z
(1 + t)ν−ω+1 a−1 h2t dxdt.
2
We can bound the integral of (1 +t)
av using the estimate from Proposition 2
with µ = ν − ω − 1. Then
Z TZ
(1 + t)ν−ω E(t; v) dxdt
T0
. (1 + T0 )ν [E(T0 ; v) + E(T0 ; u)] +
Z
T
(1 + t)ν−ω−2 E(t; u)dt
(23)
T0
+
Z
T
T0
Z
(1 + t)ν−ω−1 a−1 [(1 + t)2 h2t + h2 ] dxdt.
Denote by θ = ν − ω. Then (23) becomes the claimed inequality (15). The proof is
complete provided we show:
Existence of the weight function W . Let
W (t) = w0 (1 + t)−ω ,
(24)
where :
(i) w1 ≤ w0 ≤ w2 with w1 , w2 given by (20)(iv).
(ii) the exponent ω is chosen such that (12) is satisfied, i.e.
2(α − γ)
max 0,
< ω < 1.
2−β−γ
This choice for the weight W satisfies all the constraints of (20) as we show below:
• The inequality (20)(i). By (5) it suffices to show:
a0
w0 (1 + t)−ω ≤
inf 2 (1 + |x|)γ−α .
2c1 |x|≤R+Ct 2−β−γ
For α ≤ γ this is obvious since (1 + t)−ω → 0 as t → ∞, while in the right
hand side we have (1 + |x|)γ−α > 1. For α > γ we need to show
2
a0
w0 (1 + t)−ω ≤ (R + Ct 2−β−γ )γ−α ,
c1
but for sufficiently large times t this holds since ω > 2(α−γ)
2−β−γ by (12).
• (20)(ii) holds since for W given by (24) we have Wtt ≥ 0, Wt < 0 and the
coefficients a and c are positive.
• For the left inequality in (20)(iii) we first complete the square in v and vt , so
we are left to show
2W a − 2Wt c − W 2 c ≥ 0.
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PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
By using (5) and (24) we reduce the problem of proving the above inequality
to showing:
2w0 a0 (1 + t)−ω (1 + |x|)−α + 2w0 ωc0 (1 + t)−ω−1 (1 + |x|)−γ
− w02 c1 (1 + t)−2ω (1 + |x|)−γ ≥ 0. (25)
This can be simplified to
2w0 a0 (1 + |x|)−α+γ + 2w0 ωc0 (1 + t)−1 − w0 c1 (1 + t)−ω ≥ 0.
(26)
The assumption (12) on ω gives us that the above inequality is true for large
times.
We follow a similar approach for the right inequality in (20) (ii); we complete the square so we have:
2
W a − Wt c 2
Wa 2 W2 2
Wv
vt −
−
v + C0
v −
v ≥ 0.
2
2c
c
4
We need to show
(4C0 − 2)W a + 2Wt c − W 2 c ≥ 0
(27)
The constant C0 can be chosen sufficiently large so that we have 4C0 − 2 > 0.
By (5) and (6) in order to prove (27) it is enough to show
(4C0 − 2)w0 a0 (1 + |x|)−α+γ − c1 w02 (1 + t)−ω − 2ωw0 c1 (1 + t)−1 ≥ 0,
which is done exactly as above when proving (26).
• The condition (20)(iv) is obviously satisfied since ω < 1.
Remark 1. The proofs of the above results are obtained for T0 sufficiently large,
but the results hold for all times. In order for the inequalities to hold on intervals
(T0′ , T ) where T0′ < T0 one just needs to add the integrals from T0′ to T0 on both
sides of the inequality.
A simple induction argument applied to (23) yields the decay estimate announced
in Theorem 2.1. The above arguments allow us to also obtain pointwise decay rates
for the energy as stated in Theorem 2.2. Recall that the main claim of Theorem
2.2 was that the solution satisfies the following decay estimate:
"
k
X
k
−θ−2k−1
E(T ; ∂t u) . (1 + T )
(1 + T0 )θ+ω [
E(T0 ; ∂ti u)]
(28)
i=0
+
Z
T
T0
θ
(1 + t) E(t; u)dt +
Z
T
T0
Z
θ+1
(1 + t)
k
X
i=0
2i
(1 + t)
(∂ti h(x, t))2
a(x)
#
dxdt ,
for θ > 0 and some 0 < ω < 1. We present the proof of this inequality below.
Proof. In (14) written for ∂tk u we apply Young’s inequality with appropriate coefficients such that the damping term disappears. This yields
Z TZ
[∂sk h(x, s)]2
E(T ; ∂tk u) ≤ E(t; ∂tk u) +
dxds, 0 < t < T.
(29)
a(x)
t
HIGHER ORDER ENERGY DECAY RATES
619
Hence
Z
T
θ+2k
(1 + t)
T0
−
Z
T
E(t; ∂tk u)dt
(1 + t)θ+2k
T0
Z
t
T
Z
≥
E(T ; ∂tk u)
Z
T
(1 + t)θ+2k dt
(30)
T0
[∂sk h(x, s)]2
dxdsdt.
a(x)
Simplifying the above inequality and using (11) we obtain the desired estimate.
We conclude this section on decay estimates for the time derivatives of the solution with the following
estimate on the decay of the energy lost through the damping
Z
mechanism, i.e.
au2t dx:
Proposition 4. The solution u of (1) satisfies
Z
Z 2
h
2
1/2
dx.
aut dx . (E(t; u)E(t; ut )) +
a
(31)
Proof. We go back to lower order derivatives and estimate
Z
Z
2
aut dx = −cutt ut + div(b∇u)ut − hut dx
which by the Cauchy inequality gives
Z
1/2 Z
1/2
Z
au2t dx ≤
cu2tt dx
cu2t dx
+
Z
1/2 Z
1/2
Z 2
Z
1
h
1
2
+
b|∇u| dx
b|∇ut | dx
dx +
au2t dx.
2
a
2
2
Remark 2. An explicit decay rate can be obtained from (31) by using (28) with
k = 0 and k = 1. By ignoring the effects of the source term, one basically has that
the decay of the energy associated with the damping term is faster than the decay
of the energy. More precisely, for h = 0 one has
Z
E(t; u)
au2t dx .
.
t
This fact is in agreement with Proposition 2 which shows the same relationship for
the decay rates “averaged” in time.
4. Weighted L2 estimates for spatial derivatives. Decay estimates of higher
order spatial derivatives are important for studying nonlinear perturbations of equation (1). Such problems require more regular solutions and naturally lead to W k,p
norms with k ≥ 1 and 2 < p ≤ ∞. Although the latter norms are expected to decay
faster than H k norms, this is not easy to verify. The first difficulty is that Lp norms
are not convenient to estimate by the multiplier method when p 6= 2. Here the standard approach is to use interpolation or embedding, i.e., the Gagliardo-Nirenberg
or Sobolev inequalities. We thus reduce the question to multiplier estimates with
losses of derivatives. The variable coefficients present an additional difficulty as
x-differentiation produces terms that change the dissipative form of (1). A simple
solution is to express x-derivatives in terms of t-derivatives from the equation:
c
h
M u = utt + ut − ,
a
a
620
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
where M u = a−1 div(b∇u). The diffusion phenomenon means M u ≈ ut , so secondorder spatial derivatives are related with the first-order time derivative. Similarly
we can derive an identity for M 2 u. Thus, if one discounts the effects of the variable
coefficients (a = b = c = const.) and of the source term (h = 0) then the decay rate
for the spatial derivatives of the solution is basically given (up to order k = 4) by:
||∂xk u||2L2 . t−1 ||∂xk−1 u||2L2
(for the decay of the ||u||L2 and ||∇u||L2 as a consequence of the energy decay see
Theorem 6.1 here or in [13]). Below we give two sufficient conditions on a, b, and
c to guarantee decay estimates of M u and M 2 u, respectively. Given λ1 , λ2 ∈ [0, 1],
the coefficients satisfy
sup
x∈ supp u(·,t)
sup
x∈ supp u(·,t)
2 #
b(x) c(x) c(x)
+
∇
. (1 + t)λ1 ,
a(x) a(x) a(x) 2
1
c(x)
div b(x)∇
. (1 + t)λ2 .
a(x)
a(x)
"
(32)
(33)
Proposition 5. Let a, b, c ∈ C 2 (Rn ) be coefficients which satisfy (5) and (6), let
h ∈ C 2 ([0, ∞); H 2 (Rn )), and θ > 0.
Let M u = a−1 div(b∇u) and assume that u is a solution of (1).
(i) If (32) holds, then
a(M u)2 . (1 + t)λ1 cu2tt + au2t +
h2
.
a
(ii) If (32) and (33) hold, then
a(M 2 u)2
. (1 + t)3λ1 cu2tttt + (1 + t)λ1 cu2ttt + b|∇utt |2
+(1 + t)λ2 au2tt
2
2
h2t
h
2λ1 htt
+ + (1 + t)
+a M
.
a
a
a
Proof. (i) The first claim follows from (32) and
2
a(M u) ≤ 3a
h2
c2 2
2
u
+
u
+
tt
t
a2
a2
.
(ii) To verify the second claim, we apply M (uv) = uM v + vM u + 2(b/a)∇u · ∇v.
We have the chain of identities
2
M u =
=
c
h
M
utt + ut −
a
a
c
c
b c
h
utt M + M utt + 2 ∇ · ∇utt + M ut − M .
a a
a a
a
HIGHER ORDER ENERGY DECAY RATES
621
Using the expression for M u, we further obtain
c
c c
htt
2
utttt + uttt −
M u = utt M +
a a a
a
b c
c
ht
h
+2 ∇ · ∇utt +
uttt + utt −
−M
a a
a
a
a
c
c2
c
b c
=
utttt + 2 uttt + M + 1 utt + 2 ∇ · ∇utt
a2
a
a
a a
ht
c htt
h
− −
−M .
a
a a
a
The square of M 2 u is bounded by the sum of squares times 7:
4
c
2 c
c2
a(M 2 u)2 ≤ 7a 4 u2tttt + 4 2 u2ttt + M + 1 u2tt
a
a
a
"
2 #
2 b c 2
h2t
c2 h2tt
h
2
+7a 4 2 ∇ |∇utt | + 2 + 2 2 + M
.
a
a
a
a a
a
Finally, we rewrite last estimate to match conditions (32) and (33):
c
c 2
c3
2
2
a(M 2 u)2 .
·
cu
+
·
cu
+
M
+
1
· au2tt
tttt
ttt
a3
a
a
2
b c 2
h2t
c2 h2tt
h
2
+ ∇ · b|∇utt | +
+ 2·
+a M
.
a
a
a
a
a
a
Thus, we have
a(M 2 u)2
. (1 + t)3λ1 cu2tttt + (1 + t)λ1 cu2ttt + b|∇utt |2
2
2
h2t
h
λ2
2
2λ1 htt
+(1 + t) autt +
+ (1 + t)
+a M
.
a
a
a
It is possible to derive expressions for M k u, k ≥ 3, and find conditions on a,
b, and c which yield decay estimates of higher order spatial derivatives. However,
considering k ≤ 2 is sufficient for most applications in Rn when n ≤ 3.
We are now in position to prove Theorem 2.3.
Proof. (i) Recall the estimate of u in Proposition 2 where µ is replaced by θ + 1:
Z TZ
Z T
θ+1
2
θ+1
(1 + t) a(x)ut dxdt . (1 + T0 ) E(T0 ; u) +
(1 + t)θ E(t; u)dt
T0
T0
+
Z
T
T0
Z
h2 (x, t)
(1 + t)θ+1
dxdt.
a(x)
We also need Proposition 2.1 with k = 1:
Z T
Z
1
X
(1 + t)θ+2 E(t; ut ) dt . (1 + T0 )ν [
E(T0 ; ∂ti u)] +
T0
+
T
T0
Z
(1 + t)θ+1
(1 + t)θ E(t; u)dt
T0
i=0
Z
T
1
X
i=0
(1 + t)2i (∂ti h)2
dxdt.
a
622
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
Since λ1 ≤ 1, the claim follows from Proposition 5(i).
(ii) A simple adaptation of Proposition 2 to ut yields
Z TZ
Z
θ+3
2
θ+3
(1 + t) autt dxdt . (1 + T0 ) E(T0 ; ut ) +
T0
T
(1 + t)θ+2 E(t; ut )dt
T0
+
Z
T
T0
Z
(1 + t)θ+3
h2t
dxdt.
a
Applying Proposition 2.1 to the integral of E(t; ut ), we obtain
Z TZ
Z T
1
X
(1 + t)θ+3 au2tt dxdt . (1 + T0 )ν [
E(T0 ; ∂ti u)] +
(1 + t)θ E(t; u)dt
T0
T0
i=0
+
Z
T
T0
Z
(1 + t)θ+1
1
X
(1 + t)2i (∂ i h)2
t
i=0
a
dxdt.
It is clear from Proposition 5(ii) and λ1 , λ2 ≤ 1 that au2tt is the main term in the
upper bound of M 2 u. We readily estimate the remaining terms by Proposition 2.1
with k = 2, 3.
Remark 3. It is straightforward to establish pointwise estimates of ka1/2 M ukL2
and ka1/2 M 2 ukL2 for large t. Such results will be presented elsewhere.
5. L∞ estimates in dimension n = 3. The most important applications of
Propositions 3.4 and 4.2 concern L∞ decay estimates. We give an example in
n = 3, although we can treat all n ≤ 7 due to the embedding H 4 (Rn ) ⊂ L∞ (Rn )
for 4 > n/2. It is important to mention that the standard Sobolev estimate ku||L∞ .
(ku||L2 + k∆u||L2 ) is not suitable, since the L∞ norm is expected to decay faster
than the L2 norm.
Proposition 6. Assume that n = 3 and a, b are C 1 -functions. If
k(∇ ln b)ukL2
.
kb1/2 ∇ukL2 ,
k∇(a1/2 b−1 u)kL2
.
kb1/2 ∇ukL2 ,
k∆ukL2
.
ka−1/2 div(b∇u)kL2 ,
for every sufficiently regular u, then
kuk2L∞ . ka−1/2 div(b∇u)kL2 kb1/2 ∇ukL2 .
(34)
Hence kukL∞ is bounded in terms of E(t; u) and E(t; ut ) whenever u is a solution
of problem (1) under the assumptions of Proposition 5.
Proof. We set
a−1/2 div(b∇u) = f
and multiply with u to obtain
−∆u2 = −2|∇u|2 − 2
a1/2
∇b
fu + 2
· u∇u.
b
b
The well-known formula
u2 (x) =
1
4π
Z
−∆u2 (y)
dy,
|x − y|
HIGHER ORDER ENERGY DECAY RATES
623
which is valid for compactly supported u, yields the estimate
Z
Z
|a1/2 (y)f (y)u(y)|
|u(y)||∇ ln b(y) · ∇u(y)|
2
dy +
dy.
u (x) .
b(y)|x − y|
|x − y|
It follows from the Cauchy inequality that
Z
|a1/2 (y)f (y)u(y)|
dy
b(y)|x − y|
Z
1/2
−2
−2 2
2
≤ kf kL
a(y)b (y)|x − y| u (y)dy
.
ka−1/2 div(b∇u)kL2
Z
|∇(a1/2 (y)b−1 (y)u(y))|2 dy
1/2
(35)
,
where the second factor comes from the Hardy inequality
Z
Z
−2 2
|x − y| f (y) dy ≤ 4 |∇f (y)|2 dy, f ∈ H 1 (R3 ).
The assumptions on a and b give
Z
Z
|∇(a1/2 (y)b−1 (y)u(y))|2 dy . b(y)|∇u(y)|2 dy,
so the final estimate becomes
Z
|f (y)u(y)|
dy . ka−1/2 div(b∇u)kL2 kb1/2 ∇ukL2 .
b(y)|x − y|
(36)
The second integral in (35) admits similar estimates. From the Cauchy and
Hardy inequalities, we have
Z
|u(y)||∇ ln b · ∇u(y)|
dy . k(∇ ln b)ukL2 k∆ukL2
|x − y|
. kb1/2 ∇ukL2 ka−1/2 div(b∇u)kL2 .
(37)
Adding estimates (36) and (37) completes the proof.
It is easy to find more explicit conditions on a and b which will allow us to use
Proposition 5.1. This part is about integration by parts and Hardy’s inequality.
Corollary 1. Assume that a ∈ C 1 and b ∈ C 2 are such that
(i)
(ii)
a(x) . 1,
|∇(a
1/2
(x)b
1 . b(x),
−1
(x))| + |∇ ln b(x)| . (1 + |x|)−1 ,
and the matrix with entries
δij
∆b − bxi xj , i, j = 1, . . . , n
2
is non-negative definite. Then the conditions on a and b in Proposition 6 hold;
hence the L∞ estimate (34) also holds.
(iii)
Proof. Conditions (i), (ii), and Hardy’s inequality readily show that
k(∇ ln b)ukL2 . kb1/2 ∇ukL2 ,
k∇(a1/2 b−1 u)kL2 . kb1/2 ∇ukL2
for sufficiently regular u. Thus, it remains to verify the condition about k∆ukL2 .
Let a−1/2 div(b∇u) = f and multiply this equation with ∆u. Then
Z
Z
Z
b(∆u)2 dx = a1/2 f ∆u dx − (∇b · ∇u)∆u dx.
624
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
Integrating by parts in the second term on the right side, we obtain
Z
Z
n Z
X
1
(∇b · ∇u)∆u dx =
∆b|∇u|2 dx −
bxi xj uxi uxj dx.
2
i,j=1
Thus,
Z
2
b(∆u) dx =
Z
1
∆b|∇u|2 dx
a f ∆u dx −
2
n Z
X
+
bxi xj uxi uxj dx.
Z
1/2
i,j=1
We have from assumption (iii) that
Z
Z
b(∆u)2 dx ≤ a1/2 f ∆u dx.
Now assumption (i) and Cauchy’s inequality imply
Z
Z
(∆u)2 dx . b(∆u)2 dx . kf kL2 k∆ukL2 ,
so k∆ukL2 . ka−1/2 div(b∇u)kL2 .
6. Application in the homogeneous case c = 1, h = 0. In this section we will
derive explicit decay estimates for (1) when c(x) = 1 and h(x, t) = 0, i.e. for
utt − div(b(x)∇u) + a(x)ut = 0,
x ∈ Rn , t > 0,
(38)
as an application of the results proven here and in [13]. To state these results we
introduce the following:
Hypothesis A. Let a and b satisfy the growth conditions listed in (5). Then
there exists a subsolution A(x) which satisfies
div(b(x)∇A(x)) ≥ a(x),
x ∈ Rn ,
(39)
and has the following properties:
(a1)
(a2)
(a3)
A(x) ≥ 0 for all x,
2−α−β
A(x) = O(|x|
) for large |x|,
a(x)A(x)
µ := lim inf
> 0.
x→∞ b(x)|∇A(x)|2
(40)
(41)
(42)
As we will see below in Theorem 6.1, the quantity µ gives the decay of the
weighted L2 norm of u, while µ + 1 gives the decay of the energy of u. In several
cases we can construct explicit subsolutions A which satisfy (a1)-(a3). Some of
these cases are listed below:
1. a, b are radial functions. In this case one can solve (40) with equality to find
an exact solution A, and the value of µ is calculated by the formula
µ=
n−α
.
2−α−β
HIGHER ORDER ENERGY DECAY RATES
625
2. a, b are “separable” with the same angular component, i.e. for r = |x| and
ω = xr ∈ S n−1 , where S is the unit sphere in Rn , there exist functions a1 , b1 ,
and ζ such that
a(x) = a(r, ω) = a1 (r)ζ(ω),
b(x) = b(r, ω) = b1 (r)ζ(ω).
3. b is radial and a is arbitrary.
4. a, b are arbitrary (no radial symmetry), α + β < 2 and
x
∇b(x) ·
≥ b0 β(1 + |x|)β−1 .
|x|
For a full discussion of these special situations see Section 7 in [13].
Theorem 6.1. Assume that a and b satisfy (5) with α, β such that
α < 1,
0 ≤ β < 2,
2α + β ≤ 2.
(43)
Also, assume that Hypothesis A holds. Then for every δ > 0 the solution of (38)
satisfies
Z
A(x)
e(µ−δ) t a(x)u2 dx ≤ Cδ (k∇u0 k2L2 + ku1 k2L2 )tδ−µ ,
Z
A(x)
e(µ−δ) t (u2t + b(x)|∇u|2 ) dx ≤ Cδ (k∇u0 k2L2 + ku1 k2L2 )tδ−µ−1
for all t ≥ 1. The constant Cδ depends also on R, a, b, and n.
An immediate consequence of the above theorem which follows by (a1) is
Corollary 2. Under the assumptions of Theorem 6.1, we have
Z
a(x)u2 dx ≤ Cδ (k∇u0 k2L2 + ku1 k2L2 )tδ−µ ,
Z
E(t; u) = u2t + b(x)|∇u|2 dx ≤ Cδ (k∇u0 k2L2 + ku1 k2L2 )tδ−µ−1 .
(44)
(45)
These results in conjunction with the decay estimates presented in Sections 3–5
yield the following:
Theorem 6.2. Assume that a, b satisfy (5) and (43). Then for every δ > 0 and
T0 > 0 sufficiently large the following inequalities hold:
(i) Weighted (in time) L2 energy decay:
Z
T
T0
(1 + t)θ+2k E(t; ∂tk u) dt . (1 + T0 )ν
(ii) Pointwise (in time) energy decay:
"
k
X
E(T0 ; ∂ti u) +
i=0
E(T ; ∂tk u) . (1 + T )−θ−2k−1 (1 + T0 )ν
k
X
T θ+δ−µ − T0θ+δ−µ
. (46)
θ+δ−µ
E(T0 ; ∂ti u)
i=0
#
T θ+δ−µ − T0θ+δ−µ
+
. (47)
θ+δ−µ
626
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
(iii) Weighted kinetic energy decay (the energy associated with the damping):
Z
au2t dx . [E(t; u)E(t; ut )]1/2
(48)
"
#
k
X
T θ+δ−µ − T0θ+δ−µ
E(T0 ; ∂ti u) +
. (1 + T )−θ−2 (1 + T0 )ν
.
θ+δ−µ
i=0
(iv) Decay of spatial derivatives (recall that M u = a−1 [div(b∇u)]):
Z TZ
(1 + t)θ+1 a(M u)2 dxdt
(49)
T0
.
θ+1
(1 + T0 )
Z
T
1
X
i=0
#
E(T0 ; ∂ti u)
+
T θ+δ−µ − T0θ+δ−µ
,
θ+δ−µ
(1 + t)θ+3−λ2 a(M 2 u)2 dxdt
" 3
#
X
T θ+δ−µ − T0θ+δ−µ
θ+3
i
.
(1 + T0 )
E(T0 ; ∂t u) +
θ+δ−µ
i=0
T0
.
Z
"
(50)
Remark 4. Note from the above decay estimate (47) that the k-th order energy
has a polynomial decay of order µ + 1 + 2k − δ, exactly 2k units below the decay
of the first order energy as given by (44). The damping term has the decay rate
µ + 2 − δ, as it follows from (48), which is 2 units below the decay of the L2 norm of
u given by (45). Finally, the average decay rates of second and fourth order spatial
derivatives resemble those of ut and utt , respectively. Here (1 + t)λ2 accounts for
possibly fast oscillations in b(x)/a(x) as |x| → ∞.
Appendix A. Bounds on the support of
non-homogeneous hyperbolic equation with damping
solutions. Recall
x ∈ Rn , t > 0,
c(x)utt − div(b(x)∇u) + a(x)ut = h,
the
(51)
where the coefficients a, b, and c satisfy conditions (5) in the introduction. The
propagation speed is determined by the ratio c(x)/b(x). We assume the following:
s
c(x)
1
′
There exists q ∈ C , 0 < q (|x|) ≤
, x ∈ Rn .
(52)
b(x)
A better bound on the propagation speed is given in terms of q satisfying
s
c(x)
|∇q(x)| =
,
b(x)
see Evans [4], but this equation is globally solvable only in special cases. Fortunately
the corresponding inequality is sufficient for propagation speed estimates and solvable in most cases. It is also convenient to have bounds depending on |x|, so we
look for radial solutions q.
We consider h ∈ C((0, ∞), L2 (Rn )) and
h(x, t) = 0 if q(|x|) > t + q(R).
1
n
(53)
1
2
n
There exist a unique solution u ∈ C((0, ∞), H (R )) ∩ C ((0, ∞), L (R )) whose
HIGHER ORDER ENERGY DECAY RATES
627
support is described below.
Proposition 7. Let u be a solution of (51) with data (u, ut )|t=0 = (u0 , u1 ), such
that (u0 , u1 ) ∈ H 1 (Rn )× L2 (Rn ). If (52) holds, then u0 (x) = u1 (x) = 0 for |x| > R
implies u(x, t) = 0 for q(|x|) > t + q(R).
Applying the above result, we can find more explicit bounds when c(x)/b(x) is
not very small.
Corollary 3. Let u be a solution of (51) with data (u, ut )|t=0 = (u0 , u1 ), such that
(u0 , u1 ) ∈ H 1 (Rn ) × L2 (Rn ). Assume that b and c satisfy (5) with β + γ < 2. There
exists q satisfying (52) and
q(|x|) ≥ q0 |x|1−(β+γ)/2 ,
x ∈ Rn ,
with q0 > 0. Hence, u0 (x) = u1 (x) = 0 for |x| > R implies u(x, t) = 0 for
2/(2−β−γ)
t + q(R)
|x| >
.
q0
Proof of Proposition 7. We can assume that (u0 , u1 ) and h are more regular, so u ∈
C 1 ((0, ∞), H 1 (Rn )) ∩ C 2 ((0, ∞), L2 (Rn )); the general case follows from a standard
approximation argument. Multiplying (51) with ut , we obtain
1
c(x)u2t + b(x)|∇u|2 t − div(b(x)ut ∇u) + a(x)u2t = hut .
2
This identity will be integrated over the exterior of a suitable truncated cone.
Choose q satisfying (52) and consider the conic surface
M (T ) = {(x, t) : T > t > 0 and q(|x|) = t + q(R)},
which is the lateral boundary of the space-time region
C(T ) = {(x, t) : T > t > 0 and q(|x|) > t + q(R)}.
The outer normal to M (T ) at (x, t) is given by the (n + 1)-vector
x
1
′
p
−q (|x|) , 1 .
|x|
1 + [q ′ (|x|)]2
We integrate the energy identity over C(T ) and apply the divergence theorem:
Z
Z
1
2
2
c(x)ut + b(x)|∇u| dx
+
a(x)u2t dxdt
2 q(|x|)>t+q(R)
C(T )
t=T
Z
c(x) 2 b(x)
x
dS
2
′
+
ut +
|∇u| + b(x)q (|x|)ut
· ∇u p
2
2
|x|
1 + [q ′ (|x|)]2
M(T )
Z
Z
1
2
2
=
c(x)ut + b(x)|∇u| dx
+
hut dxdt,
2 q(|x|)>t+q(R)
C(T )
t=0
for all T > 0. The integral over M (T ) is non-negative, since
c(x) 2 b(x)
x
ut +
|∇u|2 + b(x)q ′ (|x|)ut
· ∇u ≥ 0
2
2
|x|
p
for all ut and ∇u whenever (bq ′ )2 ≤ cb, or 0 < q ′ ≤ c/b. Thus,
628
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
≤
Z
Z
1
c(x)u2T + b(x)|∇u|2 dx +
a(x)u2t dxdt
2 q(|x|)>T +q(R)
C(T )
Z
Z
1
c(x)u21 + b(x)|∇u0 |2 dx +
hut dxdt.
2 q(|x|)>q(R)
C(T )
Notice that |hut | ≤ au2t /2 + h2 /(2a). Hence,
1
2
≤
1
2
Z
q(|x|)>T +q(R)
Z
q(|x|)>q(R)
c(x)u2T + b(x)|∇u|2 dx
c(x)u21
2
+ b(x)|∇u0 |
dx +
Z
C(T )
h2
dxdt.
2a
The integrands on the right side are 0 due to our assumptions on u0 , u1 , and h.
This shows that u(x, T ) = 0 if q(|x|) > T + q(R).
Proof of Corollary
3. We already know that u(x, t) = 0 if q(|x|) > t + q(R), where
p
q ′ (|x|) ≤ c(x)/b(x). It is sufficient to find q, such that
r
c0
′
q (|x|) =
(1 + |x|)−(β+γ)/2 , q(0) = 0,
b1
where the constants are given in (5). Solving for q, we obtain
r
c0
2
q(|x|) =
(1 + |x|)1−(β+γ)/2 ≥ q0 |x|1−(β+γ)/2 ,
b1 2 − β − γ
for some q0 > 0.
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Received October 2008; revised February 2009.
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E-mail address: [email protected]