Decay estimates for a kind of linear wave equations
We consider one kind of dissipative wave equations with exponential speed of propagation. An arbitrary power decay rate for the $L^2$-norm and energy is obtained by using the multiplier method.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507606971318272 |
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| author | Li, Ming Li, Xinfu Лі, Мін Лі, Сіньфу |
| author_facet | Li, Ming Li, Xinfu Лі, Мін Лі, Сіньфу |
| author_sort | Li, Ming |
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| datestamp_date | 2019-12-05T09:25:34Z |
| description | We consider one kind of dissipative wave equations with exponential speed of propagation. An arbitrary power decay rate
for the $L^2$-norm and energy is obtained by using the multiplier method. |
| first_indexed | 2026-03-24T02:12:00Z |
| format | Article |
| fulltext |
UDC 517.9
Xinfu Li (School Sci., Tianjin Univ. Commerce, China),
Ming Li (Dep. Inform. Technology, Tianjin Trust Co. Ltd., China)
DECAY ESTIMATES FOR A KIND OF LINEAR WAVE EQUATIONS*
ОЦIНКИ ЗАТУХАННЯ ДЛЯ ОДНОГУ ТИПУ ХВИЛЬОВИХ РIВНЯНЬ
We consider one kind of dissipative wave equations with exponential speed of propagation. An arbitrary power decay rate
for the L2 -norm and energy is obtained by using the multiplier method.
Розглянуто один тип дисипативних хвильових рiвнянь з експоненцiальною швидкiстю поширення. Довiльний сте-
пеневий закон затухання отримано для L2 -норми та енергiї за допомогою методу множникiв.
1. Introduction. Consider the following damped wave equation:
utt - \mathrm{d}\mathrm{i}\mathrm{v}(b(x)\nabla u) + a(x)ut = 0, (x, t) \in \BbbR n \times (0,\infty ),
u(x, 0) = u0(x), ut(x, 0) = u1(x), x \in \BbbR n,
(1.1)
where a(x) \in C0(\BbbR n), b(x) \in C1(\BbbR n) are positive functions, and (u0(x), u1(x)) \in H1(\BbbR n) \times
\times L2(\BbbR n) have compact support
u0(x) = 0 \mathrm{a}\mathrm{n}\mathrm{d} u1(x) = 0 \mathrm{f}\mathrm{o}\mathrm{r} | x| > R.
Such a system appears in models for traveling waves in a nonhomogeneous gas with damping that
changes with the position (see [1] and the references therein). It is well known that (1.1) admits a
unique weak solution u satisfying u \in C((0,\infty ), H1(\BbbR n)) and ut \in C((0,\infty ), L2(\BbbR n)) (see [2]).
The main quantities of interest are the L2-norm and energy associated with u. In fact, the energy
arises after multiplying equation (1.1) by ut and applying the divergence theorem on \BbbR n :
1
2
d
dt
\int
\BbbR n
\bigl(
u2t + b(x)| \nabla u| 2
\bigr)
dx+
\int
\BbbR n
a(x)u2tdx = 0.
Hence the energy is a nonincreasing function of t. The important question is whether the energy
decays as t \rightarrow \infty and if so, how fast it decays. This problem has been studied intensively when b
is constant (see [3 – 15] and the references therein). When b is space dependent, the problem (1.1)
does not have constant speed of propagation anymore (see Radu et al. [1]). In [1], they considered
the case
b0(1 + | x| )\beta \leq b(x) \leq b1(1 + | x| )\beta , a0(1 + | x| ) - \alpha \leq a(x) \leq a1(1 + | x| ) - \alpha (1.2)
with a0, a1, b0, b1 > 0 and \alpha < 1, 0 \leq \beta < 2, 2\alpha + \beta \leq 2. Using the multiplier method, they
obtained the following decay estimates of L2-norm and energy:\int
\BbbR n
u2dx \leq C\delta
\bigl(
\| \nabla u0(x)\| 2L2 + \| u1(x)\| 2L2
\bigr) \Biggl\{ t\delta - n - 2\alpha
2 - \alpha - \beta , \alpha > 0,
t
\delta - n - \alpha
2 - \alpha - \beta , \alpha \leq 0,
* This work was supported by Tianjin Municipal Education Commission with the Grant no. 2017KJ173 ”Qualitative
studies of solutions for two kinds of nonlocal elliptic equations”.
c\bigcirc XINFU LI, MING LI, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7 1001
1002 XINFU LI, MING LI\int
\BbbR n
\bigl(
u2t + b(x)| \nabla u| 2
\bigr)
dx \leq C\delta
\bigl(
\| \nabla u0(x)\| 2L2 + \| u1(x)\| 2L2
\bigr)
t
\delta - n - \alpha
2 - \alpha - \beta
- 1
.
Note that the above decay rates go to infinity when \beta \rightarrow 2 - and \alpha \rightarrow 0+. But one can not propose
limit to go to the case \beta = 2 and \alpha = 0. This is because for the case \beta = 2 and \alpha = 0, the
problem has exponential speed of propagation, while power speed of propagation for the case (1.2),
see Lemma 2.1. In this paper, we consider the case \beta = 2 and \alpha = 0. By modifying the method
in [1], we prove arbitrary power decay rate of the L2-norm and energy. For simplicity, we only
consider the radial case
b(x) = b0(1 + | x| )2, a(x) = 1. (1.3)
And the results in this paper can be generalized to the general case
b0(1 + | x| )2 \leq b(x) \leq b1(1 + | x| )2, a0 \leq a(x) \leq a1.
This paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we
prove our main results.
2. Preliminaries. First, we state a result about the support of solutions for a wave equation with
variable coefficients (see [1]).
Lemma 2.1. Assume that b(x) = b0(1 + | x| )\beta and that u(x, t) satisfies (1.1) with u0(x) =
= u1(x) = 0 for | x| > R. Then u(x, t) = 0 whenever | x| > Rt, where
Rt = (1 +R)et
\surd
b0 - 1 \mathrm{f}\mathrm{o}\mathrm{r} \beta = 2, (2.1)
Rt =
\Bigl(
(1 +R)(2 - \beta )/2 + t
\sqrt{}
b0
\Bigr) 2/(2 - \beta )
\mathrm{f}\mathrm{o}\mathrm{r} \beta < 2. (2.2)
Proof. (2.2) is a direct result of Proposition 2.1 in [1] for the radial case b(x). (2.1) can be
obtained in a similar way. We only point out the differences of the proof. Following the proof of
Proposition 2.1 in [1], for \beta = 2, we obtain
q(r) = -
r\int
r0
1\sqrt{}
b0(1 + s)2
ds =
1\surd
b0
(\mathrm{l}\mathrm{n}(1 + r0) - \mathrm{l}\mathrm{n}(1 + r)) .
Hence q - 1(y) = (1 + r0)e
- y
\surd
b0 - 1 and
Rt = q - 1(q(R) - t) = (1 +R)et
\surd
b0 - 1.
Lemma 2.2. Under assumptions (1.3) there exists a solution A(x) which satisfies
\mathrm{d}\mathrm{i}\mathrm{v}(b(x)\nabla A(x)) = a(x) = 1, x \in \BbbR n, (2.3)
with the following properties:
(\mathrm{a}1) A(x) \geq 0, x \in \BbbR n;
(\mathrm{a}2) A(x) = O(\mathrm{l}\mathrm{n}(1 + | x| )) for large | x| ;
(\mathrm{a}3) \mathrm{l}\mathrm{i}\mathrm{m}| x| \rightarrow \infty
A(x)
b(x)| \nabla A(x)| 2
= +\infty .
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
DECAY ESTIMATES FOR A KIND OF LINEAR WAVE EQUATIONS 1003
Proof. As in [1], we obtain a radial solution for (2.3). In this case (2.3) becomes
b(r)
\biggl(
Arr +
n - 1
r
Ar
\biggr)
+ brAr = 1, r = | x| .
Multiplying the above equation by rn - 1, we have\bigl(
rn - 1b(r)Ar
\bigr)
r
= rn - 1.
After integration, we obtain
Ar =
r1 - n
b(r)
\left( C0 +
r\int
0
\rho n - 1d\rho
\right) .
Integrate again, and then we find a solution in the form
A(r) = C1 +
r\int
0
s1 - n
b(s)
\left( C0 +
s\int
0
\rho n - 1d\rho
\right) ds.
By letting C0 = C1 = 0, we obtain the solution which satisfies A(0) = Ar(0) = 0. By condition
(1.3), we get
A(r) =
1
nb0
\biggl(
\mathrm{l}\mathrm{n}(1 + r) +
1
1 + r
- 1
\biggr)
, A\prime (r) =
r
nb0(1 + r)2
. (2.4)
It is easy to see that (\mathrm{a}1) – (\mathrm{a}3) are satisfied by such choice of A(x) = A(| x| ).
Lemma 2.3. Under assumptions of Lemmas 2.1 and 2.2. Define
G(t) = \mathrm{s}\mathrm{u}\mathrm{p}\{ A(x)| x \in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}u(\cdot , t)\} .
Then G(t) \leq G0t, where G0 > 0 is a constant.
Proof. By (2.1), the support of u is contained in the set
| x| \leq (1 +R)et
\surd
b0 - 1.
Therefore A(x) defined in (2.4) satisfies A(x) \leq G0t.
3. Main results. In this section, we use the multiplier method to obtain weighted L2 and energy
estimates for the solution to (1.1) under conditions (1.3).
As in [1], let u = \varphi \^u, then we obtain the equation for \^u:
\^utt - \^b1\Delta \^u - \^b2 \cdot \nabla \^u+ \^a1\^ut + \^a2\^u = 0, (3.1)
where \^b1 = b, \^b2 = \nabla b+ 2b\varphi - 1\nabla \varphi , \^a1 = 1 + 2\varphi - 1\varphi t and \^a2 = \varphi - 1(\varphi tt - \mathrm{d}\mathrm{i}\mathrm{v}(b\nabla \varphi ) + \varphi t).
Multiply equation (3.1) by \varphi \^u + \theta \^ut and integrate on \BbbR n using the divergence theorem. The
boundary terms vanish since u(x, t) has compact support with respect to x. Then one has the
following lemma [1].
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
1004 XINFU LI, MING LI
Lemma 3.1. Let u be a solution of (1.1). Assume that \varphi and \theta > 0 are C2-functions. Then \^u
satisfies
d
dt
E(\^ut,\nabla \^u, \^u) + F (\^ut,\nabla \^u) +G(\^u) = 0,
where
E(\^ut,\nabla \^u, \^u) =
1
2
\int \bigl(
\theta
\bigl(
\^u2t + b| \nabla \^u| 2
\bigr)
+ 2\varphi \^ut\^u+ (\^a2\theta + \varphi t + \varphi ) \^u2
\bigr)
dx,
F (\^ut,\nabla \^u) =
1
2
\int \bigl(
- \theta t + 2
\bigl(
1 + 2\varphi - 1\varphi t
\bigr)
\theta - 2\varphi
\bigr)
\^u2tdx+
+
\int
b
\bigl(
\nabla \theta - 2\theta \varphi - 1\nabla \varphi
\bigr)
\cdot \^ut\nabla \^udx+
+
1
2
\int
b( - \theta t + 2\varphi )| \nabla \^u| 2dx,
G(\^u) =
1
2
\int
(\^a2\varphi - (\^a2\theta )t) \^u
2dx.
In the following, we choose appropriate functions \varphi and \theta to estimate E, F and G. Given any
large m > 0, define
\sigma (x) = 2mA(x) + \sigma 0, (3.2)
where A(x) is defined in (2.4), and \sigma 0 = \sigma 0(m) > 0 is chosen such that \sigma (x) - b(x)| \nabla \sigma (x)| 2 \geq
\geq 0, x \in \BbbR n. Then we have \mathrm{d}\mathrm{i}\mathrm{v}(b(x)\nabla \sigma (x)) \geq 2m. Set
\varphi (x, t) = t - me -
\sigma (x)
t , \theta (x, t) =
1
2
\biggl(
2
t
+
\sigma (x)
t2
\biggr) - 1
\varphi (x, t). (3.3)
By direct calculation, we have
\varphi t =
\biggl(
- m
t
+
\sigma (x)
t2
\biggr)
\varphi , \varphi tt =
\biggl(
m
t2
- 2\sigma (x)
t3
\biggr)
\varphi +
\biggl(
- m
t
+
\sigma (x)
t2
\biggr) 2
\varphi ,
\nabla \varphi = - \nabla \sigma (x)
t
\varphi , \mathrm{d}\mathrm{i}\mathrm{v}(b(x)\nabla \varphi ) = - 1
t
\mathrm{d}\mathrm{i}\mathrm{v}(b(x)\nabla \sigma )\varphi +
b(x)| \nabla \sigma | 2
t2
\varphi ,
\mathrm{l}\mathrm{n} \theta = \mathrm{l}\mathrm{n}
1
2
- \mathrm{l}\mathrm{n}
\biggl(
2
t
+
\sigma (x)
t2
\biggr)
+ \mathrm{l}\mathrm{n}\varphi ,
\theta t
\theta
=
\biggl(
2
t
+
\sigma (x)
t2
\biggr) - 1\biggl( 2
t2
+
2\sigma (x)
t3
\biggr)
+
\biggl(
- m
t
+
\sigma (x)
t2
\biggr)
,
\nabla \theta
\theta
= -
\biggl(
2
t
+
\sigma (x)
t2
\biggr) - 1 \nabla \sigma (x)
t2
- \nabla \sigma (x)
t
.
Lemma 3.2. Let \varphi and \theta be defined in (3.3), then there exists T = T (m) > 0 such that F \geq 0
and G \geq 0 for t \geq T.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
DECAY ESTIMATES FOR A KIND OF LINEAR WAVE EQUATIONS 1005
Proof. First, we use (3.3) and Lemma 2.3 to calculate the coefficient \^a2,
\^a2 =
\mathrm{d}\mathrm{i}\mathrm{v}(b(x)\nabla \sigma ) - m
t
+
\sigma (x) - b(x)| \nabla \sigma | 2
t2
+
m
t2
- 2\sigma (x)
t3
+
\biggl(
- m
t
+
\sigma (x)
t2
\biggr) 2
\geq
\geq m
t
- 2\sigma (x)
t3
\geq m
t
- C(m)t
t3
\geq 0
for sufficiently large t, where C(m) > 0 is a constant depending on m. In a similar way, we can
show that (\^a2)t < 0 for sufficiently large t.
Next, we calculate - \theta t + \varphi . By (3.3), we obtain
- \theta t + \varphi \geq \theta
\biggl(
m
t
- \sigma (x)
t2
- 2
t
+ 2
\biggl(
2
t
+
\sigma (x)
t2
\biggr) \biggr)
=
= \theta
\biggl(
m+ 2
t
+
\sigma (x)
t2
\biggr)
.
By the above arguments, and considering the definition of G, we have G \geq 0 for sufficiently large
t. In order to show F \geq 0, we argue as follows. By (3.3) and Lemma 2.3, we get
- \theta t + 2
\bigl(
1 + 2\varphi - 1\varphi t
\bigr)
\theta - 2\varphi \geq \theta
\biggl(
- 3m - 10
t
- \sigma (x)
t2
+ 2
\biggr)
\geq
\geq \theta
\biggl(
- 3m - 10
t
- C(m)t
t2
+ 2
\biggr)
\geq \theta
for sufficiently large t. On the other hand,
b
\bigm| \bigm| \nabla \theta - 2\theta \varphi - 1\nabla \varphi
\bigm| \bigm| 2 = \theta 2
b| \nabla \sigma | 2
t2
\bigm| \bigm| \bigm| \bigm| \bigm|
\biggl(
2
t
+
\sigma (x)
t2
\biggr) - 1 1
t
- 1
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \theta 2
b| \nabla \sigma | 2
t2
.
Since \sigma (x) \geq b(x)| \nabla \sigma | 2, we obtain\bigl(
- \theta t + 2
\bigl(
1 + 2\varphi - 1\varphi t
\bigr)
\theta - 2\varphi
\bigr)
( - \theta t + 2\varphi ) \geq b
\bigm| \bigm| \nabla \theta - 2\theta \varphi - 1\nabla \varphi
\bigm| \bigm| 2 ,
which implies that F \geq 0 for sufficiently large t.
By Lemmas 3.1 and 3.2, we have
E(\^ut,\nabla \^u, \^u) \leq E(\^ut,\nabla \^u, \^u)| t=T
df
= ET , t \geq T. (3.4)
Considering the definition of E, we get
1
2
d
dt
\int
\varphi \^u2dx+
1
2
\int
\varphi \^u2dx \leq ET .
The above inequality implies
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
1006 XINFU LI, MING LI\int
\varphi \^u2dx \leq HT + 2ET , t \geq T, (3.5)
where HT =
\int
\varphi \^u2dx| t=T . Using | 2\varphi \^ut\^u| \leq
1
2
\theta \^u2t + 2\theta - 1\varphi 2\^u2, we obtain from (3.4)\int
\theta
\bigl(
\^u2t + b| \nabla \^u| 2
\bigr)
dx \leq 4ET + 4
\int \bigl(
- \^a2\theta - \varphi t - \varphi + 2\theta - 1\varphi 2
\bigr)
\^u2dx.
By the choices of \theta , \varphi , \sigma (x) and Lemma 2.3, we have
- \^a2\theta - \varphi t - \varphi + 2\theta - 1\varphi 2 \leq - \varphi t - \varphi + 2\theta - 1\varphi 2 =
= \varphi
\biggl(
8 +m
t
+
3\sigma (x)
t2
- 1
\biggr)
\leq
\leq \varphi
\biggl(
8 +m
t
+
C(m)t
t2
- 1
\biggr)
\leq 0
for sufficiently large t. Therefore, we obtain the following lemma.
Lemma 3.3. Let \varphi and \theta be defined in (3.3), then we have\int
\theta
\bigl(
\^u2t + b| \nabla \^u| 2
\bigr)
dx \leq 4ET1 , t \geq T1,
where T1 = T1(m) > T (m) is a constant.
Note that
\^u2t =
\bigl(
- \varphi - 2\varphi tu+ \varphi - 1ut
\bigr) 2 \geq 1
2
\varphi - 2u2t - \varphi - 4\varphi 2
tu
2,
| \nabla \^u| 2 =
\bigm| \bigm| - \varphi - 2\nabla \varphi u+ \varphi - 1\nabla u
\bigm| \bigm| 2 \geq 1
2
\varphi - 2| \nabla u| 2 - \varphi - 4| \nabla \varphi | 2u2.
Combined with the estimate in Lemma 3.3, these inequalities imply
1
2
\int
\theta \varphi - 2
\bigl(
u2t + b| \nabla u| 2
\bigr)
dx \leq 4ET1 +
\int
\theta \varphi - 4
\bigl(
\varphi 2
t + b| \nabla \varphi | 2
\bigr)
u2dx.
By the choices of \varphi , \theta and Lemma 2.3, we get
\theta \varphi - 4
\bigl(
\varphi 2
t + b| \nabla \varphi | 2
\bigr)
=
= \varphi - 1 1
2
\biggl(
2
t
+
\sigma (x)
t2
\biggr) - 1
\Biggl( \biggl(
- m
t
+
\sigma (x)
t2
\biggr) 2
+
b(x)| \nabla \sigma (x)| 2
t2
\Biggr)
\leq
\leq \varphi - 1
for sufficiently large t. And in terms of u, (3.5) can be stated as\int
\varphi - 1u2dx \leq HT1 + 2ET1 , t \geq T1.
Therefore, we obtain the estimates of u as follows.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
DECAY ESTIMATES FOR A KIND OF LINEAR WAVE EQUATIONS 1007
Lemma 3.4. Let \varphi and \theta be defined in (3.3), then for t \geq T0 \geq T1 we have\int
\varphi - 1u2dx \leq C(HT0 + ET0),\int
\theta \varphi - 2
\bigl(
u2t + b| \nabla u| 2
\bigr)
dx \leq C(HT0 + ET0),
where C and T0 = T0(m) are positive constants, HT0 =
\int
\varphi \^u2dx| t=T0 and ET0 = E(\^ut,\nabla \^u, \^u)| t=T0 .
From Lemma 3.4, we have for any t \geq T0(m),\int
tme
\sigma (x)
t u2dx \leq C(HT0 + ET0), (3.6)
\int \biggl(
2
t
+
\sigma (x)
t2
\biggr) - 1
tme
\sigma (x)
t
\bigl(
u2t + b| \nabla u| 2
\bigr)
dx \leq C(HT0 + ET0). (3.7)
Note that \biggl(
2
t
+
\sigma (x)
t2
\biggr) - 1
=
t2
2t+ \sigma (x)
\geq C1(m)t, t \geq T0,
and
HT0 + ET0 \leq C2(m)
\bigl(
\| \nabla u0\| 2L2 + \| u1\| 2L2
\bigr)
,
where C1(m) and C2(m) > 0 depends also on R, b(x) and n. Note also that the estimates (3.6)
and (3.7) are trivial for small t \geq 1, thus we have our main results.
Theorem 3.1. Assume that a(x) and b(x) satisfy condition (1.3). Then for any large m > 0 the
solution of (1.1) satisfies\int
e
\sigma (x)
t u2dx \leq C0(m)
\bigl(
\| \nabla u0\| 2L2 + \| u1\| 2L2
\bigr)
t - m,\int
e
\sigma (x)
t
\bigl(
u2t + b| \nabla u| 2
\bigr)
dx \leq C0(m)
\bigl(
\| \nabla u0\| 2L2 + \| u1\| 2L2
\bigr)
t - m - 1
for all t \geq 1. Here \sigma (x) > 0 is defined in (3.2). The constant C0(m) depends also on R, b(x) and
n.
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Received 14.10.14
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
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| id | umjimathkievua-article-1754 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:00Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/72/7ec701c014597fa93b00caf3cd384472.pdf |
| spelling | umjimathkievua-article-17542019-12-05T09:25:34Z Decay estimates for a kind of linear wave equations Оцiнки затухання для одногу типу хвильових рiвнянь Li, Ming Li, Xinfu Лі, Мін Лі, Сіньфу We consider one kind of dissipative wave equations with exponential speed of propagation. An arbitrary power decay rate for the $L^2$-norm and energy is obtained by using the multiplier method. Розглянуто один тип дисипативних хвильових рiвнянь з експоненцiальною швидкiстю поширення. Довiльний степеневий закон затухання отримано для $L^2$-норми та енергiї за допомогою методу множникiв. Institute of Mathematics, NAS of Ukraine 2017-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1754 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 7 (2017); 1001- Український математичний журнал; Том 69 № 7 (2017); 1001- 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1754/736 Copyright (c) 2017 Li Ming; Li Xinfu |
| spellingShingle | Li, Ming Li, Xinfu Лі, Мін Лі, Сіньфу Decay estimates for a kind of linear wave equations |
| title | Decay estimates for a kind of linear wave equations |
| title_alt | Оцiнки затухання для одногу типу хвильових рiвнянь |
| title_full | Decay estimates for a kind of linear wave equations |
| title_fullStr | Decay estimates for a kind of linear wave equations |
| title_full_unstemmed | Decay estimates for a kind of linear wave equations |
| title_short | Decay estimates for a kind of linear wave equations |
| title_sort | decay estimates for a kind of linear wave equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1754 |
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