Some results on the global solvability for structurally damped models with a special nonlinearity
The main purpose of this paper is to prove the global (in time) existence of solution for the semilinear Cauchy problem $$u_{tt} + (-\Delta )^{\sigma} u + (-\Delta )^{\delta} u_t = |u_t|^p,\; u(0, x) = u_0(x),\; u_t(0, x) = u_1(x)$$. The parameter $\delta \in (0, \sigma]$ describes the structural...
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| author | Duong, P. T. Дуонг, П. Т. |
| author_facet | Duong, P. T. Дуонг, П. Т. |
| author_sort | Duong, P. T. |
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| description | The main purpose of this paper is to prove the global (in time) existence of solution for the semilinear Cauchy problem
$$u_{tt} + (-\Delta )^{\sigma} u + (-\Delta )^{\delta} u_t = |u_t|^p,\; u(0, x) = u_0(x),\; u_t(0, x) = u_1(x)$$.
The parameter $\delta \in (0, \sigma]$ describes the structural damping in the model varying from the exterior damping $\delta = 0$ up to the visco-elastic type damping $\delta = \sigma$. We will obtain the admissible sets of the parameter p for the global solvability of this semilinear Cauchy problem with arbitrary small initial data u0, u1 in the hyperbolic-like case $\delta \in \Bigl(\cfrac{\sigma}{2} , \sigma \Bigr)$, and in the exceptional case $\delta = 0$. |
| first_indexed | 2026-03-24T02:09:28Z |
| format | Article |
| fulltext |
UDC 517.9
P. T. Duong (Hanoi Nat. Univ. Education, Vietnam)
SOME RESULTS ON THE GLOBAL SOLVABILITY FOR STRUCTURALLY
DAMPED MODELS WITH A SPECIAL NONLINEARITY*
ДЕЯКI РЕЗУЛЬТАТИ ПРО ГЛОБАЛЬНУ РОЗВ’ЯЗНIСТЬ
ДЛЯ МОДЕЛЕЙ ЗI СТРУКТУРНИМ ЗАТУХАННЯМ
ТА НЕЛIНIЙНIСТЮ СПЕЦIАЛЬНОГО ВИГЛЯДУ
The main purpose of this paper is to prove the global (in time) existence of solution for the semilinear Cauchy problem
utt + ( - \Delta )\sigma u+ ( - \Delta )\delta ut = | ut| p, u(0, x) = u0(x), ut(0, x) = u1(x).
The parameter \delta \in (0, \sigma ] describes the structural damping in the model varying from the exterior damping \delta = 0 up to
the visco-elastic type damping \delta = \sigma . We will obtain the admissible sets of the parameter p for the global solvability of
this semilinear Cauchy problem with arbitrary small initial data u0, u1 in the hyperbolic-like case \delta \in
\Bigl( \sigma
2
, \sigma
\Bigr)
and in the
exceptional case \delta = 0.
Основною метою цiєї статтi є доведення глобального (за часом) iснування розв’язку напiвлiнiйної задачi Кошi
utt + ( - \Delta )\sigma u+ ( - \Delta )\delta ut = | ut| p, u(0, x) = u0(x), ut(0, x) = u1(x).
Параметр \delta \in (0, \sigma ] описує структурне затухання в моделi, що змiнюється вiд зовнiшнього затухання \delta = 0 до
затухання в’язкоеластичного типу \delta = \sigma . Визначено множини параметра p, допустимi з точки зору глобальної
розв’язностi даної напiвлiнiйної задачi Кошi з як завгодно малими початковими даними u0, u1 у випадку гiпербо-
лiчного типу \delta \in
\Bigl( \sigma
2
, \sigma
\Bigr)
, а також у винятковому випадку \delta = 0.
1. Introduction. The semilinear Cauchy problem
utt - \Delta u = | u| p, u(0, x) = u0(x), ut(0, x) = u1(x)
has been investigated in recent years in many papers in order to prove Strauss’ conjecture about
the critical exponent pcrit for the global solvability with arbitrarily small data (u0, u1). The next
question of interest is to understand the influence of a damping term of the form ( - \Delta )\delta ut that will
be included in the model. There are several papers that have introduced methods to deal with the
nonlinear Cauchy problem of the form
utt - \Delta v + \mu ( - \Delta )\delta vt = F (x, u, ut), v(0, x) = v0(x), vt(0, x) = v1(x).
In [2] the authors have linear decay estimates to prove global (in time) existence results for this
kind of nonlinear Cauchy problems. From these paper one can learn that several interesting intervals
of \delta generate either a parabolic or a hyperbolic behavior of the solution for the corresponding linear
model from the point of decay estimates.
More precisely, the solutions to the linear model
vtt - \Delta v + \mu ( - \Delta )\delta vt = 0, v(0, x) = v0(x), vt(0, x) = v1(x),
* This paper was supported by the Ministry-level fundamental scientific research, project B2017-SPH-33 (Vietnam
Ministry of Education and Training).
c\bigcirc P. T. DUONG, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9 1211
1212 P. T. DUONG
for \delta \in
\biggl(
0,
1
2
\biggr)
behave themselves similarly to solutions to a corresponding parabolic model. Mean-
while for \delta \in
\biggl(
1
2
, 1
\biggr)
solutions behave more like those of a corresponding hyperbolic model. In [8]
the authors have applied a strategy to study the following more general model:
vtt + ( - \Delta )\sigma v + \mu ( - \Delta )\delta vt = 0, v(0, x) = v0(x), vt(0, x) = v1(x). (1)
There were several difficulties which arose in the understanding of the last model. In order to
estimate the nonlinear term by Duhamel’s principle, some new results on the fractional Gagliardo –
Nirenberg inequality from harmonic analysis have been applied successfully. These problems ap-
peared mainly due to the fact that the fractional Laplacian is a nonlocal operator.
Classically there exist several equivalent definitions for this operator. Without any restriction on
the parameter we can define the fractional Laplacian for all positive \sigma by the Fourier transform
\scrF
\bigl(
( - \Delta )\sigma f(\xi )
\bigr)
= | \xi | 2\sigma \scrF (f)(\xi )
for all \sigma > 0. In the last expression \scrF (f) denotes the Fourier transform of the function f with
respect to x variable.
The above formal definition is not so useful for practical applications. Therefore, for some
special values of \sigma it is possible to introduce the fractional Laplacian as a singular integral. Indeed,
for \sigma \in (0, 1) we can adopt the following more convenient integral representation of the fractional
Laplacian:
( - \Delta )\sigma u(x) = cn,\sigma
\int
\BbbR n
u(x) - u(y)
| x - y| n+2\sigma
dy
for sufficient smooth u with a normalization positive constant cn,\sigma =
22\sigma \sigma \Gamma (n/2 + \sigma )
\pi n/2\Gamma (1 - \sigma )
depending
on n and \sigma .
The nonlinear Cauchy problem with the right-hand side | ut| p has some specific feature in the
comparison with the cases of nonlinearity | u| p or
\bigm| \bigm| | D| au
\bigm| \bigm| p. We do not apply only the Gagliardo –
Nirenberg inequality to obtain the global solvability for the last model, since the term | ut| p is absent
from the definition of the data spaces and the solution spaces.
The goal of this paper is to cover possible values of the parameter \sigma , \delta which were missing in
our previous paper [8] for the model with nonlinearity | ut| p. We will obtain global solvability results
for the whole range \delta \in
\biggl(
\sigma
2
, \sigma
\biggr)
, whereas such a result was obtained in [8] just only for the case
\delta =
\sigma
2
. Unlike the proof in [8] in the case \delta =
\sigma
2
, the method we will use in this paper does not
require large values of p for the global solvability of the nonlinear Cauchy problem with a class of
nonlinearities f = f(ut) on the right-hand side of the equation, since we do not apply the fractional
power rules (see [7]).
Finally, we will state and prove an interesting result in the case \delta = 0 with the same type of
nonlinearity | ut| p. This case requires a certain attention due to the special formula of the characteristic
roots of the corresponding parameter-dependent ordinary differential equation after applying the
Fourier transform to the equation in the Cauchy problem (1).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
SOME RESULTS ON THE GLOBAL SOLVABILITY FOR STRUCTURALLY DAMPED MODELS . . . 1213
Recall that by applying the partial Fourier transform with respect to the spatial variables x to the
equation in (1) the following Cauchy problem is obtained for the Fourier transform w = w(\xi , t) :=
:= \^ux\rightarrow \xi (\xi , t) = F(u)(\xi , t):
wtt + \mu | \xi | 2\delta wt + | \xi | 2\sigma w = 0, w(0, \xi ) = w0(\xi ), wt(0, \xi ) = w1(\xi ).
The characteristic equation \lambda 2 + \mu | \xi | 2\delta \lambda + | \xi | 2\sigma = 0 has the roots
\lambda 1,2(\xi ) =
1
2
\Bigl(
- \mu | \xi | 2\delta \pm
\sqrt{}
\mu 2| \xi | 4\delta - 4| \xi | 2\sigma
\Bigr)
.
Using these characteristic roots we can write down directly the detailed formula for the solution
of the linear Cauchy problem (1) by means of the fundamental solutions:
u = K0 \ast u0 +K1 \ast u1,
where \ast denotes the convolution with respect to x variable. In the last expression the fundamental
solutions Ki are given by the following formulas:
K0 = \scrF - 1
\biggl(
\lambda 1e
\lambda 2t - \lambda 2e
\lambda 1t
\lambda 1 - \lambda 2
\biggr)
, K1 = \scrF - 1
\biggl(
e\lambda 1t - e\lambda 2t
\lambda 1 - \lambda 2
\biggr)
.
Our main results are contained in the following theorems which will be formulated with different
values of the damping parameter \delta .
Theorem 1. Consider the Cauchy problem for the structurally damped model
utt + ( - \Delta )\sigma u+ \mu ( - \Delta )\delta ut = | ut| p, u(0, x) = u0(x), ut(0, x) = u1(x),
with \sigma \in
\biggl(
1,
n
2
\biggr)
, \mu > 0 and \delta \in
\biggl(
\sigma
2
, \sigma
\biggr)
. The data (u0, u1) are supposed to belong to the function
spaces (L1\cap Hs)\times (L1\cap Hs - \sigma ), where the parameter s satisfies
\biggl[
n
2
\biggr]
+1+
\biggl[
[n/2] + 1
2
\biggr]
\leq s \in \BbbN .
Then for all
p \in
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
1 +
2s
n
, s
\biggr)
;
n
[n+ 2\sigma - 2s]+
\biggr)
there exists a uniquely determined global (in time) small data solution in C([0,\infty ), Hs)\cap C1([0,\infty ),
Hs - \sigma ).
We see that in the case s >
n
2
+\sigma the condition for p is simply reduced to p > \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
1+
2s
n
, s
\biggr\}
.
This conditions on s, p in the above theorem may seem too strict, however as we will explain at the
end of the proof of Theorem 1, the analogous result with the same restriction on s, p is also valid if
we replace | ut| p in the right-hand side by more general nonlinearity function f(ut) satisfying some
growth of power type.
For the external damping model with \sigma = 1, \delta = 0 we have the following result.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
1214 P. T. DUONG
Theorem 2. Let us consider the Cauchy problem for the external damped model
utt - \Delta u+ ut = | ut| p, u(0, x) = u0(x), ut(0, x) = u1(x).
The data (u0, u1) are assumed to belong to the function space (L1 \cap Hs) \times (L1 \cap Hs - 1) with
s > 1 +
n
2
. Then for any
p > \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
2; s; 1 +
4(s - 1)
ns
\biggr\}
there exists a uniquely determined global (in time) small data energy solution from C
\bigl(
[0,\infty ), Hs
\bigr)
\cap
\cap C1
\bigl(
[0,\infty ), Hs - 1
\bigr)
.
It should be noted that we also can obtain a similar result with general \sigma > 1 with only minor
changes on the conditions for s and p.
Throughout this paper the notation f \lesssim g for two functions f = f(t), g = g(t) means | f(t)| \leq
\leq c| g(t)| for all t in the range that will be shown, with a positive constant c.
The paper is organized as follows. In Section 2 we recall some estimates for solutions to the
homogeneous linear models corresponding to the problems stated in Theorems 1 and 2. In Section 3
we shall present the proofs of our main results. As conclusion, in the last Section 4 we will give
some remarks and comments for our further study in the case of variable coefficients models.
2. Estimates of solution for linear Cauchy problems. In this section of the article we will
present several estimates for solutions to homogeneous Cauchy problems which are useful in the
study of the nonlinear Cauchy problems. These estimates were obtained already in [8] and [9]
by using singular Bessel integrals for the Fourier transform of radial functions and exploiting the
asymptotic profiles of the characteristic roots \lambda i, i = 1, 2, that were given before. For the case \delta = 0
such estimates were obtained by comparing the solution of the damped model with those for the
corresponding evolution model ut + ( - \Delta )\sigma u = 0 by studying the diffusion phenomenon.
First, we have the following result (Proposition 22 in [8]) on (L1 \cap L2) - L2 and L2 - L2 decay
estimates of solutions for the linear (homogeneous) Cauchy problem in the case \delta \in
\biggl(
\sigma
2
, \sigma
\biggr)
.
Proposition 1. Let us consider the Cauchy problem
vtt + ( - \Delta )\sigma v + ( - \Delta )\delta vt = 0, v(0, x) = v0(x), vt(0, x) = v1(x),
for \delta \in
\biggl(
\sigma
2
, \sigma
\biggr)
and data (v0, v1) \in (L1\cap H2\delta )\times (L1\cap L2). Then the solutions and their derivatives
satisfy the following (L1 \cap L2) - L2 estimates:
\| v(t, \cdot )\| L2 \lesssim (1 + t) -
n
4\delta \| v0\| L1\cap L2 + (1 + t)1 -
n
4\delta \| v1\| L1\cap L2 ,
\| vt(t, \cdot )\| L2 \lesssim (1 + t) -
n+2\sigma
4\delta \| v0\| L1\cap H2(\sigma - \delta ) + (1 + t)1 -
n+2\sigma
4\delta \| v1\| L1\cap L2 ,
\| | D| \sigma v(t, \cdot )\| L2 \lesssim (1 + t) -
n+2\sigma
4\delta \| v0\| L1\cap H\sigma + (1 + t)1 -
n+2\sigma
4\delta \| v1\| L1\cap L2 ,
\| | D| 2\delta v(t, \cdot )\| L2 \lesssim (1 + t) -
n
4\delta - 1\| v0\| L1\cap H2\delta + (1 + t) -
n
4\delta \| v1\| L1\cap L2 ,
and the L2 - L2 estimates:
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
SOME RESULTS ON THE GLOBAL SOLVABILITY FOR STRUCTURALLY DAMPED MODELS . . . 1215
\| v(t, \cdot )\| L2 \lesssim \| v0\| L2 + (1 + t)\| v1\| L2 ,
\| vt(t, \cdot )\| L2 \lesssim (1 + t) -
\sigma
2\delta \| v0\| H2(\sigma - \delta ) + \| v1\| L2 ,
\| | D| \sigma v(t, \cdot )\| L2 \lesssim (1 + t) -
\sigma
2\delta \| v0\| H\sigma + \| v1\| L2 ,
\| | D| 2\delta v(t, \cdot )\| L2 \lesssim (1 + t) - 1\| v0\| H2\delta + (1 + t) -
2\delta - \sigma
2\delta \| v1\| L2 ,
in arbitrary space dimensions n.
Under special restrictions to the dimension n and the parameters \sigma , \delta one can prove a sharper
result than Proposition 1 by using the technique from [2] (see Proposition 24 in [8]).
Proposition 2. Let us consider the Cauchy problem
vtt + ( - \Delta )\sigma v + ( - \Delta )\delta vt = 0, v(0, x) = v0(x), vt(0, x) = v1(x),
for \delta \in
\biggl(
\sigma
2
, \sigma
\biggr)
and data (v0, v1) \in (L1 \cap H2\delta )\times (L1 \cap L2) under the constrain condition n \geq 2\sigma
for the dimension n. Then the solution and its derivatives satisfy the following (L1 \cap L2) - L2
estimates:
\| v(t, \cdot )\| L2 \lesssim
\left\{ (1 + t) -
n
4\delta \| v0\| L1\cap L2 + (1 + t) -
n - 2\sigma
4\delta \| v1\| L1\cap L2 if n > 2\sigma ,
(1 + t) -
\sigma
2\delta \| v0\| L1\cap L2 + \mathrm{l}\mathrm{o}\mathrm{g}(e+ t)\| v1\| L1\cap L2 if n = 2\sigma ,
\| vt(t, \cdot )\| L2 \lesssim (1 + t) -
n+2\sigma
4\delta \| v0\| L1\cap H2(\sigma - \delta ) + (1 + t) -
n
4\delta \| v1\| L1\cap L2 ,\bigm\| \bigm\| | D| \sigma v(t, \cdot )
\bigm\| \bigm\|
L2 \lesssim (1 + t) -
n+2\sigma
4\delta \| v0\| L1\cap H\sigma + (1 + t) -
n
4\delta \| v1\| L1\cap L2 ,\bigm\| \bigm\| | D| 2\delta v(t, \cdot )
\bigm\| \bigm\|
L2 \lesssim (1 + t) -
n
4\delta - 1\| v0\| L1\cap H2\delta + (1 + t) -
n+4\delta - 2\sigma
4\delta \| v1\| L1\cap L2 .
More generally, for arbitrary \alpha > 0 it holds\bigm\| \bigm\| \partial j
t | D| \alpha v(t, \cdot )
\bigm\| \bigm\|
L2 \lesssim (1+t) -
n - 2\sigma
4\delta - \sigma j+\alpha
2\delta
\bigl(
(1+t) -
\sigma
2\delta \| v0\| L1\cap H2j(\sigma - \delta )+| \alpha | +\| v1\| L1\cap H2(j - 1)\delta +\alpha
\bigr)
. (2)
In the exceptional case \delta = 0, the method to use the asymptotic profile of the characteristic roots
does not bring any decay rate in t of the solution to the corresponding linear Cauchy problem. The
study of the diffusion phenomenon in the abstract setting for the equation utt + Bu + ut = 0 leads
to the following result (see Proposition 2.1 in [9]).
Proposition 3. The solution v = v(t, x) of the linear Cauchy problem for external damped
model or for the model with friction
vtt + ( - \Delta )\sigma v + vt = 0, v(0, x) = v0(x), vt(0, x) = v1(x),
and its derivatives satisfy the following (L1 \cap L2) - L2 estimates:
\| v(t, \cdot )\| L2 \lesssim (1 + t) -
n
4\sigma \| v0\| L1\cap L2 + (1 + t) -
n
4\sigma \| v1\| L1\cap H - \sigma ,
\| v(t, \cdot )\| \.H\sigma \lesssim (1 + t) -
n
4\sigma - 1
2 \| v0\| L1\cap \.H\sigma + (1 + t) -
n
4\sigma - 1
2 \| v1\| L1\cap L2 ,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
1216 P. T. DUONG
\| vt(t, \cdot )\| L2 \lesssim (1 + t) -
n
4\sigma - 1\| v0\| L1\cap \.H\sigma + (1 + t) -
n
4\sigma - 1\| v1\| L1\cap L2 ,
\| v(t, \cdot )\| \.Hk \lesssim (1 + t) -
n
4\sigma - k
2\sigma \| v0\| L1\cap \.Hk + (1 + t) -
n
4\sigma - k
2\sigma \| v1\| L1\cap \.Hk - \sigma
for all k \geq 0, and the L2 - L2 estimates
\| v(t, \cdot )\| L2 \lesssim \| v0\| L2 + (1 + t)\| v1\| L2 ,
\| vt(t, \cdot )\| L2 \lesssim (1 + t) - 1\| v0\| H\sigma + \| v1\| L2 ,\bigm\| \bigm\| | D| \sigma v(t, \cdot )
\bigm\| \bigm\|
L2 \lesssim (1 + t) -
1
2 \| v0\| H\sigma + (1 + t) -
1
2 \| v1\| L2 .
3. Proof of our main results. In this section we present the proofs of Theorems 1 and 2. Before
going into details we would like to mention that, unlike the case of nonlinearity
\bigm| \bigm| | D| au
\bigm| \bigm| p, the case
with | ut| p in the right-hand side will be proceeded with some necessary changes, mainly due to the
estimates of the nonlinear term in the integral
\int t
0
K1(t - \tau , x)| ut(\tau , x)| p d\tau . It is a quite different
situation, when | ut| p is replaced by | u| p. In that case, the fractional Gagliardo – Nirenberg inequality
allows us to derive the admissible range of p for the global (in time) existence of solutions.
3.1. Proof of Theorem 1. The proof follows the Banach fixed point method for the solution
mapping that will be shown to be Lipschitz for small data which are arbitrarily chosen. We introduce
now the data space A := (L1 \cap Hs)\times (L1 \cap Hs - \sigma ). The statements from Proposition 2 suggest us
to use the auxiliary space X(t) = C([0, t], Hs) \cap C1([0, t], Hs - \sigma ) with the norm
\| u\| X(t) = \mathrm{s}\mathrm{u}\mathrm{p}
\tau \in [0,t]
\Bigl(
(1 + \tau )
n - 2\sigma
4\delta \| u(\tau , \cdot )\| L2 + (1 + \tau )
n
4\delta \| ut(\tau , \cdot )\| L2+
+(1 + \tau )
n+2(s - \sigma )
4\delta \| ut(\tau , \cdot )\| \.Hs - \sigma + (1 + \tau )
n - 2\sigma +2s
4\delta \| u(\tau , \cdot )\| \.Hs
\Bigr)
.
We define the mapping N between the data and the solution in the following way:
N : u \in X(t) \rightarrow Nu = K0 \ast u0 +K1 \ast u1 +
t\int
0
K1(t - \tau , x) \ast (x) | ut(\tau , x)| pd\tau \in X(t). (3)
By standard arguments the uniqueness, local and global in time existence of solutions to the Cauchy
problem will be implied from the following pair of inequalities:
\| Nu\| X(t) \leq C\| (u0, u1)\| A + C\| u\| pX(t), (4)
\| Nu - Nv\| X(t) \leq C\| u - v\| X(t)
\bigl(
\| u\| p - 1
X(t) + \| v\| p - 1
X(t)
\bigr)
. (5)
From the estimates in Proposition 2 it is obvious that
\| \partial j
tNu(t, \cdot )\| Hk \lesssim \| (u0, u1)\| A +
t\int
0
(1 + t - \tau ) -
n+2(\sigma j+k - \sigma )
4\delta
\bigm\| \bigm\| | ut(\tau , \cdot )| p\bigm\| \bigm\| L1\cap Hk - \sigma +2j(\sigma - \delta )
with k, j such that k + j = s and j = 0, 1. We estimate the norm
\bigm\| \bigm\| | ut(\tau , \cdot )| p\bigm\| \bigm\| L1\cap Hk - \sigma +2j(\sigma - \delta ) by
the following result (see Lemma 3 in Matsumura [4] for the general nonlinearity f = f(u, ut)).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
SOME RESULTS ON THE GLOBAL SOLVABILITY FOR STRUCTURALLY DAMPED MODELS . . . 1217
Lemma 1 (cf. Lemma 3 in [4] for general case of f(v)). Suppose that f = f(v) has the form
f = | v| p, \pm v| v| p - 1, p > s with an integer s \geq [n/2] + 1 +
\biggl[
[n/2] + 1
2
\biggr]
. Then there exists a
nondecreasing nonnegative function h(\cdot ) with h(0) = 0 that is a Lipschitz function at a neighborhood
of v = 0, such that for every function v \in Hs the following estimates are valid:
\| f(v)\| Hs \lesssim \| v\| pHsh(\| v\| Hs) for all p > 1,
\| f(v)\| Lq \lesssim \| v\| p - 2/q
L\infty \| v\| 2/q
L2 h(\| v\| L\infty ) for all p \geq 2, 1 \leq q \leq 2, pq \geq 2.
The proof of this statement was presented in details by von Wahl in [12].
To estimate different norms which are included in the norm of the solution space X(t) we firstly
derive the estimates for the norms containing ut.
Differentiating the expression (3) with respect to t we obtain
\partial tNu = vt +
t\int
0
\partial t
\bigl(
K1(t - \tau , x) \ast (x) | ut(\tau , x)| p
\bigr)
d\tau ,
where by v we denote the solution of linear Cauchy problem with the same initial data u0, u1.
Using the estimates (2) with j = 1, \alpha = s for vt and for second expression in the above formula
we get
\bigm\| \bigm\| \partial t(Nu)(t, \cdot )
\bigm\| \bigm\|
\.Hs \lesssim (1 + t) -
n
4\delta \| (u0, u1)\| A +
t\int
0
(1 + t - \tau ) -
n+2s
4\delta
\bigl(
\| | ut| p\| L1 + \| | ut| p\| Hs
\bigr)
d\tau .
(6)
With s \geq [n/2] + 1 +
\biggl[
[n/2] + 1
2
\biggr]
we apply Lemma 1 with q = 1, p > 2 to have the following
useful estimates for the norms of | ut| p :\bigm\| \bigm\| | ut| p\bigm\| \bigm\| Hs \lesssim \| ut\| pHsh(\| ut\| Hs)
and
\| | ut| p\| L1 \lesssim \| ut\| p - 2/q
\infty \| ut\| 2/qL2 h
\bigl(
\| ut\| \infty
\bigr)
. (7)
By the Sobolev embedding theorem for L\infty (\BbbR n) under the given condition for s and from the
monotonicity of h, the right-hand side of the (7) is dominated by \| ut\| pHsh
\bigl(
\| ut\| Hs
\bigr)
with some
suitable multiplicative constant. Adding these last estimates together we get
\| | ut| p\| L1 + \| | ut| p\| Hs \lesssim \| ut\| pHsh
\bigl(
\| ut\| Hs
\bigr)
.
Substituting the last estimate into the expression in the right-hand side of (6) we obtain
\bigm\| \bigm\| \partial t(Nu)(t, \cdot )
\bigm\| \bigm\|
\.Hs \lesssim (1 + t) -
n
4\delta \| (u0, u1)\| A +
t\int
0
(1 + t - \tau ) -
n+2s
4\delta \| ut\| pHsh
\bigl(
\| ut\| Hs
\bigr)
d\tau .
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1218 P. T. DUONG
Putting
M(t) = \mathrm{s}\mathrm{u}\mathrm{p}
0\leq \tau \leq t
(1 + \tau )
n
4\delta \| ut(\tau , \cdot )\| Hs
and using the monotonicity of h we may conclude
\| \partial t(Nu)(t, \cdot )\| \.Hs \lesssim
\lesssim (1 + t) -
n
4\delta \| (u0, u1)\| A + h(M(t))
t\int
0
(M(\tau ))p(1 + t - \tau ) -
n+2s
4\delta (1 + \tau ) - p
\bigl(
n
4\delta
\bigr)
d\tau .
Now recalling that for \mathrm{m}\mathrm{a}\mathrm{x}\{ \alpha ;\beta \} > 1 the inequality
t\int
0
(1 + t - \tau ) - \alpha (1 + \tau ) - \beta d\tau \lesssim (1 + t) - min(\alpha ,\beta )
holds, we see that for p > 1 +
2s
n\bigm\| \bigm\| \partial t(Nu)(t, \cdot )
\bigm\| \bigm\|
\.Hs \lesssim (1 + t) -
n
4\delta \| (u0, u1)\| A + (1 + t) -
n+2s
4\delta (M(t))ph(M(t)) (8)
is valid.
By using the estimates for the corresponding linear Cauchy problem together with the second
inequality in Lemma 1 for q = 2, we also have\bigm\| \bigm\| \partial t(Nu)(t, \cdot )
\bigm\| \bigm\|
L2 \lesssim (1 + t) -
n
4\delta \| (u0, u1)\| A + (1 + t) -
n
4\delta (M(t))ph(M(t)). (9)
The estimates (8) and (9) imply the nonhomogeneous norm estimate\bigm\| \bigm\| \partial t(Nu)(t, \cdot )
\bigm\| \bigm\|
Hs \lesssim (1 + t) -
n
4\delta \| (u0, u1)\| A +
\bigl[
(1 + t) -
n+2s
4\delta + (1 + t) -
n
4\delta
\bigr]
(M(t))ph(M(t)).
The last inequality implies that for the local (in time) solution of Nu = u with small data u0, u1
suitable chosen in A the inequality
M(t) \leq c+
\bigl[
(1 + t) -
2s
4\delta + 1
\bigr]
(M(t))ph(M(t))
is valid. By a well-known argument, the last estimate implies that M(t) is bounded for small data
u0, u1 suitable chosen, i.e., \bigm\| \bigm\| ut(t, \cdot )\bigm\| \bigm\| Hs \lesssim (1 + t) -
n
4\delta . (10)
The last estimate is still weak and is not the one we need to prove. However, we will use it as an
intermediate step in the following estimate.
Next, we will strengthen estimate (10) by differentiating directly both sides in the equation (11).
For the local (in time) solution of u = Nu the estimate
\bigm\| \bigm\| \partial i
t\partial
\alpha
xu(t, \cdot )
\bigm\| \bigm\|
L2 \lesssim
\bigm\| \bigm\| \partial i
t\partial
\alpha
x v(t, \cdot )
\bigm\| \bigm\|
L2 +
t\int
0
\bigm\| \bigm\| \partial i
t\partial
\alpha
x
\bigl(
K1(t - \tau , x) \ast (x) f(ut)(\tau , x)
\bigr) \bigm\| \bigm\|
L2 d\tau
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SOME RESULTS ON THE GLOBAL SOLVABILITY FOR STRUCTURALLY DAMPED MODELS . . . 1219
is satisfied. The first term in the right-hand side has a growth (1+ t) -
n+2\alpha +2\sigma i - 2\sigma
4\delta \| (u0, u1)\| A from
the estimates for the corresponding linear cauchy problem. The second term containing an integral,
we estimate as follows:
t\int
0
\bigm\| \bigm\| \partial j
t \partial
\alpha
x
\bigl(
K1(t - \tau , x) \ast (x) f(ut)(\tau , x)
\bigr) \bigm\| \bigm\|
L2d\tau \lesssim
t\int
0
(1 + t - \tau ) -
n+2\alpha +2\sigma j - 2\sigma
4\delta
\bigm\| \bigm\| | ut| p\bigm\| \bigm\| H| \alpha | +2(j - 1)\delta
for j + | \alpha | \leq s. The term containing h
\bigl(
\| ut\| Hk - \sigma +2j(\sigma - \delta )
\bigr)
is bounded thanks to the estimate (10)
and by the monotonicity of h(\cdot ). Therefore it can be dropped out.
We finally arrive at\bigm\| \bigm\| \partial j
t u(t, \cdot )
\bigm\| \bigm\|
Hk \lesssim (1 + t) -
n+2k+2\sigma j - 2\sigma
4\delta \| (u0, u1)\| A + (1 + t) -
n+2k+2\sigma j - 2\sigma
4\delta \| ut\| pHk - \sigma +j(\sigma - \delta )
provided that p >
n+ 2s
n
for the last Matsumura technique to be applied. This estimate implies the
first inequality (4).
Now we will present the main steps how to prove the second estimate of (5), that is
\| Nu - Nv\| X(t) \leq C\| u - v\| X(t
\Bigl(
\| u\| p - 1
X(t) + \| v\| p - 1
X(t)
\Bigr)
.
For f(u) = | ut| p writing
\| Nu - Nv\| X(t) =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
K1(t - \tau , x) \ast (x)
\bigl(
f(u(\tau , x)
\bigr)
- f
\bigl(
v(\tau , x))
\bigr)
d\tau
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
X(t)
by means of the estimate for the corresponding linear Cauchy problem we estimate several norms of
the difference f(u(s, x)) - f(v(s, x)). The estimate of the L1 \cap L2-norm can be obtained by noting
that
| f(u) - f(v)| \lesssim | ut - vt|
\bigl(
| ut| p - 1 + | vt| p - 1
\bigr)
(this fact can be proved, for example, by applying Lagrange’s mean value theorem). Hence, by
Hölder’s inequality, we see that\bigm\| \bigm\| f(u(\tau )) - f(v(\tau ))
\bigm\| \bigm\|
L1 \lesssim \| ut(\tau ) - vt(\tau )\| Lp
\Bigl(
\| ut(\tau )\| p - 1
Lp + \| vt(\tau )\| p - 1
Lp
\Bigr)
,\bigm\| \bigm\| f(u(\tau )) - f(v(\tau ))
\bigm\| \bigm\|
L2 \lesssim \| ut(\tau ) - vt(\tau )\| L2p
\Bigl(
\| ut(\tau )\| p - 1
L2p + \| vt(\tau )\| p - 1
L2p
\Bigr)
.
The Lp- and L2p-norms of the difference ut - vt are estimated by the fractional Gagliardo – Nirenberg
inequality and they are dominated by \| u - v\| X(t). More precisely, to estimate the norm \| w(\tau , \cdot )\| Lkp ,
k = 1,m, with w = ut - vt, ut, vt we apply the fractional Gagliardo – Nirenberg inequality in the
form
\| w(\tau , \cdot )\| Lq \lesssim
\bigm\| \bigm\| | D| s - \sigma w(\tau , \cdot )
\bigm\| \bigm\| \theta 0,s - \sigma (q,2)
L2 \| w(\tau , \cdot )\| 1 - \theta 0,s - \sigma (q,2)
L2
where for q = p, 2p \geq 2 we need
\theta 0,s - \sigma (q, 2) =
n
s - \sigma
\biggl(
1
2
- 1
q
\biggr)
\in
\bigl[
0, 1
\bigr)
.
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1220 P. T. DUONG
This leads to the additional condition for p that requires p <
n
[n - 2(s - \sigma )]+
which is always valid
if we suppose s >
n
2
+ \sigma .
The more difficult estimates are those that should be established with the Sobolev norms in the
definition of X(t). For example, we will explain how to estimate
\bigm\| \bigm\| | ut| p - | vt| p
\bigm\| \bigm\|
H\gamma for \gamma > 0. By
the estimates (2), in order to bound the norm \| u - v\| X(t), we pay attention to two special values
\gamma = \alpha +2(j - 1)\delta with (j, \alpha ) = (0, s) or (j, \alpha ) = (1, s - \sigma ). These values correspond to \gamma = s - 2\delta
and \gamma = s - \sigma . Here we apply the following estimate for the nonhomogeneous scales (known as the
Kato – Ponce inequality, see [3]). We formulate this well-known inequality below for reference.
Denote by Js(f) = (1 - \Delta )s/2(f) = \scrF - 1
\bigl(
(1+4\pi 2| \xi | 2)s/2\scrF (f)(\xi )
\bigr)
, and Hs,p :=
\bigl\{
f \in Lp(\BbbR n :
Js(f) \in Lp(\BbbR n
\bigr\}
for s > 0 — special fractional Sobolev spaces on \BbbR n.
Proposition 4 (Kato – Ponce inequality). For all functions f \in Hs,p2 \cap Lq1 and g \in Hs,q2 \cap Lp1
it holds
\| Js(fg)\| Lr(\BbbR n) \lesssim
\bigl[
\| f\| Lp1 (\BbbR n)\| Jsg\| Lq1 (\BbbR n) + \| Jsf\| Lp2\BbbR n)\| g\| Lq2 (\BbbR n)
\bigr]
,
where s > 0 and
1
r
=
1
p1
+
1
q1
=
1
p2
+
1
q2
for 1 < r < \infty , 1 < p1, q2 \leq \infty , 1 < p2, q1 < \infty .
The next estimates for the nonlinear part are carried out with the application of Lemma 1 instead
of the composition lemma. We write g(w) = | w| p, G(w) =
1
p
g\prime (w) = w| w| p - 2. By the fundamental
theorem of calculus it follows
g(ut(s)) - g(vt(s)) = p
1\int
0
(ut(s) - vt(s))G(\theta ut + (1 - \theta )vt) d\theta .
Therefore,
\| | ut| p - | vt| p\| H\gamma \lesssim
1\int
0
\bigm\| \bigm\| (ut(s) - vt(s))G(\theta ut + (1 - \theta )vt)
\bigm\| \bigm\|
H\gamma d\theta .
Applying the Kato – Ponce inequality for the H\gamma -norm of the product fg, where f := G(\theta ut + (1 -
- \theta )vt), g := ut(s) - vt(s) in the right-hand side of the above estimate with r = p2 = q1 = 2, = p,
p1 = q2 = \infty we obtain
\bigm\| \bigm\| | ut| p - | vt| p
\bigm\| \bigm\|
H\gamma \lesssim
1\int
0
\bigm\| \bigm\| (ut(s) - vt(s))
\bigm\| \bigm\|
H\gamma
\bigm\| \bigm\| G(\theta ut + (1 - \theta )vt)
\bigm\| \bigm\|
L\infty d\theta +
+
\bigm\| \bigm\| (ut(s) - vt(s))
\bigm\| \bigm\|
L\infty
1\int
0
\bigm\| \bigm\| G(\theta ut + (1 - \theta )vt)
\bigm\| \bigm\|
H\gamma d\theta .
The norm
\bigm\| \bigm\| (ut(s) - vt(s))
\bigm\| \bigm\|
H\gamma is estimated by the Gagliardo – Nirenberg inequality and is dominated
from above by the norms that appear in the definition of the solution space X(t). The terms
\bigm\| \bigm\| G(\theta ut+
+ (1 - \theta )vt)
\bigm\| \bigm\|
L\infty and
\bigm\| \bigm\| (ut(s) - vt(s))
\bigm\| \bigm\|
L\infty are estimated from above by the Sobolev embedding.
The more interesting is the last term
\bigm\| \bigm\| G(\theta ut + (1 - \theta )vt)
\bigm\| \bigm\|
H\gamma .
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SOME RESULTS ON THE GLOBAL SOLVABILITY FOR STRUCTURALLY DAMPED MODELS . . . 1221
In the case of a general nonlinearity f(ut) that has power growth f(w) \sim | w| p (see the Remark 1
below), we apply the Lemma 1 once again, since G(w) has the power growth as | w| p - 1 too. This
brings \bigm\| \bigm\| G(\theta ut + (1 - \theta )vt)
\bigm\| \bigm\|
H\gamma \lesssim
\bigm\| \bigm\| \theta ut + (1 - \theta )vt
\bigm\| \bigm\| p - 1
H\gamma h
\bigl(
\| \theta ut + (1 - \theta )vt\| H\gamma
\bigr)
\lesssim
\lesssim
\bigl(
\| ut\| p - 1
H\gamma + \| vt\| p - 1
H\gamma
\bigr)
h
\bigl(
\| ut\| H\gamma + \| vt\| H\gamma
\bigr)
.
The factor h
\bigl(
\| ut\| H\gamma + \| vt\| H\gamma
\bigr)
can be dropped out of the integral by the monotonicity of h and
by the boundedness of \| ut\| H\gamma and \| vt\| H\gamma (we see that \gamma \leq s - \sigma in all cases of our interests).
The second estimate (5) follows from the above arguments. The condition for p now becomes
p - 1 > s - \sigma that is true since \sigma > 1. We should note that we need also the condition that s - \sigma \in \BbbN
for the validity of Lemma 1 applied to G.
However in the case f(ut) = | ut| p the estimate can be obtained directly by means of the
Corollary 2. According this composition result we may derive\bigm\| \bigm\| G(\theta ut + (1 - \theta )vt)
\bigm\| \bigm\|
H\gamma \lesssim
\bigm\| \bigm\| G(\theta ut + (1 - \theta )vt)
\bigm\| \bigm\|
Hs - \sigma \lesssim
\lesssim
\bigl(
\| G(ut)\| p - 1
Hs - \sigma + \| G(vt)\| p - 1
Hs - \sigma
\bigr)
.
The last inequality leads to the second estimate (5).
By (4) and (5) the existence of local (in time) solution follows immediately for large data and the
existence of global (in time) solutions will be guaranteed for sufficiently small data.
Theorem 1 is proved.
Remark 1. We have proved in [8] another global existence result with the same nonlinearity
| ut| p but only for the special case \delta =
\sigma
2
. In that paper, in order to estimate the integral term
including | ut| p in the Sobolev scales we had applied the following composition lemma (see [7]).
Proposition 5. Let p > 1 and v \in Hs,m, where s >
n
m
and s \in (0, p). Then the following
estimate holds:
\| | v| p\| Hs,m \leq C\| v\| Hs,m\| v\| p - 1
L\infty .
The following corollary follows from Proposition 5 immediately.
Corollary 1. Under the assumptions of Proposition 5 it holds\bigm\| \bigm\| | v| p\bigm\| \bigm\| \.Hs,m \leq C\| v\| \.Hs,m\| v\| p - 1
L\infty .
It seems that the conditions s >
\biggl[
n
2
\biggr]
+ 1 +
\biggl[
[n/2] + 1
2
\biggr]
and s integer are too strong in the
statement of Theorem 1. However we present the proof using Lemma 1 since this approach is simple
and it does not require more detailed information about the function h(\cdot ) except the monotonicity and
the Lipschitz property near 0. Additionally, the condition p > s comes simply from the assumption
that f(v) = | v| p \in Cs. Moreover, this approach also allows us to consider a broader class of the
nonlinearities f(ut). Actually, we can apply this method to obtain the following general result.
Theorem 3. Consider the Cauchy problem for the structurally damped model
utt + ( - \Delta )\sigma u+ \mu ( - \Delta )\delta ut = f(ut), u(0, x) = u0(x), ut(0, x) = u1(x),
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1222 P. T. DUONG
with \sigma \geq 1, \mu > 0, \delta \in
\biggl(
\sigma
2
, \sigma
\biggr)
and s \geq [n/2] + 1 +
\biggl[
[n/2] + 1
2
\biggr]
such that s, \sigma are integer. The
function f(v) satisfies the following conditions:
f(v) \in Cs,
| f(v)| \lesssim | v| p,
| f (\alpha )(v)| \lesssim | v| p - \alpha for 1 \leq \alpha < \mathrm{m}\mathrm{i}\mathrm{n}\{ s; p\} ,
| f (\alpha )(v)| \lesssim 1 if p \leq \alpha < s.
The data (u0, u1) are supposed to belong to the function space (L1 \cap Hs) \times (L1 \cap Hs - \sigma ) with
n > 2\sigma . Then for all
p \in
\biggl(
1 +
2s
n
,
n
(n+ 2\sigma - 2s)+
\biggr)
there exists a uniquely determined global (in time) small data solution in C([0,\infty ), Hs)\cap C1([0,\infty ),
Hs - \sigma ).
We see that in the above theorem, the condition for p is reduced to p > 1+
2s
n
when s >
n
2
+ \sigma
and we do not need the inequality p > s in order to guarantee the global existence with small data
result in Theorem 3.
The proof of Theorem 3 coincides to that of Theorem 1 with the exploitation of the following
result (Lemma 3 in [4]) which generalizes Lemma 1.
Lemma 2. Suppose that f = f(v) \in Cs, where s \geq
\biggl[
n
2
\biggr]
+ 1 +
\biggl[
[n/2] + 1
2
\biggr]
, satisfies all
assumption in the statement of Theorem 3. Then there exists a nondecreasing nonnegative locally
Lipschitz at v = 0 function h(\cdot ), h(0) = 0, which is a Lipschitz function at a neighborhood of v = 0,
such that for every function v \in Hs the following estimates are valid:
\| f(v)\| Hs \lesssim \| v\| pHsh(\| v\| Hs) for all p > 1,
\| f(v)\| Lq \lesssim \| v\| p - 2/q
L\infty \| v\| 2/q
L2 h(\| v\| L\infty ) for all p \geq 2, 1 \leq q \leq 2, pq \geq 2.
3.2. Proof of Theorem 2. The proof of Theorem 2 can be served as an illustration to show how
to use the composition lemma effectively to estimate the term containing | ut| p in Sobolev scales. In
order to prove Theorem 2 we need some additional decay estimates for the mixed derivatives of u
which are easily obtained from the representation of solution and can be proved by the same method
as for proving Lemma 2 (see [9] for details).
Proposition 6. The solutions of the linear Cauchy problem for classical damped waves satisfy
the following (L1 \cap L2) - L2 estimates for mixed derivatives:\bigm\| \bigm\| vt(t, \cdot )\bigm\| \bigm\| \.Hk \lesssim (1 + t) -
n
4 - 1 - k/2\| v0\| L1\cap \.Hk+1 + (1 + t) -
n
4 - 1 - k/2\| v1\| L1\cap \.Hk
for all k \geq 0, and the L2 - L2 estimates
\| vt(t, \cdot )\| \.Hk \lesssim (1 + t) - k - 1\| v0\| \.Hk+1 + (1 + t) - k\| v1\| \.Hk .
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SOME RESULTS ON THE GLOBAL SOLVABILITY FOR STRUCTURALLY DAMPED MODELS . . . 1223
Proof of Theorem 2. We introduce the data space A := (L2 \cap Hs) \times (L1 \cap Hs - 1) and the
solution space X(t) = C([0, t], Hs) \cap C1([0, t], Hs - 1) with the norm
\| u\| X(t) := \mathrm{s}\mathrm{u}\mathrm{p}
0\leq \tau \leq t
\Bigl(
(1 + \tau )
n
4 \| u(\tau , \cdot )\| L2 + (1 + \tau )
n
4 +1\| ut(\tau , \cdot )\| L2+
+(1 + \tau )
n
4 +
s+1
2 \| | D| s - 1ut(\tau , \cdot )\| L2 + (1 + \tau )
n
4 +
s
2 \| | D| su(\tau , \cdot )\| L2
\Bigr)
.
As in the proof of Theorem 1 we define a mapping N in the following way:
N : u \in X(t) \rightarrow Nu \in X(t) with
Nu = G0(t, x) \ast (x) u0 +G1(t, x) \ast (x) u1 +
t\int
0
G1(t - \tau , x) \ast (x) | ut(\tau , \cdot )| pd\tau . (11)
By standard arguments the uniqueness, local and global in time existence will be concluded from the
following pair of inequalities:
\| Nu\| X(t) \leq C\| (u0, u1)\| A + C\| u\| pX(t), (12)
\| Nu - Nv\| X(t) \leq C\| u - v\| X(t)
\bigl(
\| u\| p - 1
X(t) + \| v\| p - 1
X(t)
\bigr)
. (13)
We begin by estimating the L2 norm of Nu itself. To do that we apply the (L1\cap L2) - L2 estimates
on the interval
\biggl[
0,
t
2
\biggr]
and L2 - L2 estimates on the interval
\biggl[
t
2
, t
\biggr]
to conclude
\bigm\| \bigm\| Nu(t, \cdot )
\bigm\| \bigm\|
L2 \lesssim (1 + t) -
n
4 \| (u0, u1)\| A +
t/2\int
0
(1 + t - \tau ) -
n
4
\bigm\| \bigm\| | ut(\tau , \cdot )| p\bigm\| \bigm\| L1\cap L2d\tau +
+
t\int
t
2
(1 + t - \tau )
\bigm\| \bigm\| | ut(\tau , \cdot )| p\bigm\| \bigm\| L2d\tau .
We get immediately from the definitions of Lp norms that\bigm\| \bigm\| | ut(\tau , \cdot )| p\bigm\| \bigm\| L1\cap L2 \lesssim \| ut(\tau , \cdot )\| pLp + \| ut(\tau , \cdot )\| pL2p .
To estimate the norm \| ut(\tau , \cdot )\| Lkp , k = 1, 2, we apply the fractional Gagliardo – Nirenberg inequal-
ity in the form
\| w(\tau , \cdot )\| Lq \lesssim
\bigm\| \bigm\| | D| s - 1w(\tau , \cdot )
\bigm\| \bigm\| \theta 0,s - 1(q,2)
L2 \| w(\tau , \cdot )\| 1 - \theta 0,s - 1(q,2)
L2
with w(\tau , \cdot ) = ut(\tau , \cdot ), where for q \geq 2 we need
\theta 0,s - 1(q, 2) =
n
s - 1
\biggl(
1
2
- 1
q
\biggr)
\in [0, 1)
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1224 P. T. DUONG
that is, 2 \leq q if
n
2(s - 1)
< 1. Since \theta 0,s - 1(p, 2) < \theta 0,s - 1(2p, 2) we obtain on the interval
\biggl(
0,
t
2
\biggr)
the estimate
t/2\int
0
(1 + t - \tau ) -
n
4 \| | ut(\tau , \cdot )| p\| L1\cap L2d\tau \lesssim (1 + t) -
n
4 \| u\| pX(t)
t/2\int
0
(1 + \tau )
- p
\bigl(
n
4 +
s+1
2(s - 1)n
\bigl(
1
2 -
1
p
\bigr) \bigr)
d\tau .
We see that
- p
\biggl(
n
4
+ n
s+ 1
2(s - 1)
\biggl(
1
2
- 1
p
\biggr) \biggr)
< - 1 for p > 1 +
n+ 2(s - 1)
ns
.
On the interval
\biggl(
t
2
, t
\biggr)
, meanwhile, we proceed as follows:
t\int
t
2
(1 + t - \tau )
\bigm\| \bigm\| | ut(\tau , \cdot )| p\bigm\| \bigm\| Lm \lesssim (1 + t)\| u\| pX(t)
t\int
t
2
(1 + \tau )
- p
\bigl(
n
4 +
s+1
2(s - 1)n
\bigl(
1
2 -
1
2p
\bigr) \bigr)
d\tau .
The inequality
- p
\biggl(
n
4
+
s+ 1
2(s - 1)
n
\biggl(
1
2
- 1
2p
\biggr) \biggr)
< - n
4
- 2
holds for
p > 1 +
4(s - 1)
ns
.
Noting that 1 +
4(s - 1)
ns
> 1 +
n+ 2(s - 1)
ns
for s > 1 +
n
2
we arrive at the first condition
p > 1 +
4(s - 1)
ns
for the exponent p. By this approach we have proved the estimate (12) for Nu
itself.
Differentiating (11) with respect to t we obtain
\partial tNu = vt(t, x) +
t\int
0
\partial t
\bigl(
G1(t - \tau , x) \ast (x) | ut(\tau , \cdot )| p
\bigr)
d\tau ,
where we introduce v := G0(t, x) \ast (x) u0 + G1(t, x) \ast (x) u1 as the solution of the corresponding
linear Cauchy problem with the initial data u0, u1.
Using the above techniques for getting the estimate for Nu we arrive at
(1 + \tau )
n
4 +1
\bigm\| \bigm\| \partial tNu(\tau , \cdot )
\bigm\| \bigm\|
L2 \leq C
\bigm\| \bigm\| (u0, u1)\bigm\| \bigm\| A + C\| u\| pX(t) for all \tau \in [0, t]
under the same assumption for p.
Now let us turn to estimate
\bigm\| \bigm\| \partial t| D| s - 1Nu(t, \cdot )
\bigm\| \bigm\|
L2 . We use the following:
\partial t| D| s - 1Nu = | D| s - 1vt(t, x) +
t\int
0
\partial t| D| s - 1
\bigl(
G1(t - \tau , x) \ast (x) | ut(\tau , \cdot )| p
\bigr)
d\tau .
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SOME RESULTS ON THE GLOBAL SOLVABILITY FOR STRUCTURALLY DAMPED MODELS . . . 1225
Taking account of the estimate in Proposition 6 with k = s - 1 and using the (L1\cap L2) - L2 estimates
on the interval
\biggl(
0,
t
2
\biggr)
and L2 - L2 estimates on the interval
\biggl(
t
2
, t
\biggr)
we have
\bigm\| \bigm\| \partial t| D| s - 1(Nu)
\bigm\| \bigm\|
L2 \lesssim (1 + t) -
n+2(s+1)
4 \| (u0, u1)\| A+
+
t
2\int
0
(1 + t - \tau ) -
n+2(s+1)
4
\bigl(
\| | ut(\tau , \cdot )| p\| L1\cap L2 + \| | ut(\tau , \cdot )| p\| \.Hs - 1
\bigr)
d\tau +
+
t\int
t
2
(1 + t - \tau ) - s+1
\bigl(
\| | ut(\tau , \cdot )| p\| L2 + \| | ut(\tau , \cdot )| p\| \.Hs - 1
\bigr)
d\tau .
The integrals with
\bigm\| \bigm\| | ut(\tau , \cdot )| p\bigm\| \bigm\| L1\cap L2 or
\bigm\| \bigm\| | ut(\tau , \cdot )| p\bigm\| \bigm\| L2 will be handled as before if we apply the con-
dition p > 1 +
4(s - 1)
ns
. To estimate the integrals with
\bigm\| \bigm\| | ut(\tau , \cdot )| p\bigm\| \bigm\| \.Hs - 1 we apply the composition
result for p > s. In this way we may proceed further as follows:
t/2\int
0
(1 + t - \tau ) -
n
4 -
s+1
2 \| | ut(\tau , \cdot )| p\| \.Hs - 1d\tau \lesssim
\lesssim
t/2\int
0
(1 + t - \tau ) -
n
4 -
s+1
2 \| | ut(\tau , \cdot )| \| \.Hs - 1\| | ut(\tau , \cdot )| \| p - 1
L\infty d\tau \lesssim
\lesssim
t/2\int
0
(1 + t - \tau ) -
n
4 -
s+1
2 \| | ut(\tau , \cdot )| \| \.Hs - 1\| | ut(\tau , \cdot )| \| p - 1
Hs0d\tau \lesssim
\lesssim
t/2\int
0
(1 + t - \tau ) -
n
4 -
s+1
2 \| | ut(\tau , \cdot )| \| \.Hs - 1
\Bigl(
\| | ut(\tau , \cdot )| \| L2 + \| | ut(\tau , \cdot )| \| \.Hs - 1
\Bigr) p - 1
d\tau
with s - 1 > s0 >
n
2
. Using again the estimates from Proposition 3 we get
t/2\int
0
(1 + t - \tau ) -
n
4 -
s+1
2
\bigm\| \bigm\| | ut(\tau , \cdot )| p\bigm\| \bigm\| \.Hs - 1d\tau \lesssim
\lesssim (1 + t) -
n
4 -
s+1
2 \| u\| pX(t)
t/2\int
0
(1 + \tau ) -
n
4 - 1
\Bigl(
(1 + \tau ) -
n
4 - 1 + (1 + \tau ) -
n
4 -
s+1
2
\Bigr) p - 1
d\tau . (14)
It is obvious that for p > 2 and s > 1 the integral in (14) is uniformly bounded. The same argument
is applied on the interval
\biggl(
t
2
, t
\biggr)
to conclude that
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1226 P. T. DUONG
(1 + \tau )
n
4 +
s+1
2 \| | ut(\tau , \cdot )| p\| \.Hs - 1 \lesssim \| (u0, u1)\| A + \| u\| pX(t) for all \tau \in (0, t).
An analogous reasoning leads to the other estimates which are required by the definition of X(t)-
norm.
The second inequality (13) is obtained by an analogous approach that was carried out in the proof
of Theorem 1. Namely, the L1 \cap L2-norm of f(u(s, x)) - f(v(s, x)) for f(u) = | ut| p is estimated
by using \bigm| \bigm| f(u) - f(v)
\bigm| \bigm| \lesssim | ut - vt|
\bigl(
| ut| p - 1 + | vt| p - 1
\bigr)
,
and by Hölder’s inequality, which leads to\bigm\| \bigm\| f(u(\tau )) - f(v(\tau ))
\bigm\| \bigm\|
L1 \lesssim \| ut(\tau ) - vt(\tau )\| Lp
\Bigl(
\| ut(\tau )\| p - 1
Lp + \| vt(\tau )\| p - 1
Lp
\Bigr)
,\bigm\| \bigm\| f(u(\tau )) - f(v(\tau ))
\bigm\| \bigm\|
L2 \lesssim \| ut(\tau ) - vt(\tau )\| L2p
\Bigl(
\| ut(\tau )\| p - 1
L2p + \| vt(\tau )\| p - 1
L2p
\Bigr)
.
After that the Lp- and L2p-norms of the difference ut - vt are estimated by the fractional Gagliardo –
Nirenberg inequality and they are dominated by \| u - v\| X(t).
Now we will use another version of Kato – Ponce inequality which is stated in terms of homoge-
neous Sobolev spaces.
Proposition 7 (Kato – Ponce inequality for the homogeneous Sobolev spaces, see [3]). For all
functions f \in \.Hs,p2 \cap Lq1 and g \in \.Hs,q2 \cap Lp1 it holds\bigm\| \bigm\| | D| s(fg)
\bigm\| \bigm\|
Lr(\BbbR n)
\lesssim
\bigl[
\| f\| Lp1 (\BbbR n)\| | D| sg\| Lq1 (\BbbR n) + \| | D| sf\| Lp2\BbbR n)\| g\| Lq2 (\BbbR n)
\bigr]
where s > 0 and
1
r
=
1
p1
+
1
q1
=
1
p2
+
1
q2
for 1 < r < \infty , 1 < p1, q2 \leq \infty , 1 < p2, q1 < \infty .
By writing
\bigm\| \bigm\| | ut| p - | vt| p
\bigm\| \bigm\|
\.H\gamma \lesssim
1\int
0
\bigm\| \bigm\| (ut(s) - vt(s))G(\theta ut + (1 - \theta )vt)
\bigm\| \bigm\|
\.H\gamma d\theta
and by applying the Kato – Ponce inequality for the \.H\gamma -norm for the product fg with f := G(\theta ut+
+(1 - \theta )vt) and g := ut(s) - vt(s) in the right-hand side of the above estimate with suitable constants
p1, q1, p1, q2 > 0 we are able to bound the norm
\bigm\| \bigm\| G(\theta ut + (1 - \theta )vt)
\bigm\| \bigm\|
\.H\gamma ,q1
for G = u| u| p - 2 with
some constant q1 > 0 to be chosen later.
The norm in \.H\gamma ,q2 for G
\bigl(
\theta ut+(1 - \theta )vt
\bigr)
can be estimated if we apply the general composition
result which is stated for a broader class of functions. We introduce the class \mathrm{L}\mathrm{i}\mathrm{p}\mu in the following
(see [7]).
Definition 1. Let \mu > 0, N \in \BbbN 0 and 0 < \alpha \leq 1 such that \mu = N + \alpha . Then we define
\mathrm{L}\mathrm{i}\mathrm{p}\mu =
\Biggl\{
f \in CN,loc(\BbbR ) : f (j)(0) = 0, j = 0, . . . , N, and \mathrm{s}\mathrm{u}\mathrm{p}
t0 \not =t1
\bigm| \bigm| f (N)(t0) - f (N)(t1)
\bigm| \bigm|
| t0 - t1| \alpha
< \infty
\Biggr\}
.
Further we put
\| f\| Lip\mu =
N - 1\sum
j=0
| f (j)(t)|
| t| \mu - j
+ \mathrm{s}\mathrm{u}\mathrm{p}
t0 \not =t1
\bigm| \bigm| f (N)(t0) - f (N)(t1)
\bigm| \bigm|
| t0 - t1| \alpha
.
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SOME RESULTS ON THE GLOBAL SOLVABILITY FOR STRUCTURALLY DAMPED MODELS . . . 1227
It is clear that | t| \mu \in \mathrm{L}\mathrm{i}\mathrm{p}\mu , t| t| \mu - 2 \in \mathrm{L}\mathrm{i}\mathrm{p} (\mu - 1) for \mu > 1. The following helpful general
composition result for the class \mathrm{L}\mathrm{i}\mathrm{p}\mu was obtained in [7]. Let us denote \sigma p = n\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
0;
1
p
- 1
\biggr\}
.
Proposition 8 (Theorem 6.3.4 (i) in [7]). Let \sigma p < s < \mu and \mu > 1.
Then there exists some constant c such that
\| G(f)\| F s
p,q
\leq c\| G\| Lip\mu \| f\| F s
p,q
\| f\| \mu - 1
L\infty
holds for all f \in F s
p,q \cap L\infty and all G \in \mathrm{L}\mathrm{i}\mathrm{p}\mu .
Proposition 8 together with the Sobolev embedding imply immediately the following consequence
in the supercritical case s >
n
2
.
Corollary 2. Let s \in
\biggl(
n
2
, p
\biggr)
. Denote either G(u) = | u| p or G = \pm u| u| p - 1 with p > 1. Then
for all u \in Hs the following composition estimate holds:
\| G(u)\| Hs \lesssim \| u\| pHs .
With G = u| u| p - 2, by choosing pi, qi, i = 1, 2, as in the proof of Theorem 1 and by estimating
the homogeneous
\bigm\| \bigm\| G(\theta ut + (1 - \theta )vt)
\bigm\| \bigm\|
\.H\gamma from above by the nonhomogeneous norm
\bigm\| \bigm\| G(\theta ut +
+ (1 - \theta )vt)
\bigm\| \bigm\|
H\gamma Corollary 2 allows us now to obtain the second estimate (13). It is possible since
p - 1 > \gamma = s - \sigma for \sigma > 1, p > s and both of the norms \| ut\| L2 , \| ut\| \.H\gamma are included in the
norm of X(t).
Theorem 2 is proved.
Remark 2. We have applied two different approaches to deal with the nonlinearity of the form
| ut| p. The first one, which was used in the proof of Theorem 1, follows the Matsumura technique to
use some monotone function h(t) that doesn’t require the exact expression. Therefore the condition
for p could be relaxed very nicely up to p > 1 +
2s
n
for the general nonlinearity function f(ut)
satisfying a certain power growth at infinity. However the regularity s of the solution must be higher
than that which is obtained by the second approach in the proof of Theorem 2. We have applied the
Runst – Sickel composition there to estimate the term | ut| p with the weaker condition on s that is
s > 1+
n
2
. On the other hand the condition for the exponent p is more strict by following the second
approach. We see that for the solvability of the Cauchy problem in Theorem 2 the admissible values
for p must be larger than \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
2; s; 1 +
4(s - 1)
ns
\biggr\}
.
4. Further studies and concluding remarks. The Cauchy problem that has been treated in
Sections 2 and 3 could be generalized to the case of x-dependent coefficient models. We will deal
with an operator B = B(x) that behaves in some sense as b(x, t)( - \Delta )\sigma .
Consider the x-dependent functional matrix b(x, t) = b(x), where b(x) = (bij(x))1\leq i,j\leq n, bij =
= bji and bij \in C0. We assume that the (n \times n)-matrix b = b(x) satisfies the following growth
condition at infinity:
b0
\bigl(
1 + | x|
\bigr) \beta | \xi | 2 \leq b(x)\xi \cdot \xi \leq b1
\bigl(
1 + | x|
\bigr) \beta | \xi | 2
for some positive constants b0, b1.
In order to define b(x)( - \Delta )\sigma , we restrict ourselves to the following reduction of the divergence-
like form. Assume that
\surd
B is a self-adjoint operator acting on L2(\Omega ) such that the following identity
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1228 P. T. DUONG
for the scalar product is satisfied:\bigl\langle \surd
Bu,
\surd
Bv
\bigr\rangle
2
=
\Bigl\langle
b(x)( - \Delta )\sigma /2u, ( - \Delta )\sigma /2v
\Bigr\rangle
2
, u, v \in \scrD
\bigl( \surd
B
\bigr)
, (15)
where \scrD
\bigl( \surd
B
\bigr)
is the completion of C\infty
0 with respect to the norm
u \mapsto \rightarrow
\Bigl( \Bigl\langle
b(x)( - \Delta )\sigma /2u, ( - \Delta )\sigma /2u
\Bigr\rangle
2
+ \| u\| 2
\Bigr) 1/2
.
The existence of such self-adjoint operator is guaranteed by a result on the Friedrichs extension of
the nonnegative operators. Actually, the operator b(x)( - \Delta )\sigma may not be self-adjoint, however by
considering the completion of the space C\infty
0 with respect to the above norm, in which all Cauchy
sequences are converging, there exists in a unique way such an operator
\surd
B that can be denoted by\bigl(
b(x)( - \=\Delta )\sigma
\bigr) 1/2
in order to distinguish it from the usual notation
\bigl(
b(x)( - \Delta )\sigma
\bigr) 1/2
.
We see that the formula (15) is valid for such functions u, v the fractional Laplacian of which
have some behavior at infinity as | x| - \beta /2. This means
\scrD
\bigl( \surd
B
\bigr)
\supset
\bigl\{
u \in H
\delta /2
0 (\Omega ) :
\bigl(
1 + | x| \beta /2
\bigr)
| ( - \Delta )\delta /2u| \in L2(\Omega )
\bigr\}
.
We define the operator B and its fractional powers as operator functions of
\surd
B using the
spectral calculus for operators (known also as the identity resolution of the spectrum). Particularly,
B :=
\bigl( \surd
B
\bigr) 2
, where
\surd
B is the above Friedrichs construction based on
\bigl(
b(x)( - \Delta )\sigma
\bigr) 1/2
.
Let \Omega be the exterior of a compact set K in \BbbR n with smooth boundary. Now we consider the
following mixed problem in \Omega :
utt +Bu+ ut = 0, x \in \Omega , t > 0,
u(0, x) = u0(x),
ut(0, x) = u1(x),
u(t, x) = 0, x \in \partial \Omega , t \geq 0.
(16)
The decay estimates, which have a close relation to solutions to the exterior damped model with
constant coefficients, will be proved by using a generalization of Gagliardo – Nirenberg’s and Hardy’s
inequalities with arbitrary derivative orders. In order to obtain the decay estimates stated in the
Conjecture 1, which will be formulated later, we need some information for the solution to the
parabolic mixed problem
vt +Bv = 0, x \in \Omega , t > 0,
v(0, x) = v0(x),
v(x, t) = 0, x \in \partial \Omega , t \geq 0.
First of all, using the equivalence of the Nash inequality and the L\infty - L1 decay of the semigroup
\{ e - tB\} , we will show that the following inequality holds for the operator B :
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SOME RESULTS ON THE GLOBAL SOLVABILITY FOR STRUCTURALLY DAMPED MODELS . . . 1229
\| f\| 2+2(2\sigma - \beta )/n
L2 \lesssim \langle Bf, f\rangle \| f\| 2(2\sigma - \beta )/n
L1 (17)
for all function f \in L1(\Omega ) \cap \scrD (B).
In fact, we have the following application of Hölder’s inequality:
\| f\| 2L2 \leq \| f\| 1/p
L1 \| f\|
(2p - 1)/p
L(2p - 1)/(p - 1) .
Choosing now p =
n+ 2\sigma - \beta
2(2\sigma - \beta )
and raising the last inequality to 1 +
2\sigma - \beta
n
we arrive at
\| f\| 2+2(2\sigma - \beta )/n
L2 \leq \| f\| 2(2\sigma - \beta )/n
L1 \| f\| 2
L2n/(n+\beta - 2\sigma ) .
In order to estimate the norm \| f\|
L
2n
n+\beta - 2\sigma
of f by the scalar product \langle Bf, f\rangle we need the follow-
ing generalization of the classical Hardy – Littlewood result on fractional derivatives inequality of
Sobolev type which was obtained by E. Stein and G. Weiss in [11].
Let us introduce the operator T\lambda acting on the functions f \in L1
loc(\BbbR n) as follows:
T\lambda : f \mapsto \rightarrow T\lambda f =
\int
\BbbR n
f(x)
| x - y| \lambda
dy, 0 < \lambda < n.
Proposition 9 (Theorem B* in [11]). Let 0 < \lambda < n, 1 < p < \infty , \alpha < n/p\prime , \beta < n/q,
\alpha + \beta \geq 0, 1/q = (1/p) + [(\lambda + \alpha + \beta )/n] - 1. If p \leq q < \infty , then\left( \int
\BbbR n
\{ | T\lambda f | | x| - \beta \} qdx
\right) 1/q
\lesssim
\left( \int
\BbbR n
\{ | f(x)| | x| \alpha \} p
\right) 1/p
.
Now by the definition of the Riesz potential, it is clear that T\lambda is related to the fractional Laplacian
with a negative exponent by the following relation:
T\lambda = c\lambda ( - \Delta )
\lambda - n
2
for 0 < \lambda < n with a constant c\lambda > 0.
Proposition 9 can be rewritten in a more convenient form involved the fractional Laplacian with
positive exponent as follows:
\| u\| Lq \lesssim
\bigm\| \bigm\| | x| \alpha ( - \Delta )\mu u
\bigm\| \bigm\|
Lp
for 0 < \mu <
n
2
, 0 < \alpha <
n
p\prime
with p, q, \alpha , \mu being related by
1
q
=
1
p
+
\alpha - 2\mu
n
.
Choosing
q =
2n
n+ \beta - 2\sigma
, p =
n+ 2\sigma - \beta
2(2\sigma - \beta
, \alpha =
\beta
2
in the last inequality we get, for \beta \in (0, n) and \sigma \in (0, n), the following inequality:
\| f\|
L
2n
n+\beta - 2\sigma
\lesssim
\bigm\| \bigm\| | x| \beta /2\nabla \sigma f
\bigm\| \bigm\|
L2 .
Recalling the condition for B we arrive at the desired estimate (17).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
1230 P. T. DUONG
It is well known (see [1]) that (17) is equivalent to the following relation:
\| e - tB\| L\infty \lesssim \| f\| L1 \forall f \in L1(\Omega ) \forall t > 0. (18)
Usually, the inequalities (17) or (18) are not sufficient to imply the decay rate of \{ e - tB\} in other
L1 - Lq scales with arbitrary q > 1. The so-called Markov property including the positivity and the
contraction for the semigroup \{ e - tB\} must be satisfied in order to obtain further energy estimates
for the solution u of the mixed problem (16). The positivity, however, does not hold for general \sigma
even for the constant coefficient case.
For example, if we can prove the Markovian property for \{ e - tB\} , then the above Nash-type
estimates allow us to obtain the following result.
Conjecture 1. Suppose n > 2\sigma . Moreover, we assume that the initial data belong to
(u0, u1) \in \scrD
\bigl( \surd
B \cap L1(\Omega )
\bigr)
\times
\bigl(
L2(\Omega ) \cap L1(\Omega )
\bigr)
,
where B is defined as above. Then the solution of (16) satisfies the following decay estimates for
t \geq 1:
\| u\| L2 \lesssim t
- n
2(2\sigma - \beta )
\Bigl(
\| u0\| L1 + \| u1\| L1 + \| u0\| L2 + \|
\bigl( \surd
B + 1
\bigr) - 1
u1\| L2
\Bigr)
,
\|
\surd
Bu\| L2 \lesssim t
- n
2(2\sigma - \beta ) -
1
2
\Bigl(
\| u0\| L1 + \| u1\| L1 + \|
\surd
Bu0\| L2 + \| u1\| L2
\Bigr)
,
\| ut\| L2 \lesssim t
- n
2(2\sigma - \beta ) - 1
\Bigl(
\| u0\| L1 + \| u1\| L1 + \|
\surd
Bu0\| L2 + \| u1\| L2
\Bigr)
.
After getting these estimates we can follow the approach presented in Section 3 to study the
mixed problem
utt +Bu+ ut = | ut| p, x \in \Omega , t > 0,
u(0, x) = u0(x),
ut(0, x) = u1(x),
u(x, t) = 0, x \in \partial \Omega , t \geq 0.
In the next step of our study, we will try to answer the question proposed above about the Markov
property of the semigroup \{ e - tB\} . The case of t-dependent coefficients brings some interest to our
models and must be treated carefully. These results for the nonlinear problem with x-dependent
coefficient will appear in a forthcoming paper.
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Received 12.01.18
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
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| id | umjimathkievua-article-1629 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:28Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/dd/98b8b9a299b8e26ab6df157962cd48dd.pdf |
| spelling | umjimathkievua-article-16292019-12-05T09:21:25Z Some results on the global solvability for structurally damped models with a special nonlinearity Деякi результати про глобальну розв’язнiсть для моделей зi структурним затуханням та нелiнiйнiстю спецiального вигляду Duong, P. T. Дуонг, П. Т. The main purpose of this paper is to prove the global (in time) existence of solution for the semilinear Cauchy problem $$u_{tt} + (-\Delta )^{\sigma} u + (-\Delta )^{\delta} u_t = |u_t|^p,\; u(0, x) = u_0(x),\; u_t(0, x) = u_1(x)$$. The parameter $\delta \in (0, \sigma]$ describes the structural damping in the model varying from the exterior damping $\delta = 0$ up to the visco-elastic type damping $\delta = \sigma$. We will obtain the admissible sets of the parameter p for the global solvability of this semilinear Cauchy problem with arbitrary small initial data u0, u1 in the hyperbolic-like case $\delta \in \Bigl(\cfrac{\sigma}{2} , \sigma \Bigr)$, and in the exceptional case $\delta = 0$. Основною метою цiєї статтi є доведення глобального (за часом) iснування розв’язку напiвлiнiйної задачi Кошi $$u_{tt} + (-\Delta )^{\sigma} u + (-\Delta )^{\delta} u_t = |u_t|^p,\; u(0, x) = u_0(x),\; u_t(0, x) = u_1(x)$$. Параметр $\delta \in (0, \sigma]$ описує структурне затухання в моделi, що змiнюється вiд зовнiшнього затухання $\delta = 0$ до затухання в’язкоеластичного типу $\delta = \sigma$. Визначено множини параметра p, допустимi з точки зору глобальної розв’язностi даної напiвлiнiйної задачi Кошi з як завгодно малими початковими даними $u_0, u_1$ у випадку гiперболiчного типу $\delta \in \Bigl(\cfrac{\sigma}{2} , \sigma \Bigr)$, а також у винятковому випадку $\delta = 0$. Institute of Mathematics, NAS of Ukraine 2018-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1629 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 9 (2018); 1211-1231 Український математичний журнал; Том 70 № 9 (2018); 1211-1231 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1629/611 Copyright (c) 2018 Duong P. T. |
| spellingShingle | Duong, P. T. Дуонг, П. Т. Some results on the global solvability for structurally damped models with a special nonlinearity |
| title | Some results on the global solvability for structurally damped models with a special
nonlinearity |
| title_alt | Деякi результати про глобальну розв’язнiсть
для моделей зi структурним затуханням
та нелiнiйнiстю спецiального вигляду |
| title_full | Some results on the global solvability for structurally damped models with a special
nonlinearity |
| title_fullStr | Some results on the global solvability for structurally damped models with a special
nonlinearity |
| title_full_unstemmed | Some results on the global solvability for structurally damped models with a special
nonlinearity |
| title_short | Some results on the global solvability for structurally damped models with a special
nonlinearity |
| title_sort | some results on the global solvability for structurally damped models with a special
nonlinearity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1629 |
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