Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation
We study a global mixed problem for the nonlinear Schrödinger equation with a nonlinear term in which the coefficient is a generalized function. A global solvability theorem for the analyzed problem is proved by using the general solvability theorem from [K. N. Soltanov, Nonlin. Anal.: Theory, Meth....
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| author | Soltanov, K. N. Солтанов, К. Н. |
| author_facet | Soltanov, K. N. Солтанов, К. Н. |
| author_sort | Soltanov, K. N. |
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| description | We study a global mixed problem for the nonlinear Schrödinger equation with a nonlinear term in which the coefficient is a generalized function. A global solvability theorem for the analyzed problem is proved by using the general solvability theorem from [K. N. Soltanov, Nonlin. Anal.: Theory, Meth., Appl., 72, No. 1 (2010)]. We also investigate the behavior of the solution of the problem under consideration. |
| first_indexed | 2026-03-24T02:16:08Z |
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UDC 517.9
K. N. Soltanov (Hacettepe Univ., Ankara, Turkey)
GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR
OF A NONLINEAR EQUATION OF THE SCHRÖDINGER TYPE*
ГЛОБАЛЬНЕ IСНУВАННЯ ТА ДОВГОЧАСОВА ПОВЕДIНКА
НЕЛIНIЙНОГО РIВНЯННЯ ТИПУ ШРЬОДIНГЕРА
We study a global mixed problem for the nonlinear Schrödinger equation with nonlinear addition in which the coefficient
is a generalized function. We prove a global solvability theorem for the considered problem with the use of the general
solvability theorem from [Soltanov K. N. Perturbation of the mapping and solvability theorems in the Banach space //
Nonlinear Anal.: Theory, Meth. and Appl. – 2010. – 72, № 1]. Furthermore, we also investigate the behavior of the solution
of the analyzed problem.
Вивчається глобальна мiшана задача для нелiнiйного рiвняння Шрьодiнгера з нелiнiйним додаванням, в якiй коефi-
цiєнтом є узагальнена функцiя. Доведено теорему про глобальну розв’язнiсть поставленої задачi на основi загальної
теореми про розв’язнiсть з [Soltanov K. N. Perturbation of the mapping and solvability theorems in the Banach space //
Nonlinear Anal.: Theory, Meth. and Appl. – 2010. – 72, № 1]. Крiм того, дослiджено поведiнку розв’язку задачi, що
вивчається.
We consider the following problem for the nonhomogeneous nonlinear Schrödinger equation
i
∂u
∂t
−∆u+ f (x, u) = h(t, x), (t, x) ∈ R+ × Ω ≡ Q, (0.1)
u(0, x) = u0(x), x ∈ Ω ⊂ Rn, n ≥ 1, u
∣∣
R+×∂Ω = 0, (0.2)
where h(t, x) and u0(x) are complex functions, f (x, τ) is a distribution (generalized function)
with respect to variable x ∈ Ω, Ω is a bounded domain with sufficiently smooth boundary ∂Ω,
i ≡
√
−1. We investigate this problem in the case, when the function f(x, t) can be represented as
f (x, u) = q(x) |u(t, x)|p−2 u(t, x)+a(x) |u(t, x)|p̃−2 u(t, x), i.e., the function f has the growth with
respect to unknown function of the polynomial type, where a : Ω −→ R is some function and q :
Ω −→ R is a generalized function, p ≥ 2, p̃ ≥ 2, h ∈ L2(Q) (i.e., h(t, x) ≡ h1(t, x) + ih2(t, x) and
hj ∈ L2(Q), j = 1, 2).
The nonlinear Schrödinger equation of the type (0.1), and also steady-state case of the equa-
tion (0.1) arises in several models of different physical phenomena corresponding to various function
f. The equation of such type were studied in many articles under different conditions on the function
f in the dynamic case (see, for example, [2, 4, 6 – 9, 14, 17, 18, 20, 22, 24, 32, 33, 35] and the
references therein) and in the steady-state case (see, for example, [1, 3, 5, 10 – 16, 18, 19, 23 – 26, 28,
33, 34, 36] and references therein). It is known that in this case the equation (0.1) in the steady-state
case (i.e., if u is independent of t) is an equation of the semiclassical nonlinear Schrödinger type (i.e.,
NLS) (see [1, 2, 3, 10, 13] and references therein). Considerable attention has been paid in recent
years to the problem (0.1) for small ε > 0 as the coefficients of the linear part since the solutions are
known as in the semiclassical states, which can be used to describe the transition from quantum to
classical mechanics (see [3, 10 – 14, 23 – 25, 33 – 36] and references therein).
* This paper was supported by 110T558-project of TUBITAK.
c© K. N. SOLTANOV, 2015
68 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR OF A NONLINEAR EQUATION . . . 69
In the above mentioned articles the equation (0.1), and also the steady-state case was considered
with various functions f(x, u) that are mainly Carathéodory functions1 with some additional prop-
erties. Moreover, in some of these articles are presumed, that dates of considered problem possess
more smoothness and study the behavior of a solution of the posed problem with use the Fourier mod.
Although such cases when f(x, u) possesses a singularity with respect to the variable x of certain type
were also investigated (as equations Emden – Fowler, Yamabe, NLS etc.), but in all of these articles
the coefficient q(x) is a function in the usual sense (of a Lebesgue space or of a Sobolev space).
In this paper we study problem (0.1), (0.2) in the case when f have the above representation and
the function q is a generalized function and the behavior of the solution. Moreover here for the proof
of the existence theorem of the problem is used some different method, which allow us several other
possibility. It should be noted that the steady-state case of the problem of such type were studied in
[28]. Here we study problem (0.1), (02) globally in the dynamical case, which in [32] is studied for
t ∈ (0, T ), T < ∞. Here an existence theorem (Section 1) for the problem (0.1), (0.2) is proved in
the model case when f(x, u) only has the above expression (Section 4).
In Section 2 we have defined how to understand the equation (0.1) with use of representation of
certain generalized functions and properties of some special class of functions (see, for example, [27,
28]). In Section 3 we have conducted variants of the general results from [29, 31], on which the
proof of the solvability theorem is based and in Section 5 is studied the behavior of the solutions of
the considered problem under certain additions conditions.
1. Statement of the main solvability result. Let the operator f (x, u) have the form
f (x, u) = q(x)|u|p−2u+ a (x) |u(t, x)|p̃−2 u (t, x) (1.1)
in the generalized sense, where q ∈W−1,p0(Ω), p0 ≥ 2 (it should be noted that either p0 ≡ p0 (p) or
p ≡ p (p0)), a : Ω −→ R and u : Q −→ C is an element of the space of sufficiently smooth functions
that will be determined below (see Section 2). Consequently the function q(x) is a generalized
function, which has singularity of the order 1.
We will set some necessary denotations. Everywhere later the expression of the type u ∈
∈ Lm
(
R+;W 1,2
0 (Ω)
)
∩ L2
(
R+;W 1,2
0 (Ω)
)
≡ L(2,m)
(
R+;W 1,2
0 (Ω)
)
for u : Q −→ C denote the
following:
(u1, u2) ∈
(
Lm1
(
R+;W 1,2
0 (Ω)
))2
≡
(
Lm1
(
R+;W 1,2
0 (Ω)
)
, Lm1
(
R+;W 1,2
0 (Ω)
))
holds for m1 ∈ [2,m] , where u(t, x) ≡ u1(t, x) + iu2(t, x), m ≥ 2∗ consequently we can set
u(t, x) ≡ (u1(t, x), u2(t, x)), i.e., u : Q −→ R2.
Everywhere later 〈·, ·〉 and [·, ·] denote the dual form for the pair (X,X∗) of the Banach space X
and its dual space X∗, for example, in the case when X ≡ W 1,2
0 (Ω) and X ≡ Lm1
(
R+;W 1,2
0 (Ω)
)
we have
(X,X∗) ≡
((
W 1,2
0 (Ω)
)2
,
(
W 1,2
0 (Ω)
)2
)
1Let f : Ω × Rm −→ R be a given function, where Ω is a nonempty measurable set in Rn and n,m ≥ 1. Then f
is Carathéodory function if the following hold: x −→ f(x, η) is measurable on Ω for all η ∈ Rm, and η −→ f(x, η) is
continuous on Rm for almost all x ∈ Ω.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
70 K. N. SOLTANOV
and
(X,X∗) ≡
((
Lm1
(
R+;W 1,2
0 (Ω)
))2
,
(
Lm
′
1
(
R+;W−1,2(Ω)
))2
)
,
respectively, where m′1 =
m1
m1 − 1
. In the other words we will understand these expressions every-
where later as the following representations:
〈g, w〉 ≡
∫
Ω
g(x)w(x)dx, gj ∈W 1,2
0 (Ω), wj ∈W−1,2(Ω), g ≡ g1 + ig2,
and
[g, w] ≡
∞∫
0
∫
Ω
g(t, x)w(t, x)dxdt, gj ∈ Lm1
(
R+;W 1,2
0 (Ω)
)
,
wj ∈ Lm
′
1
(
R+;W−1,2(Ω)
)
respectively.
Assume the following conditions:
(i) let p̃ <
n+ 2
n− 2
if n ≥ 3, p̃ ∈ [2,∞) if n = 1, 2 and a ∈ L∞(Ω);
(ii) there exist numbers k0 ≥ 0, p2 ≥ 1 and k1 ≤ min
{
1;
p̃
p
}
such that 1 ≤ p2 <
2n
n− 2
, if
n ≥ 3, 2 ≤ p2 <∞, if n = 1, 2 and〈
a(x)|u|p̃−2u, u
〉
≥ −k0 ‖u‖2p2 − k1
〈
q(x)|u|p−2u, u
〉
(1.2)
holds for any u ∈ L(2,m)
(
R+;W 1,2
0 (Ω)
)
and a.e. t ≥ 0, where 1 > C(2, p2)2 · k0. Here C(2; p2) is
the constant of the known inequality of embedding theorems for Sobolev spaces
‖∇u‖2 ≥ C(2; p2) ‖u‖p2 ∀u ∈W 1,2
0 (Ω).
Definition 1.1. A function
u ∈ Lm
(
R+;W 1,2
0 (Ω)
)
∩ L2
(
R+;W 1,2
0 (Ω)
)
∩
{
u
∣∣∣∣ ∂u∂t ∈ L2
(
R+;L2(Ω)
)
; u(0, x) = u0
}
is called a solution of the problem (0.1), (0.2) if the following equation is fulfilled:
∞∫
0
∫
Ω
[
i
∂u
∂t
−∆u+ f (x, u)
]
ϕdxdt =
∞∫
0
∫
Ω
hϕdxdt (1.3)
for any ϕ ∈ L(2,m)
(
R+;W 1,2
0 (Ω)
)
.
It should be noted that the sense in which equation (1.3) is to be understood will be explained
below (Section 2). We have proved the following result for the considered problem.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR OF A NONLINEAR EQUATION . . . 71
Theorem 1.1. Let the function f have the representation (1.1) in the generalized sense, where
q ∈ W−1
p0 (Ω) is a nonnegative distribution (generalized function2), p0 =
2n
2(n− 1)− p(n− 2)
,
2(n− 1)
n− 2
> p > 2 if n ≥ 3, p0, p > 2 are arbitrary if n = 2, and p0, p ≥ 2 are arbitrary
if n = 1
(
in particular, if n = 3, then 2 < p < 4 and p0 =
6
4− p
)
and conditions (i), (ii)
are fulfilled. Then for any h ∈ L2(Q) and u0 ∈ W 1,2
0 (Ω) the problem (0.1), (0.2) is solvable in
L(2,m)
(
R+;W 1,2
0 (Ω)
)
∩
{
u
∣∣∣∣ ∂u∂t ∈ L2
(
R+;L2(Ω)
)
; u(0, x) = u0
}
.
For the investigation of the considered problem we used some general solvability theorems, which
are conducted in Section 3. We begin with explanation of equation (1.3).
2. The solution concept and function spaces. So we will consider the case when the function
f (x, u) has the form (1.1) where functions a, q and u are the same as above. Consequently
the function q(x) is a generalized function, which has singularity of order 1. Therefore we must
understand the equation (0.1) in the generalized function space sense, i.e.,∫
Ω
[
i
∂u
∂t
−∆u+ f (x, u)
]
ϕ(x)dx ≡
≡
∫
Ω
[
i
∂u
∂t
−∆u(t, x) + q(x) |u(t, x)|p−2 u(t, x)
]
ϕ(x)dx−
−
∫
Ω
a(x) |u(t, x)|p̃−2 u(t, x) ϕ(x)dx =
∫
Ω
h(t, x)ϕ(x)dx (2.1)
for any ϕ ≡ ϕ1 + iϕ2, ϕj ∈ D(Ω), j = 1, 2, where D(Ω) is C∞0 (Ω) and suppϕj ⊂ Ω with
corresponding topology. Here the equation (2.1) will be understood in the sense of the space
L2(R+) .
In the beginning we need to define the expression q|u|p−2u. It is known that (see, for example,
[21]) in the case when q ∈ W−1
p0 (Ω) we can represent it in the form q(x) ≡
∑n
k=0
Dkqk(x),
Dk ≡
∂
∂xk
, D0 ≡ I, qk ∈ Lp0(Ω), k = 0, 1, n, in the generalized function space sense. From here it
follows that if a solution of the considered problem belongs to the space which contains to W 1,p̃1
0 (Ω)
for some number p̃1 > 1 then we can understand the term q|u|p−2u in the following sense:〈
q |u|p−2u, ϕ
〉
≡
∫
Ω
q(x)|u(t, x)|p−2u(t, x)ϕ(x)dx (2.2)
for any ϕ ≡ ϕ1 + iϕ2, ϕj ∈ D(Ω), j = 1, 2, and a.e. t > 0. Therefore we must find the needed
number p̃1 ≥ 2. Namely we must find the relation between the numbers p0 and p̃1. So, taking into
account that for a function u ∈ Lm
(
R+;W 1,2
0 (Ω)
)
, i.e., p̃1 = 2 (as h ∈ L2(Q) by the assumption)
we have u ∈ Lm
(
R+;Lp̃
∗
1(Ω)
)
, where p̃∗1 = 2∗ =
2n
n− 2
for n ≥ 3 by virtue of the embedding
theorem, from (2.2) we get
2 See Definition 2.1 of the Section 2.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
72 K. N. SOLTANOV
〈
q |u|p−2u, ϕ
〉
≡
∫
Ω
q(x) |u(t, x)|p−2 u(t, x)ϕ(x)dx =
=
∫
Ω
n∑
k=0
∂
∂xk
qk(x) |u(t, x)|p−2 u(t, x)ϕ(x)dx =
= −
∫
Ω
n∑
k=1
qk|u|p−2u
∂ϕ
∂xk
dx−
∫
Ω
n∑
k=1
qk
∂
∂xk
(
|u|p−2u
)
ϕdx+
∫
Ω
q0 |u|p−2 uϕdx =
= I1 + I2 +
∫
Ω
q0|u|p−2uϕdx (2.3)
by virtue of the generalized function theory.
Remark 2.1. It should be noted that I2 is estimated by following way. Firstly we note that
∂
∂xk
(
|u|p−2u
)
=
∂
∂xk
(u, u)
p−2
2 u = (p− 2) (u, u)
p−4
2
(
∂u
∂xk
, u
)
u+ (u, u)
p−2
2
∂u
∂xk
=
= (p− 2) (u, u)
p−4
2
(
∂u
∂xk
, u
)
u+ |u|p−2 ∂u
∂xk
=
= (p− 2) |u|p−4
(
∂u
∂xk
, u
)
u+ |u|p−2 ∂u
∂xk
,
here as known
(
u(t, x), u(t, x)
)
= |u(t, x)|2 for u(t, x) ∈ C. Consequently we have
I2 ≡
∫
Ω
n∑
k=1
qk
∂
∂xk
(
|u|p−2u
)
ϕdx =
=
∫
Ω
n∑
k=1
qk|u|p−2 ∂u
∂xk
ϕdx+
∫
Ω
(p− 2)
n∑
k=1
qk|u|p−4
(
∂u
∂xk
, u
)
uϕdx ≤
≤
∫
Ω
n∑
k=1
qk|u|p−2
∣∣∣∣ ∂u∂xk
∣∣∣∣ |ϕ| dx+ (p− 2)
∫
Ω
n∑
k=1
qk|u|p−2
∣∣∣∣ ∂u∂xk
∣∣∣∣ |ϕ| dx =
= (p− 1)
∫
Ω
n∑
k=1
qk|u|p−2
∣∣∣∣ ∂u∂xk
∣∣∣∣ |ϕ| dx.
Here and in what follows we assume n ≥ 3. Because if n = 1, 2 then we can choose arbitrary
p ≥ 2, as will be observed below. Let us take into account that ϕj ∈ D(Ω) and n ≥ 3, then in
order for the expression in the left part of (2.3) to have the meaning, it is enough for us to take
1 ≤ p−1 ≤ 2n (p0 − 1)
p0(n− 2)
for the integral I1 and 0 ≤ p−2 ≤ n (p0 − 2)
p0(n− 2)
for the integral I2. Therefore
if 2 ≤ p ≤ 3np0 − 2 (n+ 2p0)
p0(n− 2)
then the left part of (2.3) is defined. Now, let ϕj ∈ W 1,2
0 (Ω),
j = 1, 2. Then it is sufficient to study one of the I1 and I2. Let us consider I1, from which we obtain,
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR OF A NONLINEAR EQUATION . . . 73
that 2 ≤ p ≤ 2np0 − 2 (n+ p0)
p0(n− 2)
, moreover we can choose p ≥ 2 only if p0 > n. On the other hand,
if we take into account that given p, we obtain p0 =
2n
2(n− 1)− p(n− 2)
, and consequently in order
for p0 < ∞ we must choose 2(n − 1) > p(n − 2) or p <
2(n− 1)
n− 2
. In the case when n = 3 then
p < 4 and p0 =
6
4− p
.
Thus we determined under what conditions the left part of (2.3) is defined. Hence that implies
the correctness of the statement.
Proposition 2.1. Assume f̃ be an operator defined by expression f̃(u) ≡ q|u|p−2u, where q ∈
∈W−1,p0(Ω), and u ∈ L(2,m)
(
R+;W 1,2
0 (Ω)
)
. If 2 ≤ p < 2(n− 1)
n− 2
and p0 =
2n
2(n− 1)− p(n− 2)
if n ≥ 3
(
in particular, if n = 3, then 2 ≤ p < 4 and p0 =
6
4− p
)
, then f̃ : L(2,m)
(
R+;W 1,2
0 (Ω)
)
−→
−→ L2
(
R+;W−1,2(Ω)
)
is a bounded operator.
So that exactly explains Proposition 2.1 and the representation (2.2) for any ϕj ∈ W 1,2
0 (Ω) we
consider the following class of the functions u : Ω −→ C:
Mη,W 1,β(Ω) ≡
{
u ∈ L(Ω)
∣∣∣ η(u) ∈W 1,β(Ω), η(u) ≡ |u|α/β u
}
≡ S1,α,β(Ω), (2.4)
where α ≥ 0, β > 1 are certain numbers, W 1,β(Ω) is a Sobolev space, i.e., we consider a class of
the pn-spaces3.
It is not difficult to see that if 1 ≤ α0 + β0 ≤ α1 + β1, 0 ≤ β0 < β1, α1β0 ≤ α0β1, 1 ≤ β1 then∫
Ω
|u|α0
n∑
k=1
|Dku|β0 dx ≤ c
∫
Ω
|u|α1
n∑
k=1
|Dku|β1 dx+ c1 (2.5)
holds for any u ∈ C1
0 (Ω), where constants c, c1 ≥ 0 are independent from u.
Furthermore if we will introduce the spaceM
η,W 1,β
0 (Ω)
≡
◦
S1,α,β(Ω) ≡ S1,α,β(Ω)∩{u(x)|u|∂Ω = 0}
then we get the following lemma.
Lemma 2.1. Let u ∈ W 1,2
0 (Ω) and the number p satisfy the inequation 2 < p <
2(n− 1)
n− 2
,
n ≥ 3. Then the function v(x) ≡ η (u(x)) ≡ |u(x)|p belongs to W 1,β
0 (Ω) for any β ∈ [1, p′0] , where
p0 =
2n
2(n− 1)− p(n− 2)
and p′0 =
p0
p0 − 1
=
2n
p(n− 2) + 2
.
(
It is obvious: u ∈W 1,2
0 (Ω) =⇒ v ≡
≡ |u|p ∈W 1,β
0 (Ω) for any β ∈ [1, 2) if n = 2, and for any β ∈ [1, 2] if n = 1.
)
Proof. We have∫
Ω
|u|(p−1)β |Dku|β dx ≤ k(ε)
∫
Ω
|Dku|2 dx+ ε
∫
Ω
|u|(p−1)β 2
2−β dx
for any u ∈ W 1,2(Ω) and β ∈ [1, p′0]. It is enough to consider the case β = p′0 =
2n
p(n− 2) + 2
,
because Ω ⊂ Rn is a bounded domain with sufficiently smooth boundary ∂Ω. So, from here we get
3 These are a complete metric spaces; about their properties see, for example, [30] and references therein.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
74 K. N. SOLTANOV∫
Ω
|u|(p−1)β |Dku|β dx ≤ c(ε)
∫
Ω
|Dku|2 dx+ ε
∫
Ω
|u|
(p−1)p′0
2
2−p′0 dx+ c0.
Then (p− 1) p′0
2
2− p′0
=
2n
n− 2
holds under the conditions of the Lemma 2.1. Consequently, we
obtain u ∈ W 1,2
0 (Ω) =⇒ v ≡ |u|p ∈ W 1,β
0 (Ω) for any β ∈ [1, p′0] , with choosing ε > 0 sufficiently
small and with use the embedding theorem for Sobolev spaces.
Lemma 2.1 is proved.
Corollary 2.1. Let u,w ∈ W 1,2
0 (Ω) and the number p is such that 2 < p <
2(n− 1)
n− 2
, n ≥ 3.
Then the function v(x) ≡ |u(x)|p−2 u(x)w(x) belongs to W 1,β
0 (Ω) (i.e., v ∈ W 1,β
0 (Ω)) for any
β ∈ [1, p′0] , where p0 =
2n
2(n− 1)− p(n− 2)
and p′0 =
p0
p0 − 1
.
Now we introduce a concept of the nonnegative generalized function.
Definition 2.1. A generalized function q(x) is called a nonnegative distribution (“q ≥ 0”) iff
〈q, ϕ〉 ≥ 0 holds for any nonnegative test function ϕ ∈ D(Ω).
3. General solvability results. Let X, Y be reflexive Banach spaces and X∗, Y ∗ their dual
spaces, moreover Y is a reflexive Banach space with strictly convex norm together with Y ∗ (see, for
example, references of [29]). Let f : D(f) ⊆ X −→ Y be an operator. So we conduct variant of the
main result of [29] (the more general cases can be seen in [31]). Consider the following conditions:
(a) X, Y be Banach spaces such as above and f : D(f) ⊆ X −→ Y be a continuous mapping,
moreover there is the closed ball BX
r0(x0) ⊂ X of an element x0 of D(f) that belongs to D(f)
(Br0(x0) ⊆ D(f))4.
Let the following conditions are fulfilled on the closed ball BX
r0(x0) ⊆ D(f):
(b) f is a bounded mapping on the ball BX
r0(x0), i.e., ‖f(x)‖Y ≤ µ (‖x‖X) holds for any
x ∈ BX
r0(x0) where µ : R1
+ −→ R1
+ is a continuous function;
(c) there is a mapping g : D (g) ⊆ X −→ Y ∗, and a continuous function ν : R1
+ −→ R1
nondecreasing for τ ≥ τ0 such that D(f) ⊆ D (g) , and for any SXr (x0) ⊂ BX
r0(x0), 0 < r ≤ r0,
closure of g
(
SXr (x0)
)
≡ SY ∗r (0), SXr (x0) ⊆ g−1
(
SY
∗
r (0)
)
〈f(x)− f(x0), g(x)〉 ≥ ν (‖x− x0‖X) ‖x− x0‖X , (3.1)
a.e. x ∈ BX
r0(x0) and ν(r0) ≥ δ0 > 0
holds, here δ0 > 0, τ0 ≥ 0 are constants;
(d) almost each x̃ ∈ intBX
r0(x0) possesses a neighborhood Vε (x̃), ε ≥ ε0 > 0, such that the
inequation
‖f (x2)− f (x1)‖Y ≥ Φ (‖x2 − x1‖X , x̃, ε) + ψ (‖x1 − x2‖Z, x̃, ε) (3.2)
holds for any x1, x2 ∈ Vε (x̃) ∩ BX
r0(x0), where Φ (τ, x̃, ε) ≥ 0 is a continuous function of τ and
Φ (τ, x̃, ε) = 0⇔ τ = 0 (in particular, maybe x̃ = x0, ε = ε0 = r0 and Vε (x̃) = Vr0(x0) ≡ BX
r0(x0),
consequently Φ (τ, x̃, ε) ≡ Φ (τ, x0, r0) on BX
r0(x0)), Z is a Banach space and the inclusion X ⊂ Z
is compact, and ψ (·, x̃, ε) : R1
+ −→ R1 is a continuous function at τ and ψ (0, x̃, ε) = 0;
4 Here it is enough assume: there is the closed neighborhood Uδ(x0) ⊂ X of an element x0 of D(f) that belongs to
D(f) (Uδ(x0) ⊆ D(f)) and Uδ(x0) is equivalence to BXr0(x0) for some numbers δ, r0 > 0. Consequently, it is enough
account that Ur0(x0) ≡ BXr0(x0).
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GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR OF A NONLINEAR EQUATION . . . 75
(d′) f possesses the P-property on the ball BX
r0(x0), i.e., for any precompact subset M ⊆ Im f
of Y there exists a (general) subsequence M0 ⊂ M such that there exists a precompact subset G of
BX
r0(x0) ⊂ X that satisfies the inclusions f−1 (M0) ⊆ G and f (G ∩D(f)) ⊇M0.
Theorem 3.1. Let the conditions (a), (b), (c) be fulfilled. Then if the image f
(
BX
r0(x0)
)
of the
ball BX
r0(x0) is closed (or is fulfilled the condition (d) or (d′)), then f
(
BX
r0(x0)
)
is a bodily subset
(i.e., with nonempty interior) of Y, moreover f
(
BX
r0(x0)
)
contains a bodily subset M that has the
form
M ≡
{
y ∈ Y | 〈y, g(x)〉 ≤ 〈f(x), g(x)〉 ∀x ∈ SXr0(x0)
}
.
Now we lead a solvability theorem for the nonlinear equation in Banach spaces, which is proved
using Theorem 3.1. Let F0 : D (F ) ⊆ X −→ Y and F1 : D (F1) ⊆ X −→ Y be some nonlinear
mappings such that D (F0) ∩D (F1) = G ⊆ X and G 6= ∅. Consider the equation
F (x) ≡ F0(x) + F1(x) = y, y ∈ Y, (3.3)
where y is an arbitrary element of Y.
Let BX
r (x0) ⊆ D (F0) ∩ D (F1) ⊆ X be the closed ball, r > 0 be a number. Consider the
following conditions:
1) F0 : BX
r (x0) −→ Y is a bounded continuous operator together with its inverse operator F−1
0
(as F−1
0 : D
(
F−1
0
)
⊆ Y −→ X);
2) F1 : BX
r (x0) −→ Y is a nonlinear continuous operator;
3) there are continuous functions µi : R1
+ −→ R1
+, i = 1, 2 and ν : R1
+ −→ R1 such that the
inequations
‖F0(x)− F0(x0)‖Y ≤ µ1(‖x− x0‖X) and ‖F1(x)− F1(x0)‖Y ≤ µ2(‖x− x0‖X),〈
F (x)− F (x0), g(x)
〉
≥ c
〈
F0(x)− F0(x0), g(x)
〉
≥ ν (‖x− x0‖X) ‖x− x0‖X
hold for any x ∈ BX
r (x0), moreover ν(r) ≥ δ0 holds for some number δ0 > 0, where the mapping g :
BX
r (x0) ⊆ D (g) ⊆ X −→ Y ∗ fulfills the conditions of Theorem 3.1, c > 0 is some number;
4) almost each x̃ ∈ intBX
r (x0) possesses a neighborhood BX
ε (x̃), ε ≥ ε0 > 0, such that the
inequation
‖F (x1)− F (x2)‖Y ≥ c1 ‖F0 (x1)− F0 (x2)‖Y ≥
≥ k0
(
‖x1 − x2‖X , x̃, ε
)
− k1
(
‖x1 − x2‖Z , x̃, ε
)
, X b Z,
holds for any x1, x2 ∈ BX
ε (x̃) and some number ε0 > 0, where ki (τ, x̃, ε) ≥ 0, i = 0, 1, are
continuous functions of τ for any given x̃, and such that k0 (τ, x̃, ε) = 0⇐⇒ τ = 0, k1 (0, x̃, ε) = 0,
and X b Z (i.e., X ⊂ Z is compact).
Then the following statement is true, which follows from Theorem 3.1.
Theorem 3.2. Let the conditions 1 – 3 be fulfilled. Then if F
(
BX
r (x0)
)
is closed (or is fulfilled
the condition 4 or (d′)), then the equation (3.3) has a solution in the ball BX
r (x0) for any y ∈ Y
satisfying the inequation
〈y − F (x0), g(x)〉 ≤ ν (‖x− x0‖X) ‖x− x0‖X ∀x ∈ SXr (x0).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
76 K. N. SOLTANOV
4. Proof of existence theorem of problem (0.1), (0.2). It should be noted that here we continue
the investigation of the problem studied in the article [32] where this problem is studied in the case
Q ≡ QT ≡ (0, T ) × Ω, when T < ∞ is some number. Here we will study the global existence of
this problem, i.e., in the case when Q ≡ R+ × Ω. So in the beginning we set a space and explain
the way of the investigation. As the solutions u(t, x) of the considered problem will seek in the
form u(t, x) ≡ u1(t, x) + iu2(t, x), where uj : Q −→ R, j = 1, 2 we can set this function u(t, x) as
the vector function, i.e.,
−−−−→
u(t, x) ≡ (u1(t, x);u2(t, x)) and −→u : Q −→ R2. Consequently if we write
u ∈ X
(
for example, X ≡ Lm
(
R+;W 1,2
0 (Ω)
)
∩W 1,2
(
R+;L2(Ω)
)
, m ≥ max {p, p̃ }
)
, then this we
understand as uj ∈ X, j = 1, 2 or −→u ∈ X ×X. Now we can define a solution of the problem (0.1),
(0.2) more exactly.
Definition 4.1. We say that the function u ∈ Lm
(
R+;W 1,2
0 (Ω)
)
∩W 1,2
(
R+;L2(Ω)
)
∩
{
w(t, x) |
w(0, x) = u0(x)
}
≡ X (as complex function) is a solution of the problem (0.1), (0.2) if it satisfies
the equation
i
〈
∂u
∂t
, v
〉
+ 〈∇u,∇v〉+
〈
q(x)|u|p−2u, v
〉
+
〈
a(x)|u|p̃−2u, v
〉
= 〈h, v〉
for any v ∈ L(2,m)
(
R+;W 1,2
0 (Ω)
)
and a.e. t > 0.
Let us f : X −→ Y is the operator generated by the problem (0.1), (0.2), where X is the denoted
above space, and
Y ≡ L2
(
R+;W−1,2(Ω)
)
+ Lm
′(
R+;W−1,2(Ω)
)
+ Lq̃(Q),
where q̃ =
p̃
p̃− 1
. We will show that for this operator are fulfilled all conditions of the main theorem.
We make this by sequence of steps.
1. Clearly that the conditions (a) and (b) fulfilled. Indeed, the explanations conducted in the
previous sections shows that f : X −→ Y is the continuous bounded operator. It should be noted
that the calculation of the function µ not is difficult, therefore we not will conduct this computation
here (see below Proposition 4.3).
Proposition 4.1. Let all conditions of Theorem 1.1 are fulfilled, then the operator f satisfies the
condition (c) with the operator
∂
∂t
+ I on the space W 1,2
(
R+;W 1,2
0 (Ω)
)
∩ Lm
(
R+;W 1,2
0 (Ω)
)
∩
∩ {v(t, x) | v(0, x) = u0(x)}. Moreover takes place the following inequations:
t∫
0
Im
〈
f(u),
∂u
∂s
+ u
〉
ds ≡ 1
2
‖u(t)‖22 −
1
2
‖u0‖22 +
t∫
0
∥∥∥∥∂u∂s
∥∥∥∥2
L2(Ω)
ds,
t∫
0
Re
〈
f(u),
∂u
∂s
+ u
〉
ds ≥ η ‖∇u(t)‖22 + η
t∫
0
‖∇u(s)‖22 ds+
+δ1
t∫
0
〈
q(x)|u|p−2u, u
〉
(s) ds+ δ2
〈
q(x)|u|p−2u, u
〉
(t)−
−C
(
‖∇u0‖ , ‖q‖W−1,2 , ‖u0‖2∗ , ‖a‖m , p, p̃
)
,
the constants of these inequations are determined in (4.6).
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GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR OF A NONLINEAR EQUATION . . . 77
Proof. Consider the expression
〈
f(u),
∂u
∂t
+ u
〉
, which we can write as
〈
f(u),
du
dt
〉
= i
〈
∂u
∂t
,
∂u
∂t
〉
+
1
2
d
dt
〈∇u,∇u〉+
〈
q(x)|u|p−2u,
∂u
∂t
〉
+
〈
a(x)|u|p̃−2u,
du
dt
〉
,
(4.1)
〈f(u), u〉 = i
〈
∂u
∂t
, u
〉
+ 〈∇u,∇u〉+
〈
q(x) |u|p−2 u, u
〉
+
〈
a(x)|u|p̃−2u, u
〉
, (4.2)
for any u ∈W 1,2
(
R+;W 1,2
0 (Ω)
)
. We begin by (4.2), then we get
〈f(u), u〉 =
i
2
d
dt
‖u(t)‖22 + ‖∇u‖22 +
〈
q(x)|u|p−2u, u
〉
+
〈
a(x)|u|p̃−2u, u
〉
.
Consequently, we have
Im 〈f(u), u〉 =
1
2
d
dt
‖u(t)‖22
and
Re 〈f(u), u〉 = ‖∇u‖22 +
〈
q(x)|u|p−2u, u
〉
+
〈
a(x)|u|p̃−2u, u
〉
,
as Im q(x) = 0 and Im a(x) = 0. Whence follows that
t∫
0
Im 〈f(u), u〉 ds =
1
2
‖u(t)‖22 −
1
2
‖u0‖22
and
Re 〈f(u), u〉 ≥ ‖∇u‖22 +
〈
q(x)|u|p−2u, u
〉
+
〈
a(x)|u|p̃−2u, u
〉
=⇒
=⇒ Re 〈f(u), u〉 ≥ ‖∇u‖22 + (1− k1) 〈q(x), |u|p〉 − k0 ‖u‖2p2 =⇒
=⇒ Re 〈f(u), u〉 ≥
(
1− k0C
2 (p2)
)
‖∇u (t)‖22 ,
for a.e. t > 0, as k1 ≤ 1 and k0C
2 (p2) < 1 by virtue of the condition (ii).
Thereby if we examine now (4.1) and (4.2) together then we will obtain〈
f(u),
∂u
∂t
+ u
〉
≡ i
2
d
dt
‖|u(t)|‖22 + i
∥∥∥∥∂u∂t
∥∥∥∥2
L2(Ω)
+
+ ‖∇u‖22 +
〈
q(x)|u|p−2u, u
〉
+
〈
a(x)|u|p̃−2u, u
〉
+
+
1
2
∂
∂t
‖∇u(t)‖2L2(Ω) +
1
p
∂
∂t
〈q(x), |u|p〉+
1
p̃
∂
∂t
〈
a, |u|p̃
〉
,
as far as
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
78 K. N. SOLTANOV
∂
∂t
|u|p =
∂
∂t
(u, u)
p
2 =
p
2
(u, u)
p
2
−1 ∂
∂t
(u, u) =
p
2
(u, u)
p
2
−1 2
(
u,
∂
∂t
u
)
= p
(
|u|p−2u,
∂u
∂t
)
,
in other words we have
Im
〈
f(u),
∂u
∂t
+ u
〉
≡ 1
2
d
dt
‖u(t)‖22 +
∥∥∥∥∂u∂t
∥∥∥∥2
L2(Ω)
(4.3)
and
Re
〈
f(u),
∂u
∂t
+ u
〉
≡ 1
2
∂
∂t
‖∇u(t)‖2L2(Ω) + ‖∇u(t)‖22 +
+
1
p
∂
∂t
〈q(x), |u|p〉+
〈
q(x)|u|p−2u, u
〉
+
1
p̃
∂
∂t
〈
a, |u|p̃
〉
+
〈
a(x)|u|p̃−2u, u
〉
. (4.4)
If we integrate with respect to t these equation then we get
t∫
0
Im
〈
f(u),
∂u
∂s
+ u
〉
ds ≡
t∫
0
[
1
2
d
ds
‖u(s)‖22 +
∥∥∥∥∂u∂s
∥∥∥∥2
L2(Ω)
]
ds
and
t∫
0
Re
〈
f(u),
∂u
∂s
+ u
〉
ds ≡
t∫
0
[
1
2
∂
∂s
‖∇u(s)‖22 + ‖∇u(s)‖22
]
ds+
+
t∫
0
[
1
p
∂
∂s
〈q(x), |u|p〉+
〈
q(x)|u|p−2u, u
〉]
ds+
t∫
0
[
1
p̃
∂
∂s
〈
a, |u|p̃
〉
+
〈
a(x)|u|p̃−2u, u
〉]
ds.
Thence follow
t∫
0
Im
〈
f(u),
∂u
∂s
+ u
〉
ds ≡ 1
2
‖u(t)‖22 −
1
2
‖u0‖22 +
t∫
0
∥∥∥∥∂u∂s
∥∥∥∥2
L2(Ω)
ds,
(4.5)
t∫
0
Re
〈
f(u),
∂u
∂s
+ u
〉
ds ≡ 1
2
‖∇u(t)‖22 −
1
2
‖∇u0‖22 +
+
t∫
0
‖∇u(s)‖22 ds+
t∫
0
[〈
q(x)|u|p−2u, u
〉
+
〈
a(x)|u|p̃−2u, u
〉]
ds+
+
[
1
p
〈q(x), |u|p〉+
1
p̃
〈
a(x), |u|p̃
〉]
(t)− 1
p
〈q(x), |u0|p〉 −
1
p̃
〈
a, |u0|p̃
〉
.
For estimate the
∫ t
0
Re
〈
f(u),
∂u
∂s
+ u
〉
ds we use (1.2) (i.e., the condition (ii))
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR OF A NONLINEAR EQUATION . . . 79〈
a(x)|u|p̃−2u, u
〉
≥ −k0 ‖u‖2p2 − k1 〈q(x), |u|p〉
then we obtain
t∫
0
Re
〈
f(u),
∂u
∂s
+ u
〉
ds ≥ 1
2
‖∇u(t)‖22 +
t∫
0
‖∇u(s)‖22 ds+
+δ1
t∫
0
〈
q(x)|u|p−2u, u
〉
(s)ds− k0
t∫
0
‖u‖2p2 (s)ds+ δ2
〈
q(x)|u|p−2u, u
〉
(t)−
−k0 ‖u(t)‖2p2 −
1
2
‖∇u0‖22 −
1
p
〈q(x), |u0|p〉 −
1
p̃
〈
a, |u0|p̃
〉
≥
≥ η ‖∇u(t)‖22 + η
t∫
0
‖∇u(s)‖22 ds+ δ1
t∫
0
〈
q(x)|u|p−2u, u
〉
(s)ds+
+δ2
〈
q(x)|u|p−2u, u
〉
(t)− C
(
‖∇u0‖, ‖q‖W−1,2 , ‖u0‖2∗ , ‖a‖m, p, p̃
)
(4.6)
for a.e. t > 0, where δ1 = 1 − k1 ≥ 0, δ2 = p−1 − p̃−1k1 ≥ 0, η = 1 − C (2, p2)2 k0 > 0. The
expressions (4.5) and (4.6) shows that the condition (c) of the main theorem takes place for the
operator f : X −→ Y generated by posed problem.
Proposition 4.1 is proved.
2. Now we prove an inequation used in the proof of the fulfilment of the condition (d′).
Proposition 4.2. Let all conditions of Theorem 1.1 are fulfilled, then the inequality
‖f(u)− f(v)‖Y ≥ ‖u− v‖2 (t) + ‖∇(u− v)‖2 −M max
{
‖u‖p̃−2
p̃ ; ‖v‖p̃−2
p̃
}
‖u− v‖p̃ ,
holds for any u, v ∈ X ∩ {u | u(0, x) = u0(x)} .
Proof. Let us u, v ∈ X ∩ {u | u(0, x) = u0(x)} and consider ‖f(u)− f(v)‖Y that we can
estimate as∥∥∥∥i∂(u− v)
∂t
−∆ (u− v) + q
(
|u|p−2u− |v|p−2v
)
+ a
(
|u|p̃−2u− |v|p̃−2v
)∥∥∥∥
Y
≥
≥
∥∥∥∥i∂(u− v)
∂t
−∆ (u− v) + q
(
|u|p−2u− |v|p−2v
)∥∥∥∥
Y
−
∥∥∥a(|u|p̃−2u− |v|p̃−2v
)∥∥∥
Y
. (4.7)
In order that to esimate of the first adding of right-hand side of the inequality (4.7) we act in the
following way. In beginning we set〈
i
∂(u− v)
∂t
, (u− v)
〉
−
〈
∆(u− v), (u− v)
〉
+
〈
q
(
|u|p−2u− |v|p−2v
)
, (u− v)
〉
=
=
i
2
d
dt
〈
(u− v), (u− v)
〉
+ ‖∇(u− v)‖22 +
〈
q
(
|u|p−2u− |v|p−2v
)
, (u− v)
〉
(4.8)
and study it.
Here for the last adding takes place the inequality
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
80 K. N. SOLTANOV〈
q
(
|u|p−2u− |v|p−2v
)
, (u− v)
〉
= 〈q, |u|p〉+ 〈q, |v|p〉 −
〈
q|u|p−2u, v
〉
−
〈
q|v|p−2v, u
〉
.
As the expression
∣∣〈q|u|p−2u, v
〉
+
〈
q|v|p−2v, u
〉∣∣ has the following estimation:∣∣〈q|u|p−2u, v
〉
+
〈
q|v|p−2v, u
〉∣∣ ≤ 〈q|u|p−1, |v|
〉
+
〈
q|v|p−1, |u|
〉
therefore we can consider the right-hand side of (4.8) without of the last adding.
Consequently we get the following estimation for the first adding of the right-hand side of the
inequality (4.7):∥∥∥∥i∂(u− v)
∂t
−∆ (u− v) + q
(
|u|p−2u− |v|p−2v
)∥∥∥∥
Y
≥ K
[
‖u− v‖2 (t) + ‖∇(u− v)‖2
]
, K > 0,
with taking into account the equation (4.8), the last reasons and the equation
t∫
0
〈
∂ (u− v)
∂s
, (u− v)
〉
ds =
1
2
‖u− v‖22 (t),
whereas u(0, x) = v(0, x) = u0 by choosingly, that we need make by virtue of the condition (d′) of
the main theorem.
Now consider the second adding of right-hand side of the inequality (4.7), for which we have∣∣∣〈a(|u|p̃−2u− |v|p̃−2v
)
, (u− v)
〉∣∣∣ =
∫
Ω
a
(
|u|p̃−2u− |v|p̃−2v
)
(u− v)dx ≤
≤
∫
Ω
a ϕ (u, v) |u− v|2 dx, 0 ≤ ϕ (u, v) ≤M (max {|u|, |v|})p̃−2 ,
where M > 0 be some number and ϕ (u, v) be a continuous function.
Taking into account the last inequalities in (4.7) we obtain
‖f(u)− f(v)‖Y ≥ ‖u− v‖2 (t) + ‖∇(u− v)‖2 −M max
{
‖u‖p̃−2
p̃ ; ‖v‖p̃−2
p̃
}
‖u− v‖p̃ . (4.9)
Proposition 4.2 is proved.
3. Now we will conduct a priori estimations for a solutions of the problem.
Proposition 4.3. Let all conditions of Theorem 1.1 are fulfilled, then all solutions belong to
bounded subset of the space
X ≡W 1,2
(
R+;L2(Ω)
)
∩ Lm
(
R+;W 1,2
0 (Ω)
)
∩
∩
{
v
∣∣∣ |v|p ∈ Lβ (R+;W 1,β
0 (Ω)
)}
∩
{
u
∣∣∣ u(0, x) = u0(x)
}
,
i.e., there is constants K ≡ K
(
‖h‖2,QT , ‖u0‖W 1,2 , ‖q‖W−1,2 , ‖a‖, p, p̃
)
such that ‖u‖X ≤ K.
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Proof. In the beginning we note that from examination of the expression 〈f(u), u〉 in the proof
of Proposition 4.1 we get the inequations
1
2
‖u(t)‖22 −
1
2
‖u0‖22 ≤
t∫
0
|〈h, u〉| ds ≤
t∫
0
‖h(s)‖2 ‖u(s)‖2 ds
and
‖∇u‖22 +
〈
q(x)|u|p−2u, u
〉
+
〈
a(x)|u|p̃−2u, u
〉
≤ |〈h, u〉| =⇒
=⇒ ‖∇u‖22 + (1− k1) 〈q(x), |u|p〉 − k0 ‖u‖2p2 ≤ |〈h, u〉| =⇒
=⇒
(
1− k0C
2 (p2)
)
‖∇u (t)‖22 ≤ ‖h(t)‖2 ‖u(t)‖2 ,
as k1 ≤ 1 and k0C
2 (p2) < 1 by virtue of the condition (ii). Then we get that the following
estimations are true: (
1− k0C
2(p2)
)
‖∇u(t)‖22 − ε‖u(t)‖22 ≤ c(ε)‖h(t)‖22
or
‖∇u(t)‖2 ≤ ĉ(ε) ‖h(t)‖2 for a.e. t ≥ 0,
as far as ‖u(t)‖2 ≤ C1 (mes Ω) ‖∇u(t)‖2 for any u(t) ∈ W 1,2
0 (Ω) by the embedding theorems,
where ĉ(ε) ≡ ĉ (ε,mes Ω, k0C) , C1 (mes Ω) > 0 are constants, moreover,
‖u‖
L2(R+;W 1,2
0 (Ω)) ≤ ĉ1(ε) ‖h‖
L2(R+;W 1,2
0 (Ω)) + ĉ2 ‖u0‖2 .
Thus we obtain that u(t, x) belong to the bounded subset of L2
(
R+;W 1,2
0 (Ω)
)
for given h ∈
∈ L2(Q).
Using the equations (4.3) and (4.4), and also the estimates (4.5) and (4.6) we get∣∣∣∣∣∣
t∫
0
Re
〈
h,
∂u
∂s
+ u
〉
ds
∣∣∣∣∣∣ ≥
t∫
0
Re
〈
f(u),
∂u
∂s
+ u
〉
ds ≥
≥ η2 ‖∇u(t)‖22 + η1
t∫
0
‖∇u(s)‖22 ds+ δ1
t∫
0
〈
q(x)|u|p−2u, u
〉
(s)ds+
+δ2
〈
q(x)|u|p−2u, u
〉
(t)− C
(
‖u0‖W 1,2, ‖q‖W−1,2 , ‖a‖m , p, p̃
)
(4.10)
and ∣∣∣∣∣∣
t∫
0
Im
〈
h,
∂u
∂s
+ u
〉
ds
∣∣∣∣∣∣ ≥
t∫
0
Im
〈
f(u),
∂u
∂s
+ u
〉
≡
≡ 1
2
‖u(t)‖22 −
1
2
‖u0‖22 +
t∫
0
∥∥∥∥∂u∂s
∥∥∥∥2
L2(Ω)
ds. (4.11)
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82 K. N. SOLTANOV
Then from (4.10) we get the estimation
η2 ‖∇u(t)‖22 + η1
t∫
0
‖∇u(s)‖22 ds+ δ1
t∫
0
〈
q(x)|u|p−2u, u
〉
(s)ds+
+δ2
〈
q(x)|u|p−2u, u
〉
(t)− C (‖u0‖W 1,2 , ‖q‖W−1,2 , ‖a‖m′ , p, p̃ ) ≤
≤
t∫
0
‖h‖2
(∥∥∥∥∂u∂s
∥∥∥∥
2
+ ‖u‖2
)
ds a.e. t > 0,
and from (4.11) we obtain
1
2
‖u(t)‖22 −
1
2
‖u0‖22 +
t∫
0
∥∥∥∥∂u∂s
∥∥∥∥2
L2(Ω)
ds ≤
t∫
0
‖h‖2
(∥∥∥∥∂u∂s
∥∥∥∥
2
+ ‖u‖2
)
ds for a.e. t > 0,
then with combine of last two inequations we get
η ‖∇u(t)‖22 + η
t∫
0
‖∇u(s)‖22 ds+ δ1
t∫
0
〈
q (x) |u|p−2u, u
〉
(s) ds+
+δ2
〈
q(x)|u|p−2u, u
〉
(t)− C (‖u0‖W 1,2 , ‖q‖W−1,2 , ‖a‖ , p, p̃) +
+
1
2
‖u(t)‖22 −
1
2
‖u0‖22 +
t∫
0
∥∥∥∥∂u∂s
∥∥∥∥2
2
ds ≤
≤ ε1
t∫
0
∥∥∥∥∂u∂s
∥∥∥∥2
2
ds+ ε2
t∫
0
‖u‖22 ds+ C (ε1, ε2)
t∫
0
‖h‖22 ds for a.e. t > 0,
or
η ‖∇u(t)‖22 + η̃
t∫
0
‖∇u(s)‖22 ds+ δ1
t∫
0
〈
q(x)|u|p−2u, u
〉
(s) ds+
+δ2
〈
q(x)|u|p−2u, u
〉
(t) +
1
2
‖u (t)‖22 + (1− ε1)
t∫
0
∥∥∥∥∂u∂s
∥∥∥∥2
2
ds ≤
≤ C (ε1, ε2)
∞∫
0
‖h‖22 ds+ C1
(
‖u0‖W 1,2 , ‖q‖W−1,2 , ‖a‖ , p, p̃
)
.
Moreover, from here follows
1
2
‖∇u(t)‖22 −
1
2
‖∇u(0)‖22 +
t∫
0
‖∇u(s)‖22 ds+
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GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR OF A NONLINEAR EQUATION . . . 83
+
t∫
0
〈
q(x) |u|p−2 u, u
〉
(s)ds+
t∫
0
〈
a(x)|u|p̃−2u, u
〉
(s)ds+
+
1
p
〈q(x), |u|p〉 (t)− 1
p
〈q(x), |u(0)|p〉+
1
p̃
〈
a, |u|p̃
〉
(t)− 1
p̃
〈
a, |u(0)|p̃
〉
≤
≤
(
ε−1 +
1
2
) t∫
0
‖h(s)‖22 ds+ ε
t∫
0
‖u(s)‖22 ds+
1
2
t∫
0
∥∥∥∥∂u∂s
∥∥∥∥2
2
ds+
+C
(
‖∇u0‖ , ‖q‖W−1,2 , ‖u0‖2∗ , ‖a‖m , p, p̃
)
,
by virtue of the condition (1.2).
Consequently we get the following inequation:
η ‖∇u(t)‖22 + η1(ε)
t∫
0
‖∇u(s)‖22 ds ≤
≤ ε−1
t∫
0
‖h(s)‖22 ds+ C
(
‖∇u0‖2 , ‖q‖W−1,2 , ‖u0‖2∗ , ‖h‖2 , ‖a‖m , p, p̃
)
in the other words we obtain
t∫
0
∥∥∇u(s)
∥∥2
2
ds ≤ D̃
(
ε−1, . . .
)(
η̃1(ε)
)−1
(
1− e−η̃1(ε)t
)
, (4.12)
where D̃
(
ε−1, . . .
)
= D̃
(
ε−1, ‖u0‖W 1,2 , ‖q‖W−1,2 , ‖h‖2, ‖a‖m, p, p̃
)
.
Consequently we obtain, that any solution of the considered problem under the posed conditions
satisfies the following inclusion:
u ∈W 1,2
(
R+;L2(Ω)
)
∩ L∞
(
R+;W 1,2
0 (Ω)
)
∩
∩
{
v | |v|p ∈ Lβ
(
R+;W 1,β
0 (Ω)
)}
∩ {u | u(0, x) = u0(x)} ≡ X, (4.13)
in addition the preimage of each bounded neighborhood of zero from L2(Q)×W 1,2
0 (Ω) under operator
f is the bounded neighborhood of zero of the space determined by (4.13), where m ≥ 2∗ and β > 1
is denoted by Lemma 2.1. We note that here is used the inclusion given in next remark.
Proposition 4.3 is proved.
Remark 4.1. Let Z is a Banach space, then
L2 (R;Z) ∩ L∞ (R;Z) ⊂ Lm (R;Z), 2 ≤ m <∞,
holds.
From here we get that the condition (c) is fulfilled for the operator f generated by the posed
problem and the operator g(v) ≡ ∂v
∂t
+ v for any v ∈ W 1,2
(
R+;W 1,2
0 (Ω)
)
∩ Lm
(
R+;W 1,2
0 (Ω)
)
that is dense in the requisit space defined in (4.13).
4. Now we can show that the operator f satisfies the condition (d′).
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84 K. N. SOLTANOV
Proposition 4.4. Let all conditions of Theorem 1.1 are fulfilled, then the operator f satisfies the
condition (d′).
Proof. More exactly, we will prove that the image f(X) is the closed subset of Y. As the
conditions (a), (b) and (c) of the main theorem are fulfilled for the operator f : X → Y then we get,
that f(X) contains a dense subset of the space Y by virtue of the first statement of this theorem.
So, let us the sequence {hk}∞k=1 ⊂ f(X) is the fundamental sequence in Y that converge to
an element h0 ∈ Y, since f(X) contains a dense subset of the space Y therefore for any h0 ∈ Y
there exists a sequence of such type. As the sequence {hk}∞k=1 is a bounded subset of Y, and
consequently f−1({hk}∞k=1) belong to the bounded subset M0 of X by virtue of the condition (c),
which is proved in the step 3. It is known that X is the reflexive space therefore we can choose a
subsequence
{
ukj
}∞
j=1
⊂ M0 of f−1 ({hk}∞k=1) such that ukj ∈ f−1(hkj ), kj ↗ ∞, and
{
ukj
}∞
j=1
weakly converge in X, i.e., ukj ⇀ u0 ∈ X. Moreover it is known that X b Lm
(
R+;L`(Ω)
)
is
compact, where 1 < ` <
2n
n− 2
if n ≥ 3 (see, for example, [31] and its references). Then the
sequence
{
ukj
}∞
j=1
have a subsequence, that strongly converge in the space Lm
(
R+;L`(Ω)
)
, which
for simplicity we denote also by
{
ukj
}∞
j=1
, i.e., we assume that
{
ukj
}∞
j=1
is the subsequence of such
type.
Thence use previous reasons and the (4.9) from step 2 we get ukj =⇒ u0 in L2
(
R+;W 1,2
0 (Ω)
)
and in Lm
(
R+;L2(Ω)
)
. Furthermore if take into account that in this problem first two adding are
linear continuous operator then we obtain that hkj = f
(
ukj
)
−→ f(u0) ≡ h0, which show that f(X)
is the closed subset of Y.
Proposition 4.4 is proved.
Thus we can complete of the proof of Theorem 1.1. From Propositions 4.1 – 4.4 we get,
that the operator f : X −→ Y generated by the posed problem satisfies all conditions of the main
theorem (Theorem 3.1 and also Theorem 3.2). Then using Theorem 3.1 we obtain, that the op-
erator f satisfies the statement of Theorem 3.1, therefore the statement of Theorem 1.1 is correct.
Consequently, the existence theorem for the problem (0.1), (0.2) (i.e., Theorem 1.1) is proved.
5. Behavior of solutions of problem (0.1), (0.2). We will study the behavior of the solution of
problem (0.1), (0.2) in the sense of the space W 1,2(Q)∩ Lm
(
R+;W 1,2
0 (Ω)
)
. Consider the following
functional on the space W 1,2(Q) ∩ Lm
(
R+;W 1,2
0 (Ω)
)
:
I (v(t)) = ‖∇v(t)‖2L2 ≡ ‖∇v(t)‖22 ≡
∫
Ω
|∇v(t, x)|2 dx.
So we have
2−1 d
dt
I (u(t)) =
〈
∂
∂t
∇u,∇u
〉
= −
〈
∂
∂t
u,∆u
〉
.
Here if to take account u(t, x) is the solution of the considered problem then we get
−i
〈
∂u
∂t
,
∂u
∂t
〉
−
〈
∂u
∂t
, q|u|p−2u
〉
−
〈
∂u
∂t
, a|u|p̃−2u
〉
+
〈
∂u
∂t
, h
〉
=
= i
∥∥∥∥∂u∂t
∥∥∥∥2
2
− p−1 d
dt
〈q, |u|p〉 − p̃−1 d
dt
〈
a, |u|p̃
〉
+
〈
∂u
∂t
, h
〉
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whence we have
2−1 d
dt
I (u(t)) = −
〈
∂u
∂t
, q|u|p−2u
〉
−
〈
∂u
∂t
, a |u|p̃−2 u
〉
+ Re
〈
∂u
∂t
, h
〉
and
−
∥∥∥∥∂u∂t
∥∥∥∥2
2
= − Im
〈
∂u
∂t
, h
〉
=⇒
∥∥∥∥∂u∂t
∥∥∥∥
2
≤ ‖h‖2 (5.1)
for a.e. t > 0, as I (u(t)) is a real function.
Thus we obtain
2−1I (u(t))− 2−1I (u0) ≤ −p−1
t∫
0
d
ds
〈q, |u|p〉 ds− p̃−1
t∫
0
d
ds
〈
a, |u|p̃
〉
ds+
+
t∫
0
∣∣∣∣〈∂u∂s , h
〉∣∣∣∣ ds ≤ −p−1 〈q, |u|p〉 (t) + p−1 〈q, |u0|p〉−
−p̃−1
〈
a, |u|p̃
〉
(t) + p̃−1
〈
a, |u0|p̃
〉
+
t∫
0
∥∥∥∥∂u∂s
∥∥∥∥
2
‖h‖2 (s)ds ≤
≤ k0 ‖u(t)‖2p2 − δ 〈q, |u|
p〉 (t) +
t∫
0
‖h‖22 (s)ds
whence we get using the condition (1.2) (i.e., the inequality 2−1 > C(2, p2)2 · k0) and (5.1)
I (u(t)) ≡ ‖∇u(t)‖22 ≤ 2k0 ‖u(t)‖2p2 + 2
t∫
0
‖h‖22 (s)ds+ I (u0) =⇒
=⇒
(
1− 2C (2, p2)2 k0
)
‖∇u (t)‖22 ≤ 2
t∫
0
‖h‖22 (s)ds+ I (u0)
for a.e. t ≥ 0.
Moreover in the previous section we obtained the following equations:
I0 (u(t)) ≡ ‖u‖22 =⇒ 1
2
d
dt
I0 (u(t)) = 〈ut, u〉 =
= i
〈
−∆u+ q|u|p−2u+ a|u|p̃−2u− h, u
〉
=
= i ‖∇u(t)‖22 + i 〈q(x), |u|p〉+ i
〈
a(x), |u|p̃
〉
− i 〈h, u〉 =⇒
=⇒ ‖∇u(t)‖22 + 〈q(x), |u|p〉+
〈
a(x), |u|p̃
〉
− Re 〈h, u〉 = 0,
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
86 K. N. SOLTANOV
1
2
d
dt
I0 (u(t)) = − Im 〈h, u〉
whence follows
‖∇u(t)‖22 ≤ 4 ‖h(t)‖22 for a.e. t ≥ 0.
Consequently is proved the following result.
Theorem 5.1. Let the solution u(t, x) of the problem (0.1), (0.2) is a sufficiently smooth function,
i.e., u ∈ W 1,2
(
R+;W 1,2
0 (Ω)
)
∩ Lm
(
R+;W 1,2
0 (Ω)
)
and all conditions of Theorem 1.1 are fulfilled.
Then if the coefficient k0 of the condition (1.2) satisfy the inequality 2−1 > C (2, p2)2 · k0 then the
following inequalities are true:
‖∇u(t)‖2,
∥∥∥∥∂u∂t
∥∥∥∥
2
≤ b ‖h(t)‖2 =⇒ ‖u(t)‖2 ≤ b1 ‖h(t)‖2
for a.e. t ≥ 0.
Remark 5.1. It should be noted that the next inequality follows from (4.12) in the conditions of
Theorem 5.1
‖∇u(t)‖22 ≤ D̃
(
ε−1, . . .
)
exp {−η̃1(ε)t}+ ‖∇u0‖22 .
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ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
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| id | umjimathkievua-article-1964 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
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| spelling | umjimathkievua-article-19642019-12-05T09:47:39Z Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation Глобальне існування та довгочасова поведінка нелінійного рівняння типу Шрьодінгера Soltanov, K. N. Солтанов, К. Н. We study a global mixed problem for the nonlinear Schrödinger equation with a nonlinear term in which the coefficient is a generalized function. A global solvability theorem for the analyzed problem is proved by using the general solvability theorem from [K. N. Soltanov, Nonlin. Anal.: Theory, Meth., Appl., 72, No. 1 (2010)]. We also investigate the behavior of the solution of the problem under consideration. Вивчається глобальна мішана задача для нєлінійного рівняння Шрьодінгера з нєлінійним додаванням, в якій коефіцієнтом є узагальнена функція. Доведено теорему про глобальну розв'язність поставленої задачі на основі загальної теореми про розв'язність з [Soltanov K. N. Perturbation of the mapping and solvability theorems in the Banach space // Nonlinear Anal.: Theory, Meth. and Appl. - 2010. - 72, № 1]. Крім того, досліджено поведінку розв'язку задачі, що вивчається. Institute of Mathematics, NAS of Ukraine 2015-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1964 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 1 (2015); 68–87 Український математичний журнал; Том 67 № 1 (2015); 68–87 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1964/951 https://umj.imath.kiev.ua/index.php/umj/article/view/1964/952 Copyright (c) 2015 Soltanov K. N. |
| spellingShingle | Soltanov, K. N. Солтанов, К. Н. Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation |
| title | Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation |
| title_alt | Глобальне існування та довгочасова поведінка нелінійного рівняння типу Шрьодінгера |
| title_full | Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation |
| title_fullStr | Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation |
| title_full_unstemmed | Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation |
| title_short | Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation |
| title_sort | global existence and long-term behavior of a nonlinear schrödinger-type equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1964 |
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