Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid
UDC 517.9 We consider the Cauchy problem for an evolution equation modeling bidirectional surface waves in a convecting fluid. We study the existence, uniqueness, and asymptotic properties of global solutions to the initial value problem associated withthis equation in $R^n$. We obtain some polynomi...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512235224301568 |
|---|---|
| author | Mahmoudi, H. Esfahani , A. Mahmoudi, H. Esfahani , A. |
| author_facet | Mahmoudi, H. Esfahani , A. Mahmoudi, H. Esfahani , A. |
| author_sort | Mahmoudi, H. |
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| datestamp_date | 2025-03-31T08:49:43Z |
| description | UDC 517.9
We consider the Cauchy problem for an evolution equation modeling bidirectional surface waves in a convecting fluid. We study the existence, uniqueness, and asymptotic properties of global solutions to the initial value problem associated withthis equation in $R^n$. We obtain some polynomial decay estimates of the energy. |
| doi_str_mv | 10.37863/umzh.v72i10.6032 |
| first_indexed | 2026-03-24T03:25:34Z |
| format | Article |
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DOI: 10.37863/umzh.v72i10.6032
UDC 517.9
H. Mahmoudi, A. Esfahani (School Math. and Comput. Sci., Damghan Univ., Iran)
ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO AN EVOLUTION EQUATION
FOR BIDIRECTIONAL SURFACE WAVES IN A CONVECTING FLUID
АСИМПТОТИЧНА ПОВЕДIНКА РОЗВ’ЯЗКIВ ЕВОЛЮЦIЙНОГО РIВНЯННЯ
ДЛЯ ДВОНАПРАВЛЕНИХ ПОВЕРХНЕВИХ ХВИЛЬ
У РIДИНI З КОНВЕКЦIЄЮ
We consider the Cauchy problem for an evolution equation modeling bidirectional surface waves in a convecting fluid. We
study the existence, uniqueness, and asymptotic properties of global solutions to the initial value problem associated with
this equation in \BbbR n. We obtain some polynomial decay estimates of the energy.
Розглядається задача Кошi для еволюцiйного рiвняння, що моделює двонаправленi поверхневi хвилi у рiдинi з
конвекцiєю. Вивчаються iснування, єдинiсть та асимптотичнi властивостi глобальних розв’язкiв початкової задачi,
що пов’язана з цим рiвнянням у \BbbR n. Отримано деякi полiномiальнi оцiнки спадання енергiї.
1. Introduction and preliminaries. In this paper, we study
utt - \varepsilon \Delta utt - \Delta +\Delta 2u+ \alpha \Delta ut +\Delta 2ut + ut = \Delta f(u) + \beta \Delta g(ut) (1.1)
which arises as the phase equation in the study of the stability of one-dimensional periodic patterns
in systems with Galilean invariance. Also, it was derived to describe the oscillatory instability of
convective rolls and elastic beams [3, 6, 7]. Equation (1.1) is a higher order wave model in which
the terms \alpha \Delta ut + \Delta 2ut + ut represent the frictional dissipation. Equation (1.1) can be viewed
as a generalized the Cahn – Hilliard equation with an inertial term which models nonequilibrium
decompositions caused by deep supercooling in certain glasses [8 – 10, 17].
Here u = u(t, x) is the unknown function of x = (x1, . . . , xn) \in \BbbR n, and t > 0, \varepsilon > 0, \alpha < 0
as well as \beta are real constants. In addition, the nonlinear terms g(u) and f(u) are like O(| u| p). We
assume that f and g are continuously differentiable in \BbbR , and satisfy the following hypothesis:
| (f (j))(a)| \leq kj | a| p - j and | (g(j))(a)| \leq k
\prime
j | a| p - j
for all a \in \BbbR and j = 0, 1, 2, where kj and k
\prime
j are real positive constants. It is well-known
that equation (1.1) is closely related to several wave-type equations. For example, the Boussinesq
equation
utt - \Delta +\Delta 2u = \Delta f(u), (1.2)
which was derived by Boussinesq in 1872 to describe shallow water waves. The counterpart of
equation (1.2), i.e., the improved Boussinesq equation, can be presented as follows:
utt - \Delta - \Delta utt = \Delta f(u).
In [19], the authors considered the Cauchy problem associated with the Cahn – Hilliard equation with
the inertial term
utt +\Delta u - \Delta 2u+ ut = \Delta f(u).
Combining the high/low-frequency techniques and energy methods, they obtained the global exis-
c\bigcirc H. MAHMOUDI, A. ESFAHANI, 2020
1386 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO AN EVOLUTION EQUATION . . . 1387
tence and asymptotic behavior of the solutions. In [18], the authors investigated a fourth wave
equation that is of the regularity-loss type. Based on the decay property of the solution operators, the
global existence and asymptotic behavior of solutions are derived. See also [12, 13] and references
therein for the global existence and asymptotic behavior of solutions to higher order wave-type and
dissipative hyperbolic-type equations.
In this work, we show decay estimates in time for the total energy of the Cauchy problem
associated with (1.1) and the L2-norm of the solution. To this end, we study the associated linear
problem in detail as well as the behavior of the solutions. Subsequently, by using the formula of the
variation of parameters, we implement that result to the semilinear problem.
2. Main result. Before stating the main result, we give some notations which are used in this
paper.
For 1 \leq p \leq \infty , Lp = Lp(\BbbR n) denotes the usual Lebesgue space with the norm \| \cdot \| p. We also
use \| \cdot \| as the norm of L2(\BbbR n). The inner product in L2(\BbbR n) will be indicated by \langle \cdot , \cdot \rangle . Let s be a
nonnegative integer. Then Hs = Hs(\BbbR n) denotes the Sobolev space of L2 functions, equipped with
the norm \| \cdot \| Hs . Also, Ck(I;Hs) denotes the space of k-times continuously differentiable functions
on the interval I with values in the Sobolev space Hs = Hs(\BbbR n).
To state our main result, we consider the weak solution u of (1.1) with the initial data u(0, x) =
= u0(x) and ut(0, t) = u1(x). More precisely,
\langle utt(t), \psi \rangle + \varepsilon \langle \nabla utt(t),\nabla \psi \rangle + \langle \nabla u(t),\nabla \psi \rangle + \langle \Delta u(t),\Delta \psi \rangle -
- \alpha \langle \nabla ut(t),\nabla \psi \rangle + \langle \Delta ut(t),\Delta \psi \rangle = \langle \Delta f(u(t)) + \beta \Delta g(ut(t)), \psi \rangle ,
u(0, x) = u0(x),
ut(0, t) = u1(x).
(2.1)
The energy associated with the linear problem of (1.1) (see equation (3.1)) is defined by
E(t) =
1
2
\| ut\| 2 +
\varepsilon
2
\| \nabla ut\| 2 +
1
2
\| \nabla u\| 2 + 1
2
\| \Delta u\| 2.
Theorem 2.1. Let p > 2 and n \leq 3. Suppose that (u0, u1) \in H3 \times H2 with I0 < \delta . Then
there exists \delta > 0 such that problem (2.1) has a unique global solution u \in C([0,\infty );H3) \cap
\cap C1([0,\infty );H2) \cap C2([0,\infty );H1) such that \| u(t)\| 2 \leq CI0 and E(t) \leq CI0(1 + t) - 1, where
I0 = \| u0\| 2H3 + \| u1\| 2H2 .
3. Linear equation. This section is devoted to studying the existence and uniqueness of the
weak solution of the linear equation in \BbbR n :
vtt(t, x) - \varepsilon \Delta vtt(t, x) - \Delta v(t, x) + \Delta 2v(t, x) + \alpha \Delta vt(t, x) + \Delta 2vt(t, x) + vt(t, x) = 0,
v(0, x) = v0(x), (3.1)
vt(0, x) = v1(x).
Throughout this section, we assume that n \geq 1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1388 H. MAHMOUDI, A. ESFAHANI
Theorem 3.1. For (v0, v1) \in H3+k\times H2+k, k = 0, 1, 2, the linear problem (3.1) admits unique
solution
v \in C([0,\infty );H3+k) \cap C1([0,\infty );H2+k) \cap C2([0,\infty );H1+k)
such that there holds for all \psi \in H2 that
\langle vtt, \psi \rangle + \varepsilon \langle \nabla vtt,\nabla \psi \rangle + \langle \nabla v,\nabla \psi \rangle + \langle \Delta v,\Delta \psi \rangle + \alpha \langle \nabla v,\nabla \psi \rangle + \langle \Delta vt,\Delta \psi \rangle + \langle vt(t), \psi \rangle = 0.
(3.2)
The proof of the existence is obtained by employing the semigroup theory.
Proof of Theorem 3.1. Equation (3.2) can be rewritten as
vtt + (I - \varepsilon \Delta ) - 1(\Delta 2 - \Delta + I)v = (I - \varepsilon \Delta ) - 1(( - \alpha \Delta - \Delta 2 - I)vt + v).
Define D(A) to be the subspace of all v \in H2 such that there exists y = yv \in H1 satisfying
\langle \Delta v,\Delta \psi \rangle + \langle \nabla v,\nabla \psi \rangle + \langle v, \psi \rangle = \langle y, \psi \rangle + \varepsilon \langle \nabla y,\nabla \psi \rangle
for all \psi \in H2. Therefore, it is natural to define an operator A from the definition of D(A) as
follows:
A : D(A) - \rightarrow H1,
Av = yv.
Indeed, A is formally the operator (I - \varepsilon \Delta ) - 1(\Delta 2 - \Delta + I). In addition, it is straightforward to see
from the definition of A that D(A) = H3, and there exists C > 0 such that \| v\| H3 \leq C\| Av\| H1 for
all v \in D(A).
Now, we complete the proof of the existence for the linear problem.
Let z \in H1. It follows from the Lax – Milgram lemma that there exists a unique \~z \in H1 such
that
- \langle \~z, \psi \rangle - \varepsilon \langle \nabla \~z,\nabla \psi \rangle = \langle z, \psi \rangle
for all \psi \in H1. So we define the function h : H1 - \rightarrow H1 such that for each z \in H1, h(z) is given
by the equation
- \langle h(z), \psi \rangle - \varepsilon \langle \nabla h(z),\nabla \psi \rangle = \langle z, \psi \rangle (3.3)
for all \psi \in H1. Furthermore, \varepsilon \Delta h(z) - h(z) = z and h(z) = (I - \varepsilon \Delta ) - 1( - z). For (v(t), w(t)) \in
\in H2 \times H1 with t \geq 0 such that v(0) = v0, w(0) = v1, we define
Z(t) = [v(t), w(t)] ; H(Z(t)) = [0, h(\alpha \Delta w +\Delta 2w + w) - h(v)],
in which Z(0) = (v0, v1) \in H3 \times H2. Consider the operator B : D(B) - \rightarrow H2 \times H1 defined by
B(v, w) = (w, - Av), (3.4)
where D(B) = H3 \times H2. The operator B generates a C0 semigroup of contractions in H2 \times H1.
See Lemma 5.2 in the Appendix for the proof of this fact. Consequently, equation (3.1) is equivalent
to
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO AN EVOLUTION EQUATION . . . 1389
d
dt
Z(t) = B(Z(t)) +H(Z(t)), (3.5)
Z(0) = Z0 \in D(B).
It should be mentioned that H is linear and continuous in X = H2 \times H1. Hence by the semigroups
theory, the operator \scrL := B +H is the generator of an infinitesimal C0 semigroup. Therefore, (3.5)
has a unique solution
Z \in C([0,\infty );D(\scrL )) \cap C1([0,\infty ), X).
That is, there exists a unique function v \in C([0,\infty );H3) \cap C1([0,\infty );H2) \cap C2([0,\infty ), H1) sa-
tisfying (3.2), where v = v(t) is the first component of Z = Z(t). The cases k = 1 and k = 2
of Theorem 3.1 are similar.
Theorem 3.1 is proved.
To give our estimates of the linear problem we define
I1 = \| v0\| 2H4 + \| v1\| 2H3 ,
I2 = \| v0\| 2H4 + \| v1\| 2H3 + \| | x| 2(v0 + v1)\| 2,
I3 = \| v0\| 2H4 + \| v1\| 2H3 + \| v0 + v1\| 2
L
2n
n - 4
.
(3.6)
Theorem 3.2. (i) If (v0, v1) \in H3\times H2, then the solution v \in C([0,\infty );H3)\cap C1([0,\infty );H2)\cap
\cap C2([0,\infty );H1) of the linear problem (3.2) satisfies
E(t) \leq CI0(1 + t) - 1
and
\| v(t)\| 2 \leq CI0,
where I0 and E are defined in Section 2.
(ii) If (v0, v1) \in H4 \times H3 such that | x| 2(v0 + v1) \in L2, then the solution v \in C([0,\infty );H4) \cap
\cap C1([0,\infty );H3) \cap C2([0,\infty );H2) of (3.2) satisfies
E(t) \leq CI2(1 + t) - 2
and
\| v(t)\| 2 + \| \Delta vt(t)\| 2 + \| \Delta (\nabla v(t))\| 2 \leq CI2(1 + t) - 1.
(iii) If (v0, v1) \in H4 \times H3 such that (v0 + v1) \in L
2n
n - 4 , then the solution v \in C([0,\infty );H4) \cap
\cap C1([0,\infty );H3) \cap C2([0,\infty );H2) of (3.2) satisfies
E(t) \leq CI3(1 + t) - 2
and
\| v(t)\| 2 + \| \Delta vt(t)\| 2 + \| \Delta (\nabla v(t))\| 2 \leq CI3(1 + t) - 1.
To prove Theorem 3.2, we need the following lemmas.
Lemma 3.1. Suppose that the hypotheses of Theorem 3.2 hold. Then, the solution v of problem
(3.2) satisfies
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1390 H. MAHMOUDI, A. ESFAHANI
\| vt\| 2 + (1 + t)E(t) \leq CI0 + C
t\int
0
\| \nabla vt(s)\| 2ds,
where C > 0 is a constant which does not depend on the initial data.
Proof. We know from Theorem 3.1 for (v0, v1) \in H3 \times H2 that there exists a unique function
v satisfying (3.2). Thus, we have
d
dt
E(t) + \| vt(t)\| 2 + \| \Delta vt(t)\| 2 - \alpha \| \nabla vt(t)\| 2 = 0,
and consequently
E(t) +
t\int
0
(\| vt(s)\| 2 + \| \Delta vt(s)\| 2 - \alpha \| \nabla vt(s)\| 2)ds = E(0). (3.7)
On the other hand, multiplying (3.7) by (1 + t) and integrating on [0, t], we get
(1 + t)E(t) +
t\int
0
(1 + s)(\| vt(s)\| 2 + \| \Delta vt(s)\| 2 - \alpha \| \nabla vt(s)\| 2)ds = E(0) +
t\int
0
E(s)ds. (3.8)
By substituting \psi = v in (3.2) and integrating on [0, t], we deduce
\| v(t)\| 2 +
t\int
0
\| \Delta v(s)\| 2ds+
t\int
0
\| \nabla v(s)\| ds+
t\int
0
(\| vt(s)\| 2 +\Delta vt(s)\| 2 - \alpha \nabla vt(s)\| 2)ds \leq
\leq CI0 + C
t\int
0
\| \nabla vt(s)\| 2ds. (3.9)
The proof follows from (3.7) – (3.9).
Lemma 3.2. Suppose that the hypotheses of Theorem 3.2 hold. Then the solution v of (3.2)
satisfies
\| \Delta vt(t)\| 2 + \| \Delta v(t)\| 2 + \| \Delta (\nabla v(t))\| 2 +
t\int
0
\| \nabla vt(s)\| 2ds \leq CI0,
where the constant C > 0 is independent of the initial data.
Proof. Let (\varphi 0, \varphi 1) \in H4 \times H3 and \varphi \in C([0,\infty );H4) \cap C1([0,\infty );H3) \cap C2([0,\infty );H2)
be the associated solution of the linear problem (3.2). The regularity of the solution \varphi implies for
any \beta \in \BbbN n with | \beta | \leq 1 that\Bigl\langle
D\beta \varphi tt(t), D
\beta \varphi t(t)
\Bigr\rangle
+ \varepsilon
\Bigl\langle
\nabla \varphi tt(t),\nabla D\beta \varphi t(t)
\Bigr\rangle
+
+
\Bigl\langle
\nabla D\beta \varphi (t),\nabla D\beta \varphi t(t)
\Bigr\rangle
+
\Bigl\langle
\Delta D\beta \varphi (t),\Delta D\beta \varphi t(t)
\Bigr\rangle
-
- \alpha
\Bigl\langle
\nabla D\beta \varphi t(t),\nabla D\beta \varphi t(t)
\Bigr\rangle
+
\Bigl\langle
\Delta D\beta \varphi t(t),\Delta D
\beta \varphi t(t)
\Bigr\rangle
+
\Bigl\langle
D\beta \varphi t(t), D
\beta \varphi t(t)
\Bigr\rangle
= 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO AN EVOLUTION EQUATION . . . 1391
Let
ED\beta \varphi (t) =
1
2
\| D\beta \varphi t(t)\| 2 +
\varepsilon
2
\| \nabla D\beta \varphi t(t)\| 2 +
1
2
\| \nabla D\beta \varphi (t)\| 2 + 1
2
\| \Delta D\beta \varphi (t)\| 2.
Hence,
ED\beta \varphi (t) +
t\int
0
(\| D\beta \varphi t(s)\| 2 + \| \Delta D\beta \varphi t(s)\| 2 - \alpha \| \nabla D\beta \varphi t(s)\| 2)ds = ED\beta \varphi (0).
By taking \beta = (0, . . . , 0, 1, 0, . . . , 0), with 1 in the j th position, we have, for j \in \{ 0, 1, 2, . . .\} ,\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \partial 2\partial x2j \varphi t(t)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2
+
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \partial 2\partial x2j \varphi (t)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2
+
\bigm\| \bigm\| \bigm\| \bigm\| \Delta (
\partial
\partial xj
\varphi (t))
\bigm\| \bigm\| \bigm\| \bigm\| 2 +
t\int
0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \partial 2\partial x2j \varphi t(s)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2
ds \leq
\leq C
\Bigl(
\| \varphi 0\| 2H3 + \| \varphi 1\| 2H2
\Bigr)
.
Therefore, it follows that
\| \Delta \varphi t(t)\| 2 + \| \Delta \varphi (t)\| 2 + \| \Delta (\nabla \varphi (t))\| 2 +
t\int
0
\| \nabla \varphi t(s)\| 2ds \leq C(\| \varphi 0\| 2H3 + \| \varphi 1\| 2H2).
Lemma 3.3. Let (v0, v1) \in H4 \times H3. Then the solution of (3.2) satisfies
(1 + t)\| v(t)\| 2 +
t\int
0
(1 + s)(\| \nabla v(s)\| 2 + \| \Delta v(s)\| 2 + \| vt(s)\| 2 + \| \Delta vt(s)\| 2 - \alpha \| \nabla vt(s)\| 2)ds \leq
\leq CI1 + C
t\int
0
\| \nabla vt(s)\| 2ds+ C
t\int
0
\| v(s)\| 2ds.
Proof. By Theorem 3.1, there exists the unique function v satisfying (3.2). Similar to
Lemma 3.1, we have
(1 + t)E(t) +
t\int
0
(1 + s)(\| vt(s)\| 2 + \| \Delta vt(s)\| 2 - \alpha \| \nabla vt(s)\| 2)ds \leq CI1 + C
t\int
0
\| \nabla vt(s)\| 2ds
and
d
dt
\biggl[
\langle vt, v\rangle + \varepsilon \langle \nabla vt,\nabla v\rangle -
\alpha
2
\langle \nabla v,\nabla v\rangle + 1
2
\langle \Delta v,\Delta v\rangle + 1
2
\langle v, v\rangle
\biggr]
- \| vt(s)\| 2 -
- \varepsilon \| \nabla vt(s)\| 2 + \| \nabla v(s)\| 2 + \| \Delta v(s)\| 2 = 0. (3.10)
By multiplying (3.10) by (1 + t), and integrating on [0, t] as well as utilizing (3.6), we obtain the
desired estimate.
Lemma 3.4. Under the hypotheses of Theorem 3.2(ii), the solution v of (3.2) satisfies
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1392 H. MAHMOUDI, A. ESFAHANI
t\int
0
\| v(s)\| 2ds \leq CI2.
Proof. Consider the function w defined by w(t) =
t\int
0
v(s)ds, where v is the solution of (3.2)
with initial data [v0, v1] \in H4\times H3. Then w \in C1([0,\infty );H4)\cap C2([0,\infty );H3)\cap C3([0,\infty );H2).
It can be easily found out that w is the solution of
\langle wtt(t), \psi \rangle + \varepsilon \langle \nabla wtt(t),\nabla \psi \rangle + \langle \nabla w(t),\nabla \psi \rangle + \langle \Delta w(t),\Delta \psi \rangle - \alpha \langle \nabla wt(t),\nabla \psi \rangle + \langle \Delta wt(t),\Delta \psi \rangle +
+ \langle wt(t), \psi \rangle = \langle v0 + v1, \psi \rangle + \varepsilon \langle \nabla v1,\nabla wt\rangle - \alpha \langle \nabla v0,\nabla wt\rangle + \langle \Delta v0,\Delta wt\rangle , (3.11)
in which w(0, x) = 0 and wt(0, x) = v0. Therefore, we have by substituting \psi = wt in (3.11) and
integrating on [0, t] that
\| \nabla w(t)\| 2 + \| \Delta w(t)\| 2 - \alpha
t\int
0
\| \nabla wt(s)\| 2ds+
t\int
0
\| \Delta wt(s)\| 2ds+
t\int
0
\| wt(s)\| 2ds \leq
\leq CI2 + C \langle v0 + v1, w(t)\rangle .
As a result, we get
\langle v0 + v1, w(t)\rangle =
\int
\BbbR n
| x| 2(v0(x) + v1(x))
w(t, x)
| x| 2
dx \leq
\leq 1
\varepsilon
\| | \cdot | 2(v0 + v1)\| 2 + \varepsilon
\int
\BbbR n
| w(t, x)| 2
| x| 4
dx \leq 1
\varepsilon
\| | \cdot | 2(v0 + v1)\| 2 + \varepsilon K\| \Delta w(t)\| 2,
where in the previous inequality, we used the Hardy-type inequality (see [5])\int
\BbbR n
| u(x)| 2
| x| 4
dx \leq K
\int
\BbbR n
| \Delta u(x)| 2dx, u \in H2.
By using two above estimates with \varepsilon sufficiently small, we observe that
\| \nabla w(t)\| 2 + \| \Delta w(t)\| 2 - \alpha
t\int
0
\| \nabla wt(s)\| 2ds+
t\int
0
\| \Delta wt(s)\| 2ds+
t\int
0
\| wt(s)\| 2ds \leq CI2.
Since
(1 + t)\| v(t)\| 2 +
t\int
0
(1 + s)[\| \nabla v(s)\| 2 + \| \Delta v(s)\| 2 + \| vt(s)\| 2 + \| \Delta vt(s)\| 2 - \alpha \| \nabla vt(s)\| 2]ds \leq
\leq CI1 + C
t\int
0
\| \nabla vt(s)\| 2 + C
t\int
0
\| v(s)\| 2ds \leq CI2,
then
\| v(t)\| 2 \leq CI2(1 + t) - 1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO AN EVOLUTION EQUATION . . . 1393
Lemma 3.5. Under the assumptions of Lemma 3.3, the solution of (3.2) satisfies
\| \Delta (\nabla vt(t))\| 2 + (1 + t)\| \Delta vt(t)\| 2 + (1 + t)\| \Delta (\nabla vt(t))\| 2 +
t\int
0
(1 + s)\| \nabla vt(s)\| 2ds \leq CI1.
Proof. Let (\varphi 0, \varphi 1) \in H5 \times H4 and \varphi \in C([0,\infty );H5) \cap C1([0,\infty );H4) \cap C2([0,\infty );H3)
be the solution of (3.2) with initial data (\varphi 0, \varphi 1). Then, for each \beta \in Nn with | \beta | \leq 2, we have, for
all \psi \in H2,\Bigl\langle
D\beta \varphi tt(t), \psi
\Bigr\rangle
+ \varepsilon
\Bigl\langle
\nabla D\beta \varphi tt(t),\nabla \psi
\Bigr\rangle
+
\Bigl\langle
\nabla D\beta \varphi (t),\nabla \psi
\Bigr\rangle
+
\Bigl\langle
\Delta D\beta \varphi (t),\Delta \psi
\Bigr\rangle
-
- \alpha
\Bigl\langle
\nabla D\beta \varphi t(t),\nabla \psi
\Bigr\rangle
+
\Bigl\langle
\Delta D\beta \varphi t(t),\Delta \psi
\Bigr\rangle
+
\Bigl\langle
D\beta \varphi t(t), \psi
\Bigr\rangle
= 0. (3.12)
Here, D\beta \varphi is denoted by w\beta . Therefore, the energy of w\beta (t, x) is given by
Ew\beta (t) =
1
2
\| w\beta
t \| 2 +
\varepsilon
2
\| \nabla w\beta
t \| 2 +
1
2
\| \nabla w\beta \| 2 + 1
2
\| \Delta w\beta \| 2.
We obtain by substituting \psi = D\beta \varphi t(t) in (3.12) that
d
dt
Ew\beta (t) + \| w\beta
t \| 2 + \| \Delta w\beta
t \| 2 - \alpha \| \nabla w\beta
t \| 2 = 0. (3.13)
By integrating the above identity on [0, t], we get
Ew\beta (t) +
t\int
0
(\| w\beta
t (s)\| 2 + \| \Delta w\beta
t (s)\| 2 - \alpha \| \nabla w\beta
t (s)\| 2)ds = Ew\beta (0), (3.14)
where | \beta | \leq 2. Since w\beta = D\beta \varphi , we see from the definition of the energy that
\| D2ei(Dej\varphi t(t))\| 2 +
t\int
0
\| Dei(Dej\varphi t(s))\| 2ds \leq CEw\beta (0)
for all i, j = 1, 2, . . . , n, where ei is the ith basis vector of \BbbR n. Thus,
\| \Delta (Dej\varphi t(t))\| 2 +
t\int
0
\| \nabla (Dej\varphi t(s))\| 2ds \leq CEw\beta (0).
Multiplying (3.13) by (1 + t) and integrating on [0, t], we obtain
(1 + t)Ew\beta (t) +
t\int
0
(1 + s)(\| w\beta
t (s)\| 2 + \| \Delta w\beta
t (s)\| 2 - \alpha \| \nabla w\beta
t (s)\| 2)ds =
= Ew\beta (0) +
t\int
0
Ew\beta (s)ds. (3.15)
On the other hand, by substituting \psi = D\beta \varphi (t) in (3.12) as well as integrating on [0, t], and using
(3.14), we derive that
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1394 H. MAHMOUDI, A. ESFAHANI
\| w\alpha (t)\| 2 + (1 + t)Ew\alpha (t) \leq C(\| \varphi 0\| 2H4 + \| \varphi 1\| 2H3) + C
t\int
0
\| \nabla w\beta
t (s)\| 2ds.
By combining this estimate with (3.14) and (3.15), we have
(1 + t)Ew\beta (t) +
t\int
0
(1 + s)(\| w\alpha
t (s)\| 2 + \| \Delta w\alpha
t (s)\| 2 - \alpha \| \nabla w\alpha
t (s)\| 2)ds \leq
\leq C(\| \varphi 0\| 2H4 + \| \varphi 1\| 2H3) + C
t\int
0
\| \nabla w\alpha
t (s)\| 2ds.
Moreover, by employing the definition of the energy once again, we can conclude the following
inequality as
(1 + t)\| D2ej\varphi t(t)\| 2 + (1 + t)\| \Delta Dej\varphi (t)\| 2 +
t\int
0
(1 + s)\| Dej\varphi t(s)\| 2ds \leq
\leq C(\| \varphi 0\| 2H4 + \| \varphi 1\| 2H3) + C
t\int
0
\| \nabla w\alpha
t (s)\| 2ds.
The above-mentioned estimates lead to
\| \Delta Dej\varphi t(t)\| 2 + (1 + t)\| D2ej\varphi t(t)\| 2 + (1 + t)\| \Delta Dej\varphi (t)\| 2 +
t\int
0
(1 + s)\| Dej\varphi t(s)\| 2ds \leq
\leq C(\| \varphi 0\| 2H4 + \| \varphi 1\| 2H3).
By summing on i, it follows that
\| \Delta (\nabla \varphi t(t))\| 2 + (1 + t)\| \Delta \varphi t(t)\| 2 + (1 + t)\| \Delta (\nabla \varphi (t))\| 2 +
t\int
0
(1 + s)\| \nabla \varphi t(s)\| 2ds \leq
\leq C(\| \varphi 0\| 2H4 + \| \varphi 1\| 2H3),
where \varphi is the solution of (3.2) with initial data (\varphi 0, \varphi 1) \in H5 \times H4. The density argument
completes the proof.
Proof of Theorem 3.2. (i) The proof is an immediate consequence of Lemmas 3.1 and 3.2.
(ii) We use the fact that the energy E(t) is a nonincreasing function. Thus,
d
dt
[(1 + t)2E(t)] = 2(1 + t)E(t) + (1 + t)2E\prime (t) \leq 2(1 + t)E(t), t \geq 0.
By integrating on [0, t], we can conclude that
E(t) \leq CI2(1 + t) - 2.
Lemmas 3.3, 3.4 and 3.5 give the estimates of (ii).
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO AN EVOLUTION EQUATION . . . 1395
(iii) The proof is similar to the one of (ii) under the assumptions of (iii) and the following fact
(see Lemma 3.4)
t\int
0
\| v(s)\| 2ds \leq CI3.
4. Global existence and asymptotic estimate. In this section, we study the existence of the
local solution for the semilinear problem (1.1). One can easily find out that \Delta f(u), \beta \Delta g(u) \in L2
for all u \in H2. We now consider two functions k1 : H2 \rightarrow H2 and k2 : H2 \rightarrow H2 defined by
\langle k1(u), \psi \rangle + \varepsilon \langle \nabla k1(u),\nabla \psi \rangle = \langle \Delta f(u), \psi \rangle ,
\langle k2(u), \psi \rangle + \varepsilon \langle \nabla k2(u),\nabla \psi \rangle = \langle \beta \Delta g(u), \psi \rangle .
(4.1)
The functions k1, k2 \in H2 are well-defined from the Lax – Milgram lemma. Also, there is C > 0
such that
\| k1(u)\| H2 \leq C\| \Delta f(u)\| L2 ,
\| k2(u)\| H2 \leq C\| \beta \Delta g(u)\| L2
for all u \in H2. In addition, as a result of the elliptic regularity and the uniqueness of (4.1), we can
obtain the following inequalities as
\| k1(u1) - k1(u2)\| H2 \leq C\| \Delta f(u1) - \Delta f(u2)\| L2 ,
\| k2(u1) - k2(u2)\| H2 \leq C\| \beta \Delta g(u1) - \beta \Delta g(u2)\| L2 .
Denote U(t) = (u(t), v(t)) and F (U(t)) = (0, k1(u)+k2(v)), where k1 and k2 are defined in (4.1).
Consider now the problem
d
dt
U(t) = B(U(t)) + F (U(t)),
U(0) = U0,
where B : D(B) \subset X - \rightarrow X is defined in Section 2, U0 = (u0, u1), X = H2 \times H1 and F (\cdot ) as
above. The following result follows from the well-known classical semigroup theorem.
Theorem 4.1 [2]. Let (u0, u1) \in H3\times H2. Then there exists Tm > 0, and a unique solution u \in
\in C([0, Tm);H3)\cap C1([0, Tm);H2)\cap C2([0, Tm);H2) of (2.1) with u(0, x) = u0(x) and ut(0, x) =
= u1(x). Moreover, Tm = +\infty or Tm < +\infty and
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow Tm
\| (u(t), ut(t))\| H3\times H2 = +\infty .
Moreover, we have, for (u0, u1) \in H4 \times H3, that
u \in C([0, Tm);H4) \cap C1([0, Tm);H3) \cap C2([0, Tm);H2).
Proof. Define the operator B : H3 \times H2 - \rightarrow H2 \times H1 with B(u, v) = (v, - Au) and A =
= (I - \varepsilon \Delta ) - 1(\Delta 2 - \Delta + I). By Lemma 5.1 in the Appendix, we need to show that the function
F : D(B) = H3 \times H2 \rightarrow D(B) given by F (u, v) = (0, k1(u) + k2(v)) is Lipschitz, with the graph
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1396 H. MAHMOUDI, A. ESFAHANI
norm on bounded sets, where k1, k2 are functions defined in (4.1). The definition of (4.1) implies
that k1(u) + k2(v) \in H2. Thus, F : D(B) - \rightarrow D(B) is well-defined. Since f and g are Lipschitz,
consequently, it is obvious that F is Lipschitz on bounded sets.
We are ready now to prove Theorem 2.1.
Proof of Theorem 2.1. Define the norms
\| (u, v)\| E = \| v\| + \| \nabla v\| + \| \Delta u\| ,
\| (u, v)\| F = \| u\| + \| \Delta v\| + \| \Delta (\nabla u)\| .
We use the local solution given by Theorem 4.1, and combine it with the decay estimates of the
linear problem (3.2). The solution of (2.1) can be written by the Duhamel principle as
U(t) = S(t)U0 +
t\int
0
S(t - s)F (u(s))ds, (4.2)
where U(t) = (u(t), ut(t)), U0 = (u0, u1), F (U(s)) = (0, k1(u(s)) + k2(ut(s))), and k1, k2 are
defined in (4.1). Also, S(t) indicates the semigroup associated with the linear problem. Define
I0 = \| k1(u(s)) + k2(ut(s))\| 2H2 for each s \in [0, t] and t \in [0, Tm), with Tm given by Theorem 4.1.
From the Gagliardo – Nirenberg inequality and the Sobolev embedding, we have
I0 \leq \| k1(u)\| 2H2 + \| k2(ut)\| 2H2 \leq C\| \Delta f(u)\| 2 + C\| \Delta g(ut)\| 2 \leq
\leq C\| \Delta (u)p\| 2 + c\| \Delta (ut)
p\| 2 \leq
\leq C\| up\| 2H2 + C\| (ut)p\| 2H2 \leq C\| u\| 2p
H2 + C\| ut\| 2pH2 \leq
\leq C
\sum
| \beta | =2
(\| D\beta u\| 2p + \| D\beta ut\| 2p). (4.3)
Furthermore, we have, from the estimates
\| S(t)U0\| E \leq CI
1/2
0 (1 + t) - 1/2,
\| S(t)U0\| F \leq CI
1/2
0 ,
that
\| S(t - s)F (U(s))\| E \leq CI
1/2
0 (s)(1 + t - s) - 1/2,
\| S(t - s)F (U(s))\| F \leq CI
1/2
0 (s)
(4.4)
for s \in [0, t] and t \in [0, Tm). Choose K large enough such that K > C to be fixed later, where C
is the same as in (4.4). Suppose, by contradiction, that the estimates
(1 + t)
1
2 \| U(t)\| E \leq KI
1
2
0 ,
\| U(t)\| F \leq KI
1
2
0
fail for all t \in [0, Tm). By choosing K sufficiently large, there exists T0 \in (0, Tm) such that, for all
t \in [0, T0), we get
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO AN EVOLUTION EQUATION . . . 1397
(1 + t)
1
2 \| U(t)\| E < KI
1
2
0 ,
\| U(t)\| F \leq KI
1
2
0 .
(4.5)
Moreover, the estimates
(1 + T0)
1
2 \| U(T0)\| E = KI
1
2
0
and
\| U(T0)\| F = KI
1
2
0
hold. By estimate (4.4) and (4.2), we have
\| U(t)\| E \leq CI
1
2
0 (1 + t)
- 1
2 + C
t\int
0
(1 + t - s)
- 1
2 I
1
2
0 (s)ds.
We obtain by (4.3) that
\| U(t)\| E \leq CI
1
2
0 (1 + t)
- 1
2 + C
\sum
| \alpha | =2
t\int
0
(1 + t - s)
- 1
2 (\| D\alpha u(s)\| 2p + \| D\alpha ut(s)\| 2p)ds.
By applying (4.5) and for all t \in [0, T0], we can conclude that
\| U(t)\| E \leq CI
1
2
0 (1 + t)
- 1
2 + C
t\int
0
(1 + t - s)
- 1
2 KpI
p
2
0 (1 + s)
- p
2 ds.
Hence, we obtain, for all t \in [0, T0],
\| U(t)\| E \leq CI
1
2
0 (1 + t)
- 1
2 + CpCK
pI
p
2
0 (1 + t)
- 1
2 .
In the previous inequality, we used the following elementary inequality (see [11, 16])
t\int
0
(1 + t - s)
- 1
2 (1 + s) - \beta ds \leq C\beta (1 + t)
- 1
2 , \beta > 1.
If we take K is sufficiently large such that K > C, and \delta > 0 such that \delta \leq
\biggl(
K - C
CpCKp
\biggr) 2
p - 1
, then
we see that, for all t \in [0, T0],
\| U(t)\| E \leq KI
1
2
0 (1 + t)
- 1
2 , (4.6)
provided I0 < \delta . On the other hand, we have from (4.4),
\| U(t)\| F \leq CI
1
2
0 + C
t\int
0
I
1
2
0 (s)ds. (4.7)
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1398 H. MAHMOUDI, A. ESFAHANI
Also, we note, from (4.3) and (4.7), that
\| U(t)\| F \leq CI
1
2
0 + C
\sum
| \alpha | =2
t\int
0
(\| D\alpha u(s)\| 2p + \| D\alpha ut(s)\| 2p)ds \leq
\leq CI
1
2
0 + C
t\int
0
KpI
p
2
0 (1 + s)
- p
2 ds \leq CI
p
2
0 + C1CK
pI
p
2
0
for all t \in [0, T0], where C1 a positive constant. If K > C and I0 < \delta such that \delta satisfies
0 < \delta \leq
\Biggl(
K - C
C1CKp
\Biggr) 2
p - 1
, then, we obtain, for all t \in [0, T0],
\| U(t)\| F \leq KI
1
2
0 . (4.8)
Hence, estimates (4.8) and (4.6) contradict (4.5). This proves the validity of (4.6). Therefore, there
exists a constant \delta > 0 such that I0 < \delta and the solution U(t) satisfies
\| U(t)\| H3\times H2 \leq C
for all t \in [0, Tm]. Hence, Tm = +\infty and the solution is global. In addition, the decay estimates of
(4.6) hold and the proof is now complete.
5. Appendix. We show here that the operator B, defined in (3.4), generates a C0 semigroup of
contraction in H2 \times H1. This will be deduced from the following result [14].
Lemma 5.1. Let B be a linear operator with the dense domain D(B) in a Hilbert space H. If
B is dissipative and 0 \in \rho (B), the resolvent set of B, then B is the infinitesimal generator of a C0
semigroup of contraction in H.
Lemma 5.2. The operator B : D(B) - \rightarrow H2 \times H1 generates a C0 semigroup of contraction
in H2 \times H1.
Proof. We show that B satisfies the assumptions of Lemma 5.1. Let (v, w) \in D(B). Therefore,
v \in H3, w \in H2, and
(B(v, w), (v, w))H2\times H1 = ((w, - Av), (v, w))H2\times H1 = (w, v)H2 + ( - Av,w)H1 \leq
\leq C(\langle w, v\rangle + \langle \nabla w,\nabla v\rangle + \langle \Delta v,\Delta w\rangle + \langle - Av,w\rangle + \varepsilon \langle \nabla ( - Av),\nabla w\rangle ) =
= C(\langle w, v\rangle + \langle \nabla w,\nabla v\rangle + \langle \Delta v,\Delta w\rangle - \langle (I - \varepsilon \Delta )v, w\rangle ) =
= C(\langle w, v\rangle + \langle \nabla w,\nabla v\rangle + \langle \Delta v,\Delta w\rangle -
\bigl\langle
(\Delta 2 - \Delta + I)v, w
\bigr\rangle
) =
= C(\langle w, v\rangle + \langle \nabla w,\nabla v\rangle + \langle \Delta v,\Delta w\rangle - \langle \Delta v,\Delta w\rangle + \langle \Delta w, v\rangle - \langle w, v\rangle ) =
= C(\langle w, v\rangle - \langle \Delta w, v\rangle + \langle \Delta v,\Delta w\rangle - \langle \Delta v,\Delta w\rangle + \langle \Delta w, v\rangle - \langle w, v\rangle ) = 0.
Hence, B is dissipative. Next, we show that 0 \in \rho (B). We first show for (f, g) \in H2 \times H1
that there exists (v, w) \in D(B) such that B(v, w) = (f, g), and consequently w = f \in H2 and
- Av = g \in H1. Let y = - g \in H1. Then, by the Lax – Milgram lemma, there exists a unique
function v \in H2 satisfying
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO AN EVOLUTION EQUATION . . . 1399
\langle \Delta v,\Delta \psi \rangle + \langle \nabla v,\nabla \psi \rangle + \langle v, \psi \rangle = \langle y, \psi \rangle + \varepsilon \langle \nabla y,\nabla \psi \rangle
for all \psi \in H2. Hence, we see that v \in D(A) as well as Av = y, and the operator B is onto. The
fact \| v\| H3 \leq C\| Av\| H1 implies that the operator B is one-to-one. On the other hand, we have, for
v \in D(A) such that - Av = g,
\| B - 1(f, g)\| 2X = \| B - 1(B(v, w))\| 2X = \| (v, w)\| 2X = \| v\| 2H2 + \| w\| 2H1 \leq
\leq \| v\| 2H3 + \| f\| 2H1 \leq C\| Av\| 2H1 + \| f\| 2H2 = C\| - g\| 2H1 + \| f\| 2H2 \leq C\| (f, g)\| 2X ,
where X := H2 \times H1. Hence, 0 \in \rho (B) and B - 1 is continuous. Also, D(B) = H3 \times H2 is dense
in H2 \times H1. Using again the facts D(A) = H3, and \| v\| H3 \leq C\| Av\| H1 for all v \in D(A) and
some C > 0, the proof of Lemma 5.2 follows.
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| id | umjimathkievua-article-6032 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:34Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-60322025-03-31T08:49:43Z Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid Mahmoudi, H. Esfahani , A. Mahmoudi, H. Esfahani , A. Boussinesq equation Asymptotic Behavior Sobolev spaces Convecting fluid Boussinesq equation Asymptotic Behavior Sobolev spaces Convecting fluid UDC 517.9 We consider the Cauchy problem for an evolution equation modeling bidirectional surface waves in a convecting fluid. We study the existence, uniqueness, and asymptotic properties of global solutions to the initial value problem associated withthis equation in $R^n$. We obtain some polynomial decay estimates of the energy. УДК 517.9 Aсимптотична поведiнка розв’язкiв еволюцiйного рiвняння для двонаправлених поверхневих хвиль&nbsp;у рiдинi з конвекцiєю Розглядається задача Кошi для еволюцiйного рiвняння, що моделює двонаправленi поверхневi хвилi у рiдинi з конвекцiєю. Вивчаються iснування, єдинiсть та асимптотичнi властивостi глобальних розв’язкiв початкової задачi, що пов’язана з цим рiвнянням у $R^n$.&nbsp; Отримано деякi полiномiальнi оцiнки спадання енергiї. Institute of Mathematics, NAS of Ukraine 2020-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6032 10.37863/umzh.v72i10.6032 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 10 (2020); 1386 - 1399 Український математичний журнал; Том 72 № 10 (2020); 1386 - 1399 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6032/8760 |
| spellingShingle | Mahmoudi, H. Esfahani , A. Mahmoudi, H. Esfahani , A. Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid |
| title | Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid |
| title_alt | Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid |
| title_full | Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid |
| title_fullStr | Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid |
| title_full_unstemmed | Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid |
| title_short | Asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid |
| title_sort | asymptotic behavior of solutions to an evolution equation for bidirectional surface waves in a convecting fluid |
| topic_facet | Boussinesq equation Asymptotic Behavior Sobolev spaces Convecting fluid Boussinesq equation Asymptotic Behavior Sobolev spaces Convecting fluid |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6032 |
| work_keys_str_mv | AT mahmoudih asymptoticbehaviorofsolutionstoanevolutionequationforbidirectionalsurfacewavesinaconvectingfluid AT esfahania asymptoticbehaviorofsolutionstoanevolutionequationforbidirectionalsurfacewavesinaconvectingfluid AT mahmoudih asymptoticbehaviorofsolutionstoanevolutionequationforbidirectionalsurfacewavesinaconvectingfluid AT esfahania asymptoticbehaviorofsolutionstoanevolutionequationforbidirectionalsurfacewavesinaconvectingfluid |