Long-Time Behavior of Solution for Autonomous Suspension Bridge Equations with State-Dependent Delay ()
1. Introduction
In this paper, we are concerned with the following autonomous suspension bridge with state-dependent delay
(1)
where
,
denotes the deflection in the downward direction,
is a mapping defined on solutions with values in some interval
,
presents the retardation time,
represents the viscous damping,
,
is the initial data on the interval
,
denotes state-dependent delay term and
,
is an external force term.
With regard to partial differential equation with delay (constant and time-dependent delay), there are many results [1] - [6] . For example, Wang in [3] studied dynamics of wave equation with delay by means of pullback asymptotically compactness for the multi-valued processes. Aouadi [5] proved the global and exponential attractors for extensible thermoelastic plate with time-varying delay by establishing quasi-stability estimates. Wang and Ma in [6] considered the existence of pullback attractors for suspension bridge equations with constant delay by using contractive function methods. In order to describe the real world, a new class of state-dependent delay models was introduced and studied recently. When the delay term depends on unknown variables in an equation, we call it a state-dependent delay differential equation. Partial differential equations with state-dependent delay have been essentially less investigated, see the discussions in the papers [7] [8] where they considered the parabolic case, and the results about the systems with state-dependent delay are not so rich as that for other kinds of delay differential equations so far. Chueshov and Rezounenko [9] considered dynamics of second order of in time evolution equations with state-dependent delay where they gave the abstracts results of system with state-dependent delay.
Inspired by the above-mentioned papers, our main goal is to study the existence of global attractor for autonomous suspension bridge equations with state-dependent delay. Compared with the dynamics of suspension bridge equation with constant or time-dependent delay, the new problem encountered in this paper is that the appearance of the state-dependent delay term firstly will lead to the solution of system is not unique, in order to guarantee the uniqueness of solutions, we prove the well-posedness of solution in a certain appropriate C1-type space. Secondly, in the proof of the dissipative property, we need an additional term in energy functional as a compensator for the delay term. In the end, we obtain the existence of global and exponential attractor using quasi-stability method which is different from [10] where the authors considered the non-autonomous suspension bridge equations with state-dependent delay by using contractive function.
The rest of this article consists of four Sections. In the next Section, we give functions setting and iterate some useful lemmas and abstracts. In Section 3, we show the well-posedness of the solution for (1). Finally, the existence of global attractor and exponential attractor for (1) is proved in Sections 4 and 5.
2. Preliminaries
Firstly, define
,
where
, then
is a strictly positive self-adjoint operator, and introduce the scale of Hilbert spaces generated the powers of A as follows:
denote
,
,
, and the scalar product and the norm of
and V as follows:
By the Poincaré inequality, we have
where
is the first eigenvalue of A.
We will denote by
the Banach space
, endowed with the sup-norm. For an element
, its norm is
.
Introduce the phase space
its norm is
Secondly, in order to prove the well-posedness of solution and dissipative of system (see [9] [10] for details), assume that nonlinear term f satisfies the following dissipative conditions (2) (3) and growth conditions (4):
(2)
where
.
(3)
and
(4)
where
.
For every
, there exists
, such that
(5)
Finally, due to the appearance of the state delay term, while proving the well-posedness of solution, it is necessary to suppose that the mapping
is locally Lipschitz
(6)
for every
,
.
Remark [9] The main example of state-depenent delay term is
where
maps
into some interval
. We note that this delay term M is not locally Lipschitz in the classical space of continuous functions
, see [6] [9] [10] [11] [13] for details.
Definition 1 [12] [14] Let
be three reflexive Banach spaces with X compactly embedded in Y,
and let
be a dynamical system given by an evolution operator
We call the dynamical system
quasi-stable on
. If there exists a compact seminorm
on X and nonnegative scalar functions
and
, locally bounded in
, and
with
, such that
(7)
and
(8)
Proposition 2 [12] [14] Let the hypothesis in Definition 1 be in force. Suppose that the dynamical system
is quasi-stable on every bounded forward invariant set B in H. Then
is asymptotically smooth.
Theorem 3 [12] [14] A dissipative dynamical system
has a compact global attractor if and only if it is asymptotically smooth.
Theorem 4 [12] [14] If dynamical system
possesses a compact global attractor
and is quasi-stable on
. Then the attractor
has finite fractal dimension.
Theorem 5 [12] [14] Suppose that dynamical system
is dissipative and quasi-stable on some bounded absorbing set B. In addition, assume that there exists an extended space
, such that mapping
is Hölder continuous in
for each
, that is, there exist
and
, such that
Then the dynamical system
possesses a generalized fractal exponential attractor whose dimension is finite in the space
.
3. Well-Posedness
Definition 6 A vector function
is said to be a weak solution of the problem (1) on the interval
, if
satisfies:
1)
;
2)
, we have that
Lemma 7 Suppose that f satisfy (2) - (3) and (5),
. For any
, there exists
, such that
. Then the solution
of Equation (1) satisfies the following estimates
(9)
where
.
Proof Similar to reference [9] , we set
we can prove that
(10)
where
. Then (9) hold true.
Set
, rewrite (1) as the following first order differential equation in the space
where
, define the operator
and the mapping
as follows:
We can show that the operator
generates exponentially stable C0-semigroup
in
(see [15] ).
Definition 8 A mild solution of (1) on an interval
is defined as a function
where
and
satisfies
Theorem 9 [9] [10] Assume that f satisfy (4) and (6),
. Then for any
,
, there exists
, and a unique mild solution
of (1.1) on the interval
,
or
.
Theorem 10 [9] [10] (Well-posedness) Let assumptions (2) - (6) hold true and
. Then for any
,
, there exists a unique global mild solution
of (1) on the interval
. Moreover, for any
and
there exists a positive constant
, such that
Owing to Theorem 10, we can define an evolution operator as following:
(11)
by the formula
, where
is the solution of (1), satisfying
.
4. Global Attractor
In this section, firstly, we prove the system
has a bounded set; secondly, we will show that the semigroup
corresponding to (1) is asymptotically compact; Finally, the existence of global attractor system (1) is obtained.
Theorem 11 (Dissipative) Assume that the assumptions in Lemma 7 be in force. Then dynamical system
is dissipative, i.e. there exists
,
, such that
Proof Similar to Lemma 7, there holds
now setting
instead of t (where
) in (10),
(12)
Hence, by (12)
It implies that there exists
, any ball
with
is a bounded absorbing set B of
.
In order to prove
is asymptotically smooth, furthermore assume that there exists
, the delay term satisfies for any
,
, there exists
, such that
, one has
(13)
Remark In this paper, the one-dimensional suspension bridge equation with state-dependent delay is considered, but we still need to assume that state-dependent delay term satisfies condition (13), when we establish quasi-stability estimates (8) to verify asymptotical compactness of semigroup
.
Lemma 12 (Quasi-stability) Suppose that (2) - (6) and (13) hold true and
. Then there exists
and
, such that solutions
of (1), initial data
satisfying
(14)
and there holds the following quasi-stability estimates
(15)
where
.
Proof Assume that
are solutions of Equation (1), set
is solution of the following equation
(16)
According to Theorem 11, it is obviously that (14) i.e. true.
We define energy functional
Multiplying (16) with
and integrating it on
, it yields
(17)
By Differential mean value theorem and
.
(18)
where
, and
(19)
Applying condition (13), we obtain
(20)
Combining (18) - (20), from (17), there holds
(21)
For any
, choosing
is enough large, such that
(22)
Multiplying (16) with
and integrating it on
, it yields
(23)
Similar to (18) - (20), we have
and
Substitute above inequalities into (23),
(24)
Furthermore, by using Young and Hölder inequalities, we have
By the definition of energy functional and from (24), it implies that
(25)
Integrating (21) on
, combining with (22), we obtain
(26)
Set
in (21) and combining with (22), we can see that
(27)
and
(28)
Adding (25) (28), and supposing that
, it yields
(29)
Adding
to (29) and substituting (21) into (29),
(30)
Valuation
, from (27)
(31)
substituting (31) into (30),
(32)
where
depends on
.
Set
from (32),
(33)
Subsequently, we compute the last term, set
in (21) and combining with (22),
Substituting above inequality into (33),
By the definition of energy functional
, we have
, choosing
, such that
So we obtain that
Furthermore,
set
, there exists
, such that
By using reference [16] , Repeat the above steps, we conclude that (15) is true.
Remark Maximize (15) on
, it has
(34)
where
is compact semi-norm in Y, taking
,
.
Theorem 13 (Global attractor) Suppose that (2) - (6) and (13) hold true and
. Then the dynamical system
generated by problem (1) has a compact global attractor with finite fractal dimension.
Proof By Theorem 11,
is dissipative. So we only need to prove
is asymptotically smooth. Firstly, by Lemma 12 and (34), quasi-stability inequality (8) holds true, that is
is quasi-stable on any bounded positively invariant sets. According to Proposition 2, then
is asymptotically smooth. It follows from Theorem 3 that dynamical system
has a compact global attractor.
Secondly, we introduce the auxiliary space
its norm
Notice that when
,
, so the space
is an extended space Y.
Let
, denote
the set of functions
which solve (1) with initial data
. We also define the mapping
by the formula
(35)
where u is the solution of (1) with initial data from B. Then the following inequality holds true
(36)
where
,
is a compact seminorm on the
space
. The proof of inequality (36) can see reference [9] . We take
and choose
such that
, where
is the global attractor. One can see that set
is strictly invariant. So we can get the finite dimensionality of the set
in
( [16] , Theorem 2.15). Finally, we consider the restriction mapping
it is clear that mapping
is Lispschitz continuous. Since
and Lispschitz do not increase fractal dimension of a set, we can deduce that
5. Exponential Attractor
Theorem 14 (Exponential attractor) Let the assumptions of Theorem 13 be in force. Then the dynamical system
possesses a generalized fractal exponential attractor
whose dimension is finite in extended space
where
, denotes the closure of H with respect to the norm
.
Proof Let B be a forward invariant bounded absorbing set for
. Then according to (36), we can obtain quasi-stability property for the mapping
defined in (35) on
. Choosing
in (36) such that
and deduce that the mapping
possesses a fractal exponential attractor
( [16] , Corollary 2.23). Subsequently, using (1) we can see that
for all
. This allows us to show that
is a Hölder continuous in t in the space
,
(37)
Now we consider the restriction mapping
and sets
,
. On can see that
is forward invariant. Since
is finite dimensionality,
is Lipschitz from
into
. So the property in (37) implies that
has a finite fractal dimension in
and
is an exponential attracting set for
.
6. Conclusion and Suggestions
In the last several decades, many engineers, physicists and mathematicians intensely focused on studying the collapse of the Tacoma narrow bridge. They tried their best to explain such an amazing event. Lazer and McKenna [17] suggested that a one-dimensional simply supported beam suspended by hangers was modelled as a suspension bridge, which described the vibration of the roadbed in the vertical direction, and the long-time behavior of this suspension bridge model without delay effects were studied by many authors. But we considered the long-time behavior of suspension bridge equation model with state-dependent delay in the paper, compared with constant delay and time-varying delay, differential equations with state-dependent delay are more complex, but they are closer to simulating the real phenomena. However, the theoretical methods of differential equations with state-dependent delay are not as rich as those of other types of delay differential equations, so there are relatively few studies on PDE with state-dependent delay, and they mainly investigated the long-time behavior of the solution of parabolic equations with state-dependent delay. Under suitable assumptions, we consider the long-time behavior of the system by establishing quasi-stability inequality, and obtain the existence of global attractor, exponential attractor, and also discuss the fractal dimension of the attractor in this paper. Therefore, our work can provide theoretical support for the numerical calculation and simulation of suspension bridge equations, viscoelastic beam equations and nonlinear hyperbolic equations in engineering and mathematical physics, and ensure that the numerical calculation and simulation of the problems studied can be carried out smoothly. However, when proving the existence of the global attractor, compared with the contractive function method used in [10] , the damping coefficient (22) needs to be large enough. On the other hand, the existence of attractors is proved in weak topological spaces, but the regularity or asymptotic structure of attractors in strong topological Spaces needs further consideration. Finally, in this paper, we only consider one of the factors affecting the suspension bridge—time delay factor, and then we should consider other factors affecting the stability of the suspension bridge subsequently, such as random factors.
Acknowledgements
We thank the referees for their insightful comments and suggestions all of which improved the presentation. This work was partly supported by the NSFC (11961059, 12161071). Natural Science Foundation of Gansu Province (22JR11RM165, 23JRRM730). Doctor research funding of Longdong University (XYBYZK2112, XYBYZK2113).