Abstract
In this paper, we study a strongly damped plate or beam equation. By using spatial sequence techniques and energy estimate methods, we obtain an existence theorem of the solution to abstract strongly damped plate or beam equation and to a nonlinear plate or beam equation.
MSC: 35L05, 35L20, 35D30, 35D35.
Keywords:
existence; solution; plate; beam; strongly damped1 Introduction
We consider the following nonlinear strongly damped plate or beam equation:
where Δ is the Laplacian operator, Ω denotes an open bounded set of
It is well known that flexible structures like suspension bridges or overhead power transmission lines can be subjected to oscillations due to various causes. Simple models for such oscillations are described with second and fourthorder partial differential equations as can be seen for example in [18]. The problem (1.1) can be applied in the mechanics of elastic constructions for the study of equilibrium forms of the plate and beam, which has a long history. The abstract theory of Eq. (1.1) was investigated by several authors [914].
The main objective of this article is to find proper conditions on f and g to ensure the existence of solutions of Eq. (1.1). This article uses the spatial sequence techniques, each side of the equation to be treated in different spaces, which is an important way to get more extensive and wonderful results.
The outline of the paper is as follows. In Section 2 we provide an essential definition and lemma of solutions to abstract equations from [1518]. In Section 3, we give an existence theorem of solutions to abstract strongly damped plate or beam equations. In Section 4.10, we present the main result and its proof.
2 Preliminaries
We introduce two spatial sequences:
where H,
Furthermore, L has eigenvectors
and
We consider the following abstract equation:
where
Definition 2.1[15]
We say
for all
Lemma 2.2[18]
Let
3 Existence theorem of abstract equation
Let
(A1) There is a
(A2) Functional
(A3) B satisfies
for
Theorem 3.1If
then, for all
(1) If
(2) If
then Eq. (2.4) has a global weak solution
(3) Furthermore, if
for
Proof Let
Clearly,
By using Galerkin method, there exists
for
for
Firstly, we consider
We get
Let
From (3.2), (3.13) and (3.14), we obtain
Let
which implies that
According to (2.2), (2.4), (2.5) and (3.4), we obtain that
Let
Since
We set v the following variable:
where
and,
In view of (3.17) and (3.18), we have
We know that
and
Let
According to (2.2) and (2.5), we obtain that
Let
Since
which implies that
Secondly, we consider
From (3.3), we have
where
By using the Gronwall inequality, it follows that
which implies that for all
From (3.20) and (3.21), it follows that
Let
which implies that
The remaining part of the proof is same as assertion (1).
Lastly, assume (3.9) holds. Let
From (3.21), the above inequality implies
We see that for all
4 Main result
Now, we consider the nonlinear strongly damped plate or beam equation (1.1). Set
We assume
where
Theorem 4.1Under the assumptions (4.1)(4.6), ifφsatisfies the bounded condition of Eq. (1.1), for
Proof We introduce spatial sequences
where the inner products of
where
Linear operator
It is known that
Let
which implies conditions (A1), (A2) of Theorem 3.1.
From (4.3), we have
From (4.5) and (4.6), we obtain that
We will show (3.3) as follows. From (4.4) and (4.5), for
which implies condition (A3) of Theorem 3.1. From Theorem 3.1, Eq. (1.1) has a solution
Lastly, we show that
Then, for any
where
Then, it follows that
From (4.2) and (4.5), we have
By using the Sobolev embedding theorem, it follows that from (4.7) and (4.8) the right
of the above inequality is bounded. Then,
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Acknowledgement
The authors are very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced the presentation of the manuscript. Supported by the National Natural Science Foundation of China (NO. 11071177), the NSF of Sichuan Science and Technology Department of China (NO. 2010JY0057) and the NSF of Sichuan Education Department of China (NO. 11ZA102).
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