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 damped
We consider the following nonlinear strongly damped plate or beam equation:
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 fourth-order partial differential equations as can be seen for example in [1-8]. 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 [9-14].
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 [15-18]. 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.
We introduce two spatial sequences:
We consider the following abstract equation:
3 Existence theorem of abstract equation
(A3) B satisfies
then Eq. (2.4) has a global weak solution
From (3.2), (3.13) and (3.14), we obtain
According to (2.2), (2.4), (2.5) and (3.4), we obtain that
We set v the following variable:
In view of (3.17) and (3.18), we have
We know that
According to (2.2) and (2.5), we obtain that
From (3.3), we have
By using the Gronwall inequality, it follows that
From (3.20) and (3.21), it follows that
The remaining part of the proof is same as assertion (1).
From (3.21), the above inequality implies
4 Main result
Now, we consider the nonlinear strongly damped plate or beam equation (1.1). Set
Proof We introduce spatial sequences
which implies conditions (A1), (A2) of Theorem 3.1.
From (4.3), we have
which implies condition (A3) of Theorem 3.1. From Theorem 3.1, Eq. (1.1) has a solution
Then, it follows that
From (4.2) and (4.5), we have
The authors declare that they have no competing interests.
All authors typed, read and approved the final manuscript.
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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