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Existence of solutions to strongly damped plate or beam equations
Boundary Value Problems volume 2012, Article number: 76 (2012)
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.
We consider the following nonlinear strongly damped plate or beam equation:
where Δ is the Laplacian operator, Ω denotes an open bounded set of () with a smooth boundary ∂ Ω and u denotes a vertical displacement at .
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:
where H, , , are Hilbert spaces, X is a linear space, and , are Banach spaces. All embeddings of (2.1) are dense. Let
Furthermore, L has eigenvectors satisfying
and constitutes a common orthogonal basis of H and .
We consider the following abstract equation:
where is a mapping, and is a bounded linear operator satisfying
We say is a global weak solution of Eq. (2.4) provided that
for all and .
Let, X be a Banach space. If (), then, satisfying
3 Existence theorem of abstract equation
Let . Assume:
(A1) There is a functional such that
(A2) Functional is coercive, i.e.,
(A3) B satisfies
Theorem 3.1 Ifis bounded and continuous, and DF is monotone, i.e.,
then, for all, the following assertions hold.
If satisfies (A 1) and (A 2), then Eq. (2.4) has a global weak solution(3.5)
If satisfies (A 1)-(A 3), and in such that(3.6)
then Eq. (2.4) has a global weak solution
Furthermore, if satisfies(3.9)
Proof Let be a common orthogonal basis of H and , satisfying (2.3). Set
Clearly, , .
By using Galerkin method, there exists satisfying
for , and
Firstly, we consider . Let in (3.12). Taking into account (2.2)and (3.1), it follows that
Let . From (2.1) and (2.2), it is known that is an orthogonal basis of . We find that in , and in . From that is an imbedding, it follows that
From (3.2), (3.13) and (3.14), we obtain
which implies that in is uniformly weakly convergent from that is a compact imbedding.
According to (2.2), (2.4), (2.5) and (3.4), we obtain that
Let . From (3.15), we get
Since is dense in , the above equality (3.16) holds for .
We set v the following variable:
where , λ is a real, if , and if . Thus the equality (3.16) is read as
In view of (3.17) and (3.18), we have
We know that
Let . (3.19) can be read as
According to (2.2) and (2.5), we obtain that
Let . It follows that
Since is dense, the above inequality can be rewritten as
which implies that is a global weak solution of Eq. (2.4).
Secondly, we consider . Let in (3.12). In view of (2.2) and (3.1), it follows that
From (3.3), we have
By using the Gronwall inequality, it follows that
which implies that for all ,
From (3.20) and (3.21), it follows that
which implies that in is uniformly weakly convergent from that is a compact imbedding.
The remaining part of the proof is same as assertion (1).
Lastly, assume (3.9) holds. Let in (3.12). It follows that
From (3.21), the above inequality implies
We see that for all , is bounded. Thus . □
4 Main result
Now, we consider the nonlinear strongly damped plate or beam equation (1.1). Set
where , corresponds to .
Theorem 4.1 Under the assumptions (4.1)-(4.6), if φ satisfies the bounded condition of Eq. (1.1), for, then there exists a global strong solution for Eq. (1.1)
Proof We introduce spatial sequences
where the inner products of , and are defined by
where such that is an embedding.
Linear operator and is defined by
It is known that and L satisfy (2.2), (2.3) and (2.5). Define by
Let , where F is the same as in (4.2). We get
which implies conditions (A1), (A2) of Theorem 3.1.
From (4.3), we have
From (4.5) and (4.6), we obtain that is a compact operator. Then, B satisfies (3.6) and (3.7).
We will show (3.3) as follows. From (4.4) and (4.5), for , it follows that
which implies condition (A3) of Theorem 3.1. From Theorem 3.1, Eq. (1.1) has a solution
Lastly, we show that . By Definition 2.1, u satisfies
Then, for any , it follows that
where . Let . From (4.9), we have
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, exists almost everywhere in , and , . □
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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).
The authors declare that they have no competing interests.
All authors typed, read and approved the final manuscript.
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Luo, H., Li, L. & Ma, T. Existence of solutions to strongly damped plate or beam equations. Bound Value Probl 2012, 76 (2012). https://doi.org/10.1186/1687-2770-2012-76
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