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

Definition 2.1[15]

We say is a global weak solution of Eq. (2.4) provided that

for all and .

Lemma 2.2[18]

Let, Xbe a Banach space. If (), then, satisfying

Let . Assume:

(A1) There is a functional such that

(A2) Functional is coercive, i.e.,

(A3) B satisfies

for .

Theorem 3.1Ifis bounded and continuous, andDFis monotone, i.e.,

then, for all, the following assertions hold.

(1) Ifsatisfies (A1) and (A2), then Eq. (2.4) has a global weak solution

(2) Ifsatisfies (A1)-(A3), andinsuch that

then Eq. (2.4) has a global weak solution

(3) Furthermore, ifsatisfies

for, then.

Proof Let be a common orthogonal basis of H and , satisfying (2.3). Set

Clearly, , .

By using Galerkin method, there exists satisfying

for , and

for .

Firstly, we consider . Let in (3.12). Taking into account (2.2)and (3.1), it follows that

We get

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

Let

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

and,

In view of (3.17) and (3.18), we have

We know that

and

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

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 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 . □

Now, we consider the nonlinear strongly damped plate or beam equation (1.1). Set

We assume

where , corresponds to .

Theorem 4.1Under the assumptions (4.1)-(4.6), ifφsatisfies 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 , . □

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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