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Existence of solutions to strongly damped plate or beam equations

Hong Luo1*, Li-mei Li1 and Tian Ma2

Author affiliations

1 College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan, 610066, China

2 College of Mathematics, Sichuan University, Chengdu, Sichuan, 610041, China

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Citation and License

Boundary Value Problems 2012, 2012:76  doi:10.1186/1687-2770-2012-76


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/76


Received:11 April 2012
Accepted:3 July 2012
Published:20 July 2012

© 2012 Luo et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 damped

1 Introduction

We consider the following nonlinear strongly damped plate or beam equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M1">View MathML</a>

(1.1)

where Δ is the Laplacian operator, Ω denotes an open bounded set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M3">View MathML</a>) with a smooth boundary Ω and u denotes a vertical displacement at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M4">View MathML</a>.

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.

2 Preliminaries

We introduce two spatial sequences:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M5">View MathML</a>

(2.1)

where H, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M8">View MathML</a> are Hilbert spaces, X is a linear space, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M10">View MathML</a> are Banach spaces. All embeddings of (2.1) are dense. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M11">View MathML</a>

(2.2)

Furthermore, L has eigenvectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M12">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M13">View MathML</a>

(2.3)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M12">View MathML</a> constitutes a common orthogonal basis of H and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M8">View MathML</a>.

We consider the following abstract equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M16">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M17">View MathML</a> is a mapping, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M19">View MathML</a> is a bounded linear operator satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M20">View MathML</a>

(2.5)

Definition 2.1[15]

We say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M21">View MathML</a> is a global weak solution of Eq. (2.4) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M22">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M23">View MathML</a>

(2.6)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M25">View MathML</a>.

Lemma 2.2[18]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M26">View MathML</a>, Xbe a Banach space. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M27">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M28">View MathML</a>), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M29">View MathML</a>, satisfying

3 Existence theorem of abstract equation

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M31">View MathML</a>. Assume:

(A1) There is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M32">View MathML</a> functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M33">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M34">View MathML</a>

(3.1)

(A2) Functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M33">View MathML</a> is coercive, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M36">View MathML</a>

(3.2)

(A3) B satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M37">View MathML</a>

(3.3)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M38">View MathML</a>.

Theorem 3.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M39">View MathML</a>is bounded and continuous, andDFis monotone, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M40">View MathML</a>

(3.4)

then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M22">View MathML</a>, the following assertions hold.

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M42">View MathML</a>satisfies (A1) and (A2), then Eq. (2.4) has a global weak solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M43">View MathML</a>

(3.5)

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M44">View MathML</a>satisfies (A1)-(A3), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M45">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M46">View MathML</a>such that

(3.6)

(3.7)

then Eq. (2.4) has a global weak solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M49">View MathML</a>

(3.8)

(3) Furthermore, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M44">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M51">View MathML</a>

(3.9)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M52">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M53">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M54">View MathML</a> be a common orthogonal basis of H and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M8">View MathML</a>, satisfying (2.3). Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M56">View MathML</a>

(3.10)

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M58">View MathML</a>.

By using Galerkin method, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M59">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M60">View MathML</a>

(3.11)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M61">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M62">View MathML</a>

(3.12)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M63">View MathML</a>.

Firstly, we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M42">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M65">View MathML</a> in (3.12). Taking into account (2.2)and (3.1), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M66">View MathML</a>

We get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M67">View MathML</a>

(3.13)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M68">View MathML</a>. From (2.1) and (2.2), it is known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M69">View MathML</a> is an orthogonal basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M6">View MathML</a>. We find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M71">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M8">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M73">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M6">View MathML</a>. From that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M75">View MathML</a> is an imbedding, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M76">View MathML</a>

(3.14)

From (3.2), (3.13) and (3.14), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M77">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M78">View MathML</a>

(3.15)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M80">View MathML</a> is uniformly weakly convergent from that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M81">View MathML</a> is a compact imbedding.

According to (2.2), (2.4), (2.5) and (3.4), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M82">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M83">View MathML</a>. From (3.15), we get

(3.16)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M85">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M86">View MathML</a>, the above equality (3.16) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M87">View MathML</a>.

We set v the following variable:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M89">View MathML</a>, λ is a real, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M90">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M91">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M92">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M93">View MathML</a>. Thus the equality (3.16) is read as

(3.17)

and,

(3.18)

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

(3.19)

We know that

and

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M99">View MathML</a>. (3.19) can be read as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M100">View MathML</a>

According to (2.2) and (2.5), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M101">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M102">View MathML</a>. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M103">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M104">View MathML</a> is dense, the above inequality can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M105">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M106">View MathML</a> is a global weak solution of Eq. (2.4).

Secondly, we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M44">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M65">View MathML</a> in (3.12). In view of (2.2) and (3.1), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M109">View MathML</a>

From (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M110">View MathML</a>

(3.20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M111">View MathML</a>.

By using the Gronwall inequality, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M112">View MathML</a>

(3.21)

which implies that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M113">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M114">View MathML</a>

From (3.20) and (3.21), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M115">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M116">View MathML</a>

(3.22)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M118">View MathML</a> is uniformly weakly convergent from that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M81">View MathML</a> is a compact imbedding.

The remaining part of the proof is same as assertion (1).

Lastly, assume (3.9) holds. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M120">View MathML</a> in (3.12). It follows that

From (3.21), the above inequality implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M122">View MathML</a>

(3.23)

We see that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M124">View MathML</a> is bounded. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M125">View MathML</a>. □

4 Main result

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M126">View MathML</a>

(4.1)

We assume

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M133">View MathML</a> corresponds to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M134">View MathML</a>.

Theorem 4.1Under the assumptions (4.1)-(4.6), ifφsatisfies the bounded condition of Eq. (1.1), for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M135">View MathML</a>, then there exists a global strong solution for Eq. (1.1)

Proof We introduce spatial sequences

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M137">View MathML</a>

where the inner products of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M8">View MathML</a> are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M141">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M142">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M75">View MathML</a> is an embedding.

Linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M145">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M146">View MathML</a>

It is known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M147">View MathML</a> and L satisfy (2.2), (2.3) and (2.5). Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M148">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M149">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M150">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M152">View MathML</a>

From (4.5) and (4.6), we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M153">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M154">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M155">View MathML</a>

which implies condition (A3) of Theorem 3.1. From Theorem 3.1, Eq. (1.1) has a solution

(4.7)

(4.8)

Lastly, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M158">View MathML</a>. By Definition 2.1, u satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M159">View MathML</a>

Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M160">View MathML</a>, it follows that

(4.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M162">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M163">View MathML</a>. From (4.9), we have

Then, it follows that

From (4.2) and (4.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M166">View MathML</a>

By using the Sobolev embedding theorem, it follows that from (4.7) and (4.8) the right of the above inequality is bounded. Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M167">View MathML</a> exists almost everywhere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M168">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/76/mathml/M170">View MathML</a>. □

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).

References

  1. Lazer, AC, McKenna, PJ: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev.. 32, 537–578 (1990). Publisher Full Text OpenURL

  2. Lazer, AC, McKenna, PJ: Large scale oscillatory behavior in loaded asymmetric systems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 4, 243–274 (1987)

  3. McKenna, PJ, Walter, W: Nonlinear oscillations in a suspension bridges. Arch. Ration. Mech. Anal.. 98, 167–177 (1987)

  4. McKenna, PJ, Walter, W: Traveling waves in a suspension bridge. SIAM J. Appl. Math.. 50, 702–715 (1990)

  5. Ahmed, NU, Biswas, SK: Mathematical modeling and control of large space structures with multiple appendages. Math. Comput. Model.. 10, 891–900 (1988). Publisher Full Text OpenURL

  6. Ahmed, NU, Harbi, H: Mathematical analysis of dynamical models of suspension bridges. SIAM J. Appl. Math.. 58, 853–874 (1998). Publisher Full Text OpenURL

  7. Krol, MS: On a Galerkin-averaging method for weakly nonlinear wave equations. Math. Methods Appl. Sci.. 11, 649–664 (1989). Publisher Full Text OpenURL

  8. van Horssen, WT: An asymptotic theory for a class of initial-boundary value problems for weakly nonlinear wave equations with an application to a model of the galloping oscillations of overhead transmission lines. SIAM J. Appl. Math.. 48, 1227–1243 (1988). Publisher Full Text OpenURL

  9. Medeiros, LA: On a new class of nonlinear wave equation. J. Math. Anal. Appl.. 69, 252–262 (1979). Publisher Full Text OpenURL

  10. Nakao, M: Decay of solutions of some nonlinear evolution equations. J. Math. Anal. Appl.. 60, 542–549 (1977). Publisher Full Text OpenURL

  11. Nishihara, K: Exponentially decay of solutions of some quasilinear hyperbolic equations with linear damping. Nonlinear Anal.. 8, 623–636 (1984). Publisher Full Text OpenURL

  12. Patcheu, SK: On a global solution and asymptotic behaviour for the generalized damped extensible beam equation. J. Differ. Equ.. 135, 299–314 (1997). Publisher Full Text OpenURL

  13. Pereira, DC: Existence uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation. Nonlinear Anal.. 14, 613–623 (1990). Publisher Full Text OpenURL

  14. Kim, JA, Lee, K: Energy decay for the strongly damped nonlinear beam equation and its applications in moving boundary. Acta Appl. Math.. 109, 507–525 (2010). Publisher Full Text OpenURL

  15. Ma, T, Wang, SH: Bifurcation Theory and Applications, World Scientific, Singapore (2005)

  16. Ma, T, Wang, SH: Stability and Bifurcation of Nonlinear Evolution Equations, Science Press, China (2007) in Chinese

  17. Ma, T, Wang, SH: Phase Transition Dynamics in Nonlinear Sciences, New York, Springer (2012)

  18. Ma, T: Theories and Methods for Partial Differential Equations, Science Press, China (2011) in Chinese