Unboundedness of Solutions of Timoshenko Beam Equations with Damping and Forcing Terms

Timoshenko beam equations with external damping and internal damping terms and forcing terms are investigated, and boundary conditions (end conditions) to be considered are hinged ends (pinned ends), hinged-sliding ends, and sliding ends. Unboundedness of solutions of boundary value problems for Timoshenko beam equations is studied, and it is shown that the magnitude of the displacement of the beam grows up to∞ as t → ∞ under some assumptions on the forcing term. Our approach is to reduce the multidimensional problems to one-dimensional problems for fourth-order ordinary differential inequalities.


Introduction
The most fundamental beam equations are of the following form:   2   2 +   4   4 = 0 (0 <  < ,  > 0) with the length , the mass density , the cross-sectional area , the modulus of elasticity , and the flexural rigidity  (see [1, page 416]).Taking account of the rotary inertia and the deflection due to shear, we obtain the following fourth-order beam equation for the transverse vibrations of prismatic beams on elastic foundations: )  4   2  2 +   4   4   =  (, ) (see [1, page 433] and Wang and Stephens [2, page 150]).
However, there appears to be no known unboundedness results for beam equations.The objective of this paper is to provide unboundedness results for (5) by reducing the multi-dimensional problems to one-dimensional problems for ordinary differential inequalities of fourth-order.
In Section 2 we treat the hinged ends and reduce unboundedness problem for (5) to that for ordinary differential inequalities.Sections 3 and 4 are devoted to the hingedsliding ends and sliding ends, respectively.In Section 5, we study fourth-order differential inequalities, and we derive unboundedness results for (5) in Section 6.

Hinged Ends
In this section, we treat the case where the ends of the beam are hinged, so that solutions  = (, ) are required to satisfy the boundary condition Proof.Suppose to the contrary that there exists a solution  of the boundary value problem ( 5), (HE) which is bounded on  × [0, ∞).Then, there exists a constant  > 0 such that that is, First we consider the case where − ≤ (, ) on  × [0, ∞).

Hinged-Sliding Ends
In this section, we deal with the case of hinged-sliding ends, for which the boundary condition takes the form Theorem 3. Every solution  of (5) satisfying (HSE) is unbounded on  × [0, ∞) if for any constant M > 0, all solutions () of the fourth-order differential inequalities 2 )   () 2 )   () As in the proof of Theorem 2, we observe that ∫  0 () is a solution of (19) which is bounded from below.This contradicts the hypothesis.The case where (, ) ≤  on  × [0, ∞) can be treated similarly, and we find that ∫  0 (−)() is a solution of (20) which is bounded from below.The contradiction establishes the theorem.

Sliding Ends
We study the case of sliding ends for which the boundary condition takes the form It is easy to see that Hence, we have the inequality where () = ∫ L 0 .Therefore, we conclude that () is a solution of (23) which is bounded from below.This contradicts the hypothesis.The case where (, ) ≤  on  × [0, ∞) can be treated analogously, and we observe that ∫  0 (−) is a solution of (24) which is bounded from below.This is a contradiction and the proof is complete.

Fourth-Order Ordinary Differential Inequalities
We deal with the ordinary differential inequality of the fourth order and derive sufficient condition for every solution () of ( 28) to be unbounded from below.It is assumed that , ℓ, , and  are nonnegative constants, and () is a continuous function on [0, ∞).

Unboundedness Results for Timoshenko Beam Equations
Combining Theorems 2-4 with Theorem 5, we present unboundedness results for the three types of boundary value problems for (5) under consideration.
Theorem 6.Every solution  of (5) satisfying Proof.The hypothesis (39) implies that every solution () of ( 7) is not bounded from below via Theorem 5. Since the hypothesis (40) implies that lim inf we observe that every solution () of ( 8) is not bounded from below.The conclusion follows from Theorem 2. The proof is complete.
We combine Theorems 3 and 4 with Theorem 5 to establish the following two theorems.