Global Asymptotic Stability of Solutions to Nonlinear Marine Riser Equation

Abstract

This paper studies initial boundary value problem of fourth-order nonlinear marine riser equation. By using multiplier method, it is proven that the zero solution of the problem is globally asymptotically stable.

1. Introduction

The straight-line vertical position of marine risers has been investigated with respect to dynamic stability [1]. It studies the following initial boundary value problem describing the dynamics of marine riser:

(1.1)

(1.2)

where is the flexural rigidity of the riser, is the "effective tension", is the coefficient of the Coriolis force, is the coefficient of the nonlinear drag force, and is the mass line density. represents the riser deflection.

By using the Lyapunov function technique, Köhl has shown that the zero solution of the problem is stable.

In [2], Kalantarov and Kurt have studied the initial boundary value problem for the equation

(1.3)

under boundary conditions (1.2). Here , and are given positive numbers, is given real number, is a function, and . It is shown that the zero solution of the problem (1.3)-(1.2) is globally asymptotically stable, that is, the zero solution is stable and all solutions of this problem are tending to zero when . Moreover the polynomial decay rate for solutions is established.

There are many articles devoted to the investigation of the asymptotic behavior of solutions of nonlinear wave equations with nonlinear dissipative terms (see, e.g. [3, 4]), where theorems on asymptotic stability of the zero solution and estimates of the zero solution and the estimates of the rate of decay of solutions to second order wave equations are obtained.

Similar results for the higher-order nonlinear wave equations are obtained in [5].

In this study, we consider the following initial boundary value problem for the multidimensional version of  (1.1):

(1.4)

(1.5)

(1.6)

where is a bounded domain with sufficiently smooth boundary . , and are given positive numbers, and , are given real numbers.

Following [2, 5], we prove that all solutions of the problem (1.4)–(1.6) are tending to zero with a polynomial rate as . In this work, stands for the norm in .

2. Decay Estimate

Theorem 2.1.

Suppose that , and are arbitrary positive numbers, and number satisfies

(2.1)

where is the first eigenvalue of the operator with the homogeneous Dirichlet boundary condition. is an arbitrary positive number when and

(2.2)

Then the following estimate holds:

(2.3)

where A depends only on the initial data and the numbers , , , , and .

Proof.

We multiply (1.4) by and integrate over :

(2.4)

Since

(2.5)

we obtain

(2.6)

Let . Multiplying (1.4) by , integrating over and adding to (2.6), we obtain

(2.7)

Using the method integrating by parts, we get

(2.8)

Hence we obtain

(2.9)

Let

(2.10)

Then we have from (2.9)

(2.11)

where . Using Cauchy-Schwarz and Young's inequalities, we can get the following estimate:

(2.12)

It is not difficult to see that

(2.13)

Using inequalities (2.12) and (2.13) in (2.11), we obtain

(2.14)

Let

(2.15)

then

(2.16)

From (2.14), we get

(2.17)

Let

(2.18)

(2.19)

From (2.6), we have

(2.20)

Therefore is a Lyapunov functional. From (2.20), we find that

(2.21)

Since , we obtain

(2.22)

If is nonnegative, then we have

(2.23)

where .

If is negative, then, using (2.13), we have

(2.24)

where .

Therefore if either nonnegative or negative then it is clear that

(2.25)

where and . Using (2.25), we obtain from (2.17)

(2.26)

Integrating (2.26) with respect to , we can get

(2.27)

(2.28)

Using Poincare's and Cauchy-Schwarz inequalities, we can estimate from below:

(2.29)

thus for

(2.30)

the following estimate holds:

(2.31)

where

(2.32)

Therefore,

(2.33)

(2.34)

Now we can estimate the right-hand side of (2.34) from below. Due to Holder inequality and (2.22), we obtain

(2.35)

where is a positive constant depending on the initial data and the parameters of (1.4).

Using the Holder inequality and the Sobolev imbedding , we obtain

(2.36)

where is a positive constant depending on . Due to (2.22) and

(2.37)

we obtain

(2.38)

where

(2.39)

Therefore

(2.40)

It follows then that for large values of , , the following estimate is valid:

(2.41)

where . Hence we have from (2.19)

(2.42)

From this inequality it follows that the zero solution (1.4)–(1.6) is globally asymptotically stable.