(Department of Apllied Mathematics, Huashang College Guangdong University of Finance &Economics, Guangzhou 511300, China)

Abstract: In this paper,the wave equation defined in a semi-infinite cylinder is considered,in which the nonlinear damping and source terms is included. By setting an arbitrary parameter greater than zero in the energy expression, the fast growth rate or decay rate of the solution with spatial variables is obtained by using energy analysis method and differential inequality technique. Secondly, we obtain the asymptotic behavior of the solution on the external domain of the sphere. In addition, in this paper we also give some useful remarks which show that our results can be extended to more models.

Keywords: Wave equation; Energy analysis; Semi-infinite cylinder; Spatial asymptotic properties

§1. Introduction

Wave equation is an important partial differential equation, which is derived from Maxwell’s equations. It mainly describes various wave phenomena in nature, including shear and longitudinal waves, such as sound, light and water waves. It has a wide range of applications in acoustics, electromagnetics, and fluid mechanics. Therefore, the wave equation has always been the focus of people’s attention. Most of them are concerned about the existence, uniqueness and stability of the solution and the energy decay with time.

In [14], Jorge and Narciso considered a model

wherefis a given function andis nonlocal coefficient. The authors obtained the existence of a global attractor with finite Hausdorff and fractal dimensions. Zhang et al. [16]considered a wave equation with nonlocal nonlinear damping and source terms. By constructing a stable set and using the multiplier technique, a general energy decay property for solutions with time was proved. For more papers, one can see [1–3,5,10].

In this paper, we study a more general wave equation with nonlinear damping and source terms

wherem,n,α,β1,β2>2, a1,a2>0 andfi(·,·):R2→R, i=1,2 are known functions. In addition,we introduce a functionF(u,v) which is defined as

whereF(0,0)=0.

Different from the above literature,we focus on the spatial asymptotic properties of equations(1.1) and (1.2). First, we define equations (1.1) and (1.2) in a semi-infinite cylinder, i.e.,

whereDis a bounded simply-connected region in(x1,x2)-plane with piecewise smooth boundary?D. Assuming that the solutions of the equations satisfy the zero boundary condition on the side of the cylinder and the nonlinear condition at the finite end of the cylinder, the growth or decay estimates are obtained. This type of study is known as Phragmén-Lindel¨of type alternative results and has received long-term attentions. However, these studies mainly focused on parabolic equations (see [4,6–9,11]). Our innovation is to set an arbitrary positive constant in the energy function, thus obtaining a more accurate decay rate than that of literatures.

In particular, we note that the paper [12] studied the spatial selectivity of a class of partial differential equations on the external domain of a sphere. The authors defined a unbounded region

It was proved that the solutions either growth exponentially or decays exponentially with the radius of the sphere. Inspired by [12], we extend the result in [12] to the equations (1.1) and(1.2). In addition, in this paper we also give some useful remarks which show that our results can be extended to more models.

The paper is organized as follows. In section 2, we give the main results. In section 3, the main results are proved. In section 4, we give a conclusion of this paper.

§2. Main results

2.1. Spatial asymptotic properties of (1.1) and (1.2) in R

In this section, we suppose that the equations (1.1) and (1.2) are defined in a semi-infinite cylindrical pipe. The cross-section ofRatx3=zis denoted as

Clearly,D(0)=D.

The equations (1.1) and (1.2) have the following initial-boundary conditions

whereTis a positive constant andgi, i=1,2 are known functions.

To get our main result, we first define an ”energy” function

The main results can be written as

Theorem 2.1.Suppose that u and v are solutions of equations (1.1) and (1.2) with the initialboundary conditions (2.1)-(2.6) and the equations are defined in R, where2<α

If for all z ≥0such that F(z,t)<0, then the solution must decay exponentially, i.e.,

where c1(ω)is a monotone increasing function of ω.

Remark 2.1.From the theorem 2.1, the rate of growth or decay depends the constant ω which is an arbitrary positive constant. So we have that the rate will be bigger than that of the literature.The result about growth is new in the linear case for cylinders.

Remark 2.2.However if the generator of R does not parallel to the x3-axis, it will be more meaningful. In this case, we define?a as

where D(x3)is a bounded simply-connected region which is parallel to(x1,x2)-plane and depends on x3, e.g.,

Although when one study the spatial behavior of various equations, the Poincaré inequality on the cross sections was often used. We note that our analysis does not make use of this inequality.Therefore the theorem 2.1 still holds for the initial-boundary problems of the present paper.

Remark 2.3.To make the decay result in(2.9)explicit, we have to derive the bound for?F(0,t). We give the bound for ?F(0,t)in the following theorem.

Theorem 2.2.Suppose that u and v are solutions of equations (1.1) and (1.2) with the initialboundary conditions (2.1)-(2.6) and the equations are defined in R. The functions f1and f2satisfy

If for all z ≥0such that F(z,t)<0, then the total energy ?F(0,t)can be bounded by known data.

2.2. Spatial asymptotic properties of (1.1) and (1.2) in ?(τ0)

Now, we suppose that the equations (1.1) and (1.2) are defined in ?(τ0). The spherical surface with radiusris denoted as

The equations (1.1) and (1.2) also have the following initial-boundary conditions

Now, we establish a new energy function

wherex=(x1,x2,x3).

Our main result can be written as

Theorem 2.3.Suppose that u and v are solutions of equations (1.1) and (1.2) with the initialboundary conditions (2.11)-(2.13) and the equations are den=fined in?(τ0). If ?r0≥0such that F(r0,t)≥0, then the solution must grow exponentially as r →∞, i.e.,

where c2(ω)is a monotone increasing function of ω. If for all r ≥0such that F(r,t)<0, then the solution must decay exponentially as r →∞, i.e.,

Remark 2.4.In fact, if the sphere B(τ0)is replaced by an ellipsoid, then theorem 2.3 is still valid. The ellipsoid can be defined as

The exterior region of the sphere can be defined as

Remark 2.5.Furthermore, if we define

where f(x1,x2,x3)is a smooth boundary surface of a bounded convex region in three-dimensionalspace, then theorem 2.3 is still valid for

§3. Proofs of main results

3.1. Proof of Theorem 2.1

Proof.Letz0be a point atx3-coordinate axis such that 0≤z0

By (1.1) we have

Similar, we have

Inserting (3.1) and (3.2) into (2.7) and then we have

Next, our purpose is to derive a inequality

from (2.7) and (3.3), where

To do this, we use the H¨older inequality and the Young inequality to obtain

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Similar, we have

Combing (3.6)-(3.8) and (2.7), we have

Similar to (3.9), we have

Inserting (3.9) and (3.10) into (2.7), we can get (3.4).

Now, we consider (3.4) for two cases.

Case I. If?z0≥0 such thatF(z0,t)≥0, then sincewe have

Therefore, (3.4) can be written as

or

Integrating (3.11) fromz0toz, we have

Integrating (3.3) fromz0tozand combining (3.12), we can obtain (2.8).

Case II. If?z ≥0 such thatF(z,t)<0. Therefore, (3.4) can be written as

Integrating (3.13) from 0 toz, we have

This show that

Integrating (3.3) fromzto∞, we have

Combining (3.15) and (3.16) we can obtain (2.9).

3.2. Proof of Theorem 2.2

Proof.To make the decay estimates explicit, we require bound for the total energy. We first clarify the expression of?F(0,t). We write (2.7) atz=0 to have

LetS1andS2be any sufficiently smooth function satisfying the same initial and boundary conditions asuandvrespectively, e.g.,

whereσ1andσ2are arbitrary positive constants. Therefor

Applying the Schwarz inequality in (3.18), we obtain

where we have used the condition (2.10).

Inserting (3.19)-(3.24) into (3.18), we have

where

Similar, we have

On the other hand, from (3.16) we obtain

Inserting (3.25) and (3.26) into (3.17), combining (3.27) we have

or

From (3.28) we can obtain Theorem 2.2.

3.3. Proof of Theorem 2.3

Proof.Using (2.14) and the equations (1.1), (1.2), (2.11)-(2.13), we have

Through the calculation similar to (3.6)-(3.10), we can get the result from (2.14)

Through the analysis similar to (3.4), we can easily get Theorem 2.3.

§4. Concluding remarks

In this paper, we have considered several situations where the solutions of equations (1.1)-(1.2) either grow or decay exponentially or polynomially. We emphasize that the Poincaré inequality on the cross sections is not used in this paper. So our results also hold for the two-dimensional case. On the other hand, there are many deeper problems to be studied in this paper. First of all, we note that Leseduarte and Quintanilla [?] imposed dynamical nonlinear boundary conditions on the lateral side of the cylinder and proved a Phragmén-Lindel¨of alternative for the solutions. Yang and Zhou [15] studied a similar initial-boundary problem and obtained existence of the solution for heat equation. Our idea is to impose nonlinear conditions on the side of the cylinder in this paper, so our problem will become more complex and such research is more meaningful. In addition, we can continue to study the continuous dependence of coefficients in the equation as in [13]. These are the issues we will continue to study in the future.

Acknowledgements

The author would like to express his sincere gratitude to professor Y. Liu from Guangdong University of Finance for his valuable suggestions and comments.