Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 901631, 14 pages

http://dx.doi.org/10.1155/2011/901631

## Oscillation Properties for Second-Order Partial Differential Equations with Damping and Functional Arguments

Department of Mathematics, Qufu Normal University, Shandong, Qufu 273165, China

Received 19 September 2011; Accepted 21 October 2011

Academic Editor: NorioΒ Yoshida

Copyright Β© 2011 Run Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Using an integral averaging method and generalized Riccati technique, by introducing a parameter , we derive new oscillation criteria for second-order partial differential equations with damping. The results are of high degree of generality and sharper than most known ones.

#### 1. Introduction

Consider the second-order partial delay differential equation where is the Laplacian in and is a bounded domain in with a piecewise smooth boundary .

Throughout this paper, we assume that(H_{1});(H_{2});(H_{3}), and are nonnegative constants, , ;(H_{4}) are convex in with , and , .

We say that a continuous function belongs to the function class , denoted by , if , where , satisfy

Furthermore, the continuous partial derivative exists on , and there is , such that

Various results on the oscillation for the partial functional differential equation have been obtained recently. We refer the reader to [1β3] for parabolic equations and to [4β11] for hyperbolic equations.

Recently, Li and Cui [12] studied the equation of the form with Robin boundary condition where is the unit exterior vector to and is a nonnegative continuous function on and obtained the following result.

Theorem A (see [12, Theorem 2.2]). *Suppose that , let*(C_{1})*, suppose that there exists some and there exist two functions satisfying,*(C_{2})*,*(C_{3})*, and for every ,*(C_{4})*,
**where , and . Then every solution of the problem (1.4), (1.5) is oscillatory in .*

In 2008, Rogovchenko and Tuncay [13] established new oscillation criteria for second-order nonlinear differential equations with damping term
without an assumption that has been required in related results reported in the literature over the last two decades. Motivated by the ideas in [12, 13], by introducing a Parameter , we will further improve Theorems A and derive new interval criteria for oscillation of (1.1). We suggest two different approaches which allow one to remove condition (C_{2}) in Theorem A. A modified integral averaging technique enables one to simplify essentially the proofs of oscillation criteria.

#### 2. Main Results

Theorem 2.1. *Suppose that there exists a function such that for some and for some ,
**
where
**
then every solution of the problem (1.1), (1.5) is oscillatory in .*

*Proof. *Suppose to the contrary that there is a nonoscillatory solution of the problem (1.1), (1.5) which has no zero on for some . Without loss of generality, we assume that and in . Integrating (1.1) with respect to over the domain , we have
From Greenβs formula and the boundary condition (1.5), we have
where denotes the surface element on . Moreover, from (H_{2}), (H_{4}) and Jensenβs inequality, we have
where .

Set
In view of (2.5)β(2.7), (2.4) yields that
Note that (H_{4}), (2.8) yields that
Put
where is given by (2.2), then
that is,
where is defined by (2.3). Multiplying (2.12) by and integrating from to , we have, for some and for all ,
Writing the latter inequality in the form
Using the properties of , we have
and for all ,
By (2.16),
which contradicts (2.1). This proves Theorem 2.1.

Consider a Kamenev-type function defined by , where is an integer. Obviously, belongs to the class , and . Then, we can get the following results.

Corollary 2.2. *Suppose that there exists a function such that for some integer and some ,
**
where and are as defined in Theorem 2.1. Then every solution of the problem (1.1), (1.5) is oscillatory in .*

Theorem 2.3. *Suppose that
**
Assume that there exist functions and such that, for all and for some ,
**
where are as defined in Theorem 2.1 and suppose further that
**
where . Then every solution of the problem (1.1), (1.5) is oscillatory in .*

*Proof. *Suppose to the contrary that there is a nonoscillatory solution of the problem (1.1), (1.5) which has no zero on for some , without loss of generality, we assume that and in .

As in the proof of Theorem 2.1, (2.14) holds for all , we have
Therefore, for ,
It follows from (2.20) that
for all and for any . Then, for all ,
Now, we claim that
Suppose the contrary, that is,
By (2.19), there is a positive constant , satisfying
Let be any arbitrary positive number, then from (2.28) we get that there exists a such that, for all ,
Using integration by parts, for all , we get
By (2.29), there exists a such that, for all ,
It follows from (2.31) that for all ,
Since is an arbitrary positive constant,
which contradicts (2.26). Consequently, (2.27) holds. And from (2.25), we obtain
which contradicts (2.21). This completes the proof of Theorem 2.3.

Choosing as in Corollary 2.2, by Theorem 2.3, we can obtain the following corollary.

Corollary 2.4. *Let and be as in Theorem 2.1, assume further that there exist functions and such that, for all , for some integer , and for some ,
**
and (2.21) hold. Then every solution of the problem (1.1), (1.5) is oscillatory in .*

Theorem 2.5. *Suppose that there exists a function such that for some and for some ,
**
where
**
Then every solution of the problem (1.1), (1.5) is oscillatory in .*

*Proof. *As in Theorem 2.1, without loss of generality, we assume that a nonoscillatory solution of the problem (1.1), (1.5) satisfies and in , , . Define a generalized Riccati transformation
where is given by (2.38). Then
Using an elementary inequality
for all and for all , we conclude from (2.40) that
Multiplying (2.42) by and integrating from , we obtain, for some and for all ,
Therefore, for all , we have
Following the same lines as in the proof of Theorem 2.1, we have
which contradicts the assumption (2.19).

This completes the proof.

Theorem 2.6. *Let (2.19) holds. Assume that there exist functions and such that, for all , any , and for some ,
**
and (2.21) holds, where are defined as in Theorem 2.6 and , then every solution of the problem (1.1), (1.5) is oscillatory in .*

Theorem 2.7. *Let all assumptions of Theorem 2.6 be satisfied except that condition (2.46) be replaced by
**
Then every solution of the problem (1.1), (1.5) is oscillatory in .*

*Remark 2.8. *By introducing the parameter in Theorem 2.3, we derive new oscillation criteria of the problem (1.1), (1.5) which are simpler than that in Theorem A; furthermore, modifications of the proofs through the refinement of the standard integral averaging method allowed us to shorten significantly the proofs of Theorem 2.3. We can also derive a number of oscillation criteria with the appropriate choice of the function and , here, we omit the details.

#### 3. Examples

Now, we consider these following examples.

*Example 3.1. *Consider the partial differential equation
with the boundary condition
where .

Here, , , , , , , .

Let
then
Let , for any ,
Therefore, Corollary 2.2 holds, then every solution of the problem (3.1), (3.2) oscillates in .

*Example 3.2. *Consider the partial differential equation
with the boundary condition (3.2), where .

Here , , , , , , .

Let , then and .

Choose , , a straightforward computation yields
Let . It is not difficult to see that
By Corollary 2.4, we obtain that every solution of problem (3.6), (3.2) oscillates in .

Note that in this example,
so the condition (C_{2}) would not have been satisfied with the same choices of .

#### Acknowledgments

This research was supported by National Science Foundation of China (11171178), the fund of subject for doctor of ministry of education (20103705110003), and the Natural Science Foundations of Shandong Province of China (ZR2009AM011 and ZR2009AL015).

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