Journal of Function Spaces
Volume 2014 (2014), Article ID 176429, 15 pages
http://dx.doi.org/10.1155/2014/176429

Research Article

Global Existence and Blow-Up for the Euler-Bernoulli Plate Equation with Variable Coefficients

Jianghao Hao and Jie Lan

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 4 October 2014; Accepted 8 December 2014; Published 25 December 2014

Academic Editor: Hugo Leiva

Copyright © 2014 Jianghao Hao and Jie Lan. 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.

We prove the local existence, blow-up, global existence, and stability of solutions to the initial boundary value problem for Euler-Bernoulli plate equation with variable coefficients.

1. Introduction

Let be a bounded domain in with smooth boundary and . We will consider the following initial-boundary value problem:

where the constants , , , , , and

where are functions in satisfying

where is a positive constant. is the so-called conormal derivative and is the unit normal of pointing towards the exterior of and .

In physical terms the entries are related to coefficients of elasticity. Let be an matrix for , and let be the natural coordinate system. For each , we define the inverse matrix of by and the inner product and norm over the tangent space by

Then is a Riemannian manifold with metric . Denote the gradient operator in metric by . Then we have

It is well known that the equation

with Dirichlet boundary condition has been studied by many authors. The interaction between the damping and the source term was first discussed by Levine [1, 2] in the linear damping case . He showed that the solutions with negative initial energy blow up in finite time. This result was improved by Kalantarov and Ladyzhenskaya [3] to study more situations. Georgiev and Todorova [4] extended Levine’s results to the nonlinear damping case and if . In their work, the authors proved that global existence for arbitrary initial data when , while blow-up result is showed when and the initial energy associated with the solutions is sufficiently negative. The blow-up result is extended by Levine and Serrin [5] to the case of negative initial energy and (and to abstract evolution problems). Hao et al. [6] studied the nonlinear wave equations with damping and source term. By using the energy compensation method, the authors proved the growth orders of the nonlinear strain term, the damping term, and the source term. They also prove that the solution to the problem exists globally and blows up, respectively, and the estimate for the blow-up time is given. For results of the same nature, we refer the reader to [5, 7–9] and the references therein. Chen and Zhou [10] considered a semilinear Petrovsky equation with damping and source terms. It was proved that the solution blows up in finite time if the positive initial energy satisfies a suitable condition and for the linear damping case the solution blows up in finite time even for vanishing initial energy. Zhou [11] gave an elegant argument to the solution to nonlinear evolution equations with vanishing initial energy. Yao [12] discussed the initial-boundary value problem for Euler-Bernoulli plate with variable coefficients and gave the observability inequality.

The Cauchy problem of (6) has been studied in the last decade; see [13–17]. Global existence for the Cauchy problem with arbitrarily chosen data in the energy space was proved by Todorova [15] for . The case has a richer structure. The negativity of initial energy was used to prove blow-up in many papers. Zhou [16, 17] considered the Cauchy problem for a nonlinear wave equation with linear damping and source terms and showed that the solution blows up in finite time even for vanishing initial energy, and global existence and large time behavior also were discussed.

For the dynamics of marine risers with damping, Köhl [18] used the Lyapunov function technique to show that the zero solution of the system is stable. Kalantarov and Kurt [19] proved that the zero solution for the considered problem is globally asymptotically stable, and an estimate of the rate of decay of the solution is obtained.

The systems of plate equation have been studied by numerous authors; see [20–27]. Messaoudi [26] considered the semilinear Petrovsky equation with the damping and the source term and showed that the solution blows up in finite time if the growth order of damping term is larger than the growth order of source term and the energy is negative, whereas the solution is global if the growth order of damping term is not larger than the growth order of source term. Hoffmann and Rybka [24] studied the analyticity of the nonlinear term forces convergence of solutions for two equations of continuum mechanics. They showed that any solution with appropriate boundary and initial conditions has a limit as goes to infinity. Avalos et al. [20] are interested in the case of thermoelastic plate and they established stability of the rest state. Eden and Milani [21] discussed the exponential attractors for extensible beam equation. Note that [24] differs from [20, 21] because [24] included a viscous term which played an important role in their consideration. Guesmia [22] considered the system of plate equation with damping and proved that the solution decays exponentially if the damping term behaves like a linear function, whereas the decay is of a polynomial order otherwise. Li and Wu [25] discussed the plate stabilization problem with infinite damping and showed that the energy of the problem decays exponentially provided that the negative damping is sufficiently small. For the Cauchy problem of multidimensional generalized double dispersion equation, Xu and Liu [27] proved the existence and nonexistence of global weak solution by potential well method.

Our purpose in this paper is to give the local existence, blow-up, global existence, and stability of the solution to the initial-boundary value problem (1).

We will write denoting the usual norm and denoting the usual norm and denoting

Let

By a weak solution of system (1) we mean a function satisfying

for any .

Our paper is organized as follows. In Section 2, we prove the local existence of the solution to the initial-boundary value problem (1). Section 3 contains the statements and the proof of the blow-up of the solution to problem (1) with . Section 4 is devoted to the blow-up result for problem (1) with . In Section 5, we prove the global existence of the solution for problem (1). The last section is devoted to the asymptotic stability of the solution for problem (1).

2. Local Existence Result

In this section we establish a local existence result for the solution to problem (1) under suitable conditions on and .

First, we give the following local existence result.

Theorem 1 (local existence). Suppose that

Then for initial data , there exists a unique weak solution of problem (1) satisfying

for small enough.

Proof.      
Step 1. For given, we consider the local existence of the problem

We take sequences to approximate and , respectively, and take a sequence to approximate . Then we consider the problems

Using the same arguments as in [28], we get the existence of a sequence of unique solutions of (14) satisfying

For large and , we let denote the set of all functions which satisfy

It follows from the trace theorem that is nonempty if is sufficiently large and is small. For example, if we let

then for suitable large and small. In the present section, we always make this assumption.
Step 2. We proceed to show that the sequence is Cauchy in . For this aim, we set

and then satisfies

We multiply the equation of (19) by and integrate over ; then it follows that

To estimate from the above second term on the right side of (20), we use the inequality

for and . Note that    denote positive constants depending only on , , , , , and rather than the initial data in this section. To estimate the second term on the left side of (20), we use the inequality

for and with , . Then (20)–(22) yield

From Hölder’s inequality and the Sobolev embedding theorem the last term of (23) takes the form

Using Young’s inequality, we have

Combining (23) and (25), we obtain, for any ,

It follows from (26) and Gronwall’s inequality that, for any ,

Furthermore, we have

Since , , and are Cauchy in , , and , respectively, we conclude that , , , and are Cauchy in , , , and , respectively.
Step 3. We now prove that the limit is a weak solution of (13).
To this end, we multiply equation of (14) by and integrate over ; then we obtain that

As , the following hold:

Then it follows that

is an absolutely continuous function on ; thus for almost all , is a weak solution of problem (13). To prove uniqueness, we denote that , are the corresponding solutions of problem (13) to , , respectively. Then satisfies

This shows that for . The uniqueness follows.
Step 4. We denote by the map which carries into ; that is,

where is the solution of problem (13). We establish a priori estimate below to show that maps into itself if is sufficiently large and is sufficiently small relative to . We then equip with the complete metric defined by

and show that is strict contraction if is sufficiently small. The contraction mapping principle thus implies that has a unique fixed point which is obviously a solution to problem (1). For this purpose, we multiply the equation of (13) by and integrate over to get that

From Hölder’s inequality, it follows that, for ,

By choosing large enough and then sufficiently small, we obtain

This shows that maps into itself.
Step 5. We verify that is a contraction if is sufficiently small. Let , and set and . Clearly, is the solution of the problem

Multiplying the equation of  (38) by and integrating it over , we have

Noticing and using the same arguments as (20)–(28), we get the estimate

where is a positive constant independent of and . By choosing so small that

then the map is a contraction. By the contraction mapping principle, the map has a unique fixed point which is obviously a solution . It is clear that is the desired solution of problem (1), and the proof of Theorem 1 is completed.

In the following we denote the maximal existence time of the solution by and the energy of system (1) by

Multiplying the equation of (1) by and integrating over , we obtain, for ,

Denote is the best embedding constant such that

Next we present two lemmas which will be used in the following sections.

Lemma 2. Suppose that satisfies (10); then one has

for .

Proof. From the fact that , we get

if . Otherwise,

Lemma 2 follows from (46) and (47).

Lemma 3. Suppose that the conditions of Theorem 1 hold, and let be the solution of problem (1) with the initial data satisfying and , where

Then there exists , such that

Proof. By Sobolev-Poincaré inequality and the property of the operator , we have

Thus we get

where

It is easy to verify that the function has a maximum at and the maximum value is . From the definition of , we see that is increasing in and decreasing in and as . From the assumptions and , we know that there exists , such that

which completes the proof of Lemma 3.

3. Blow-Up Result for

In this section we establish the blow-up result for the following problem by the energy compensation method:

Our technique of proof follows closely the argument of [7] with the modifications needed for our problem. Denoting the energy of system (54) by

multiplying the equation of (54) by , and integrating over , we obtain, for ,

Next we give the blow-up result.

Theorem 4. Assume that the conditions of Theorem 1 hold, , and is the solution of (54) with the initial data satisfying either one of the following conditions:

(ii);(iii) and , in which

and is the embedding constant satisfying

Then the solution blows up in finite time.

Proof. In the following, we discuss two cases.
Case 1  . We set

where can be chosen later. Since

and , hence

Next we define

where , are to be determined positive constants and is small enough. Integrating by parts and using the equation in (54), we obtain

To estimate the fourth term of (63), we use Young’s inequality,

where can be time dependent, since the integral is taken over the variable. Take so that

for sufficiently large to be specified later. Hence together with the definitions of and , it follows that

Furthermore, by Lemma 3 we have

Thus it follows from (66) and (67) that

By taking , we obtain

Note that, from the definition of and (48), we obtain

Thus by (70) and the inequality

we obtain

Selecting , then together with Lemma 2, we get that

Combining (69) and (73), we have

From (74) and the equality

we obtain

Then we can choose large enough such that the coefficients of and in (76) are strictly positive, and hence we have

where is the minimum of these coefficients. For fixed , we pick small enough such that and

Hence

Thus for , is a nondecreasing function, and we have, for all ,

Here we estimate from above. In the following of this section, we denote by the general positive constant which depends on , , , and .
We use Hölder’s inequality to estimate the term

which implies

Using Young’s inequality

with , , and , we have

By choosing

which implies , then using (70) and the inequality

with and , we deduce

Consequently, we have, for all ,

It follows from (79) and (88) that

A simple integration of (89) over then yields

Therefore (90) shows that blows up in a finite time given by the estimate

So the solution blows up in a certain finite time.
Case 2  . We can take in the definition of ; that is,

Then we can get our result by the same arguments as in Case 1. This completes the proof of Theorem 4.

Remark 5. The earliest blow-up time can be estimated by

and the larger the is, the quicker the blow-up takes place.

4. Blow-Up Result for

In this section we discuss the blow-up result for the following problem:

with the concavity method; see Levine [1, 2].

Denoting the energy of system (94) by

and multiplying the equation of (94) by and integrating over , we obtain, for ,

Our result is as follows.

Theorem 6. Suppose satisfies (10), is the solution of problem (94), and the initial data satisfies either one of the following conditions:

(ii);(iii) and ;(iiii) and , where

and is the embedding constant satisfying

Then the solution blows up in finite time , and

Proof. Define

where , , and are positive constants which are specified latter.
It is not difficult to see that for all . Furthermore, we have

Combining (100) and (101), we obtain

It follows from the fact

that

Noticing that

then we get that

where is defined by

By the definition of , we may also write

From (96), we find

Then we obtain

In the following, we discuss three cases.
Case 1  . By taking , we have

Furthermore, we have

Denoting , we obtain that

We can select suitable positive constants and such that

and then we have

for some . Moreover, there exists a such that and

and then we have

Case 2  . By taking we have

Furthermore, we have

Denoting , we obtain that

Since , then we have

Furthermore, we choose suitable positive constants and such that

and then from the fact that we have

for some . By the similar arguments as we did in the proof of Case 1, we can prove the desired limit.
Case 3  . It follows from Lemma 3 that there exists , such that

Thus together with (110) and (124), it follows that

By taking , we get

Furthermore, we have

Denoting , we obtain that

Furthermore, we choose suitable positive constants and such that

and then we have

for some . By the similar arguments as we did in the proof of Case 1, we can prove the desired limit. Theorem 6 is established.

5. Global Existence Result

In this section we give the result of the global existence of the solution to problem (1) which is similar to [26].

Theorem 7. Suppose , satisfy (10)-(11) and , the initial data , and is the solution of problem (1). Then the solution is global.

Proof. Set

After a simple computation, we get that

From Young’s inequality, it follows that, for any ,

Since , then

where is a constant depending on the domains and and is a constant depending on . Then we get

here and after in this section,    denote positive constants. Indeed, if , then we can choose suitable small such that , and hence

and otherwise, if , from (134), we obtain

By integrating (135) we have, for ,

where . Theorem 7 is established.

6. Global Asymptotic Stability Result

In this section we establish the global asymptotic stability result to problem (1) with small enough and or (linear source term or no source term).

Theorem 8. Suppose satisfies (11), the initial data , and be the solution of problem (1) with and or ( is the constant such that ). Then the solution is asymptotically stable, and the following decay estimate holds for sufficiently large:

where the generic constant depends on , , , , and the initial data .

Proof. For and or , using (48) with and (3), we have

where and . Then thanks to (43), we have

Combining (140) and (141), we obtain

Therefore, we get the following estimate:

Multiplying the equation of (1) by and integrating over , we obtain

Multiplying (145) by a small enough positive constant to be determined and adding to the energy identity

we get that

Set

Then the following inequality holds:

Integrating (149) with respect to over , we obtain

Combining the nondecreasing property of , we get the estimate

From Young’s inequality and the definition of and (140), we have

for small enough. Therefore, the following estimate holds:

Using Hölder’s inequality, (43), (140), and (143), we obtain

where is a positive constant depending on , and

where is the constant such that . It follows from (153)–(155) that

Therefore, there are positive constants    such that

Hence together with (140), we have, for sufficiently large, there exist positive constants , such that

In particular, in the case , we obtain from (149) with

Multiplying by a small enough positive constant to be determined and adding to (159), we obtain

Using (48) with and (3), we have

It is not difficult to see that

for some positive constant . Together with (160)–(162), we can get the following estimate:

We can choose and    sufficiently small with such that the following estimate holds:

Then we can get that

Thus due to (152), we have

where the generic constants depend on , , , and the initial data . This completes the proof of Theorem 8.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by NNSF of China (61374089), NSF of Shanxi Province (2014011005-2), Shanxi Scholarship Council of China (2013-013), and Shanxi International Science and Technology Cooperation Projects (2014081026).

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