Global Existence and Blow-Up of Solutions for Nonlinear Klein-Gordon Equation with Damping Term and Nonnegative Potentials

Abstract

This paper is concerned with the nonlinear Klein-Gordon equation with damping term and nonnegative potentials. We introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions. Using the potential well argument, we obtain a new existence theorem of global solutions and a blow-up result for solutions in finite time.

1. Introduction

In this paper, we consider the nonlinear Klein-Gordon equation with damping term and a real valued potential where , , and is a complex-valued function of , is the Laplace operator on , and is called the damping term [1]. Ha and Nakagiri [1] studied the local existence for the Cauchy problem (1). Here we are interested in the sharp criteria for global existence and blow-up of solutions of the Cauchy problem (1).

The Klein-Gordon equation is a relativistic version of the Schrödinger equation, which describes relativistic electrons. Levine [2], Ball [3], Payne and Sattinger [4], Zhang [5], and Gan and Zhang [6] applied the potential well theory and studied the blowing up properties of the nonlinear Klein-Gordon equations. In [7], Huang and Zhang studied the global existence and blow-up of solutions for the nonlinear Klein-Gordon equation with linear damping term . In [8, 9], the authors studied the existence of global solutions and decay for the energy of solution for the Klein-Gordon equation.

The case of nonlinear damping and source terms is considered by many authors. For instance, Georgiev and Todorova [10] prove that if , a global weak solution exists for any initial data; while the solution blows up in finite time when the initial energy is sufficiently negative. Ikehata [11] considers the solutions of (1) with small positive initial energy, using the so-called potential well theory introduced by Payne and Sattinger in [4]. Todorova and Vitillaro [12] prove that for any given numbers there exist infinitely many data in the energy space such that the initial energy , the gradient norm , and the solution of (1) blows up in finite time.

In this paper, we consider the interaction between the nonlinear damping and source terms for the Cauchy problem (1). For the local well-posedness of the Cauchy problem (1), the readers may refer to [13, 14]. We have considered the global existence and the finite time blowing up. The potential well theory, which was introduced by Liu [15] and has been used for Schrödinger equations in [16–18], was applied to study the Cauchy problem (1). Based on the results, we show the sharp criteria for global existence and blowing up of its solutions. Applying the perturbed energy method we prove the uniform stabilization for . And using concavity arguments we prove that the blow-up solutions exist for . The results can be extended to the case of more general nonlinearities under suitable assumptions. We extend parts of results in [7] and obtain several new results for system (1).

This paper is organized as follows. In Section 2, a family of potential wells are introduced and a series of properties are given. In Section 3, the invariant sets under the flow of problem (1) and the vacuum isolating behavior of solutions for and are discussed. In Section 4, the global existence and blowing up of solutions for problem (1) are proved. In Section 5, the theorem on asymptotic behavior of solutions when is proved.

2. Potential Wells and Their Properties

For the Cauchy problem (1), we define the energy space as becomes a Hilbert space, continuously embedded in , which is endowed with the inner product whose associated norm is denoted by .

Throughout this paper, we make the following assumptions on :

We define the energy functions and two functionals Then we define the potential well as follows: and the outside set of the corresponding potential well where

For , we define

Lemma 1. If , then . Particularly, if , then , where is an embedding constant of .

Proof. If , then It follows that .

Lemma 2. If , then . Particularly, if , then .

Proof. If , then . From we obtain .

Lemma 3. Assume that (2) holds; then(1) for . In particular, we have , .(2).

Proof. (1) For , we get and which yields for .
(2) Let be a minimizer; that is, . For any , define by that is, Then, for each , there exists a unique satisfying (18) which implies . Since implies , we get . Therefore, Noting that we have
On the other hand, let , be a minimizer; that is, . It follows that Therefore, the conclusion follows from the above discussion.

Lemma 4. Assume that (2) holds; then(1), . is continuous on .(2) is increasing on , is decreasing on , and takes the maximum at .

Proof. From Lemma 3, we obtain

3. The Invariant Sets of Solutions

Lemma 5. Assume that , , , , and are the solutions of the function .(1)If or , then for any .(2)If , then for any .

Proof. (1) Let be any solution of the Cauchy problem (1) with which gives . If , then from the definition of we obtain . If , then . Therefore .
Next, we prove . If it is not true, then there must exist a and a such that ; that is, From the energy inequality, we have Then is impossible. On the other hand, if , , then we obtain . By the definition of , we have , which contradicts (28). Hence is true.
(2) First we prove . From the energy inequality we have Using yields for . Therefore we obtain .
Next, we show that for and . If it is false, there exist a and a such that ; that is, However, from the conservation law we get that is impossible. If , then for . At the same time, Lemma 2 yields that , , and . Hence by the definition of , we have , which contradicts . So we obtain

Lemma 6. Assume , , , , and are the solutions of the function . Then and are invariant sets under the flow generated by (1), .

Proof. Let and satisfy (1). From (6), (7), and (8), we have To check , we need to prove If (34) is not true, by continuity, there would exist a such that because of . It follows that . This is impossible for and . Thus (34) is true. So is invariant under the flow generated by (1).
Similarly, we show that is also invariant under the flow generated by (1). This completes the proof of Lemma 6.

Lemma 7. Let the initial data , and be a local solution of the Cauchy problem (1) on . If there exists a and a , then the inequality is fulfilled for .

Proof. By the definition of , we have According to Lemma 6, we have for . From (36) and the identity (8), we get which completes the proof of Lemma 7.

In order to extend the case to we give the following lemma.

Lemma 8. Let , , and (2) hold. Assume ; then the solutions of problem (1) satisfy

Proof. Let be any solution of problem (1) with and , the existence time. From (6), (7), and (9), we have and get for . Hence, using we see that either or (38) hold. If , then for (otherwise there exists a such that ), which contradicts the condition .

Theorem 9. Let , , , and (2) hold. Assume or and . Then the solutions of problem (1) belong to for .

Proof. Let be any solution of problem (1) with or and , the existence time. From (6), we have for . From (41) we see that if , then for . If and , then by Lemma 8 we obtain . Therefore, by (41) we have for . Hence, for the above two cases, we have for .

4. Sharp Condition for Global Existence and Blow-Up

Definition 10 (weak solution). The function with is called a weak solution of problem (1), such that in , in and for all and .

Theorem 11. Let , , and (2) hold. Suppose that , , or ; then Cauchy problem (1) has a global weak solution for some with .

Proof. Let be a system of base functions in . Construct the approximate solution of problem (1) satisfying Multiplying (44) by and summing for , we have Integrating with respect to , we get For and or , we have From we have . Hence from , we obtain .
From (45) and (46), for sufficiently large , we obtain and . Similar with the proof of Lemma 5, from (51), for sufficiently large and , we can prove and
Thus we obtain then Using (54) and the method of compact, we obtain that is a global weak solution of problem (1). From Lemma 5, we have for .

Theorem 12. Let , , and (2) hold. Assume that , , or , where and are two roots of equation . Then the problem (1) admits a unique global solution and for and .

Proof. From Theorem 11, we see that to prove Theorem 12 we only need to prove . Indeed, if it is not true, then there exists a such that . Since implies , we obtain , which contradicts for .

Theorem 13. Let and (2) hold. Assume .(1)If , there exists such that ; then the solution of Cauchy problem (1) blows up in a finite time.(2)If , there exists such that ; then the solution of problem (1) globally exists on . Moreover, for , satisfies

Proof. By , we have .
Firstly, we prove (1) of Theorem 13. From the energy identity we have for all .
Denoting , we have Using the Hölder inequality and the interpolation inequality, we obtain with . From , we have which together with Lemma 6 give Using the Young inequality and , we have since then where the constant is chosen as follows.
Since , we choose the constant so that This guarantees . Then, using this choice and Lemma 7 we get If the constant is fixed, we choose the constant such that Finally, using the inequality (60), (63) and Lemma 7 we have where . Since (56), integrating (67) over we have which concludes that there exists a such that . Hence, is increasing for (which is the interval of existence). Since , there exists a such that is increasing for . When is large enough, the quantities and are small enough. Otherwise, assume that there is such that for all . By integrating the inequality, we obtain a contradiction with (56) and .
Thus in these cases, the quantity will eventually become positive. Therefore for large enough, from (63) and (65) we have Using the Hölder inequality, we get Since from (71) we have . Therefore is concave for sufficiently large , and there exists a finite time such that From assumption on , we obtain Thus one gets and We complete the proof of (1) of Theorem 13.
Next, we prove (2) of Theorem 13.
From (6), (7), and (55), we obtain . It follows that satisfies which will be proved by contradiction. If (76) is not true, then we have . Thus there exists such that and which implies .
On the other hand, for , and (55) yield which is contradictory to Lemma 4.
Therefore, by and Lemma 6, we have and . Thus namely, Therefore we have established the bound of in for and thus the solution of (1) exists globally on .
From (76), , and Lemma 4, we have the estimate (55).
Thus, we complete the proof of Theorem 13.

5. Asymptotic Behaviour of Solutions

We now state and prove the following theorem on asymptotic behavior of solutions when .

Theorem 14. Let in problem (1). Assume , , or . For the global solution of the problem (1) given in Theorem 13, we have for some positive constants and .

Proof. Let be a global solution of the problem (1); then by Theorem 13, we obtain for and , where and () are two roots of equation . Differentiating (7) with respect to and multiplying the obtained equality by , we have Integrating (82) with respect to , we get It follows from and then Moreover, taking in (42), we obtain which implies From (83), (85), and (87), we get From (88) and (89), it follows that where ; then Furthermore, from we obtain Let ; then From (93), we obtain where , . Choosing sufficiently small and together with Gronwall inequality, we have where .

Conflict of Interests

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