This article is devoted to study the existence of global solutions and finite time blow-up of local solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term. By applying the potential well and energy estimation method, in low initial energy and critical initial energy, we derive some sufficient conditions which are global existence and explosion of the solutions for this type of Klein-Gordon equation.
It is well known that the Klein-Gordon equation is an important wave equation that arises in relativistic quantum mechanics and quantum fields [1,2]. It is used to model many physics phenomena, for example, the motion of electric charges in an electric or magnetic field. Particularly, if an electric field was generated by multiple charges with different signs, each charge will be subjected to force from these field sources. So the following question naturally comes to our mind, which is what happens to the properties of the solution for this Klein-Gordon equation with multiple nonlinear source terms. Based on this concern, for simplicity, this article is devoted to investing the existence of global solutions and finite time blow-up of local solutions for the initial-boundary value problem of the nonlinear Klein-Gordon equation with two variable coefficient nonlinear source terms, which has the following form:
where is a bounded smooth boundary domain, , denotes the Laplace operator on , . . Moreover, Here,
In recent decades, a number of researchers are interested in the theory of existence and nonexistence of global solutions of the Klein-Gordon equation with nonlinear source term, and many important results have been obtained [3,4,5, 6,7,8]. For a special case, in , Li and Zhang studied the global existence for the solutions of the following Cauchy problem:
By applying the variational method, they obtain the necessary and sufficient conditions of the existence of global solutions for with .
Besides, in , Gan et al. considered the following system:
where . They established a sharp threshold about the solution global existence and explosion by introducing a cross-constrained variational method.
Ginibre and Velo  studied the Cauchy problem for the nonlinear Klein-Gordon equation with the following type:
where , . They proved the uniqueness of weak solutions and the existence and uniqueness of global strongly continuous solutions with nonlinear Klein-Gordon equation in the energy space under , for details see .
In , Lu and Miao consider the following Cauchy problem for the nonlinear combined Klein-Gordon equation:
where . They give a threshold of blow-up and global well-posedness by a modified variational approach in energy space . Here, .
In , Wang considered nonexistence of global solutions for the following system:
The nonlinear term , which satisfies the condition that there is a real number subject to any ,
where . He applied the concavity method to obtain sufficient condition of this system local solutions blow-up when the initial energy is arbitrarily high.
Furthermore, in , Xu considered the Cauchy problem of the nonlinear Klein-Gordon equation with dissipative term and nonlinear source, which has the following form:
where satisfies the following condition:
There exists a damping term , so that classical convexity method of  cannot be directly applied to derive the finite time blow-up of solutions. He successfully introduced a family of potential wells and proved the global existence, finite time blow up as well as the asymptotic behavior of the solutions for system (4).
Gazzola and Squassina  studied the behavior of solutions of the superlinear hyperbolic equation with (possibly strong) linear damping, which has the following form:
where is an open bounded Lipschitz subset of , , denotes homogeneous Dirichlet boundary condition first eigenvalue of the operator , and
They have shown the global existence of solutions with initial data and uniformly bounded in the natural phase space for system (5). In addition, they also obtained blow-up results in correspondence with initial data having arbitrarily large initial energy, for details please see .
By analyzing the aforementioned articles and reviewing some current literature, for example [15,17, 18,19], we find that the form of the nonlinear source term is relatively simple and that few scholars carried out their study on complex forms. On consideration of the above content, once we encounter the initial boundary value problem of a wave equation with summation form and variable coefficient nonlinear source terms, what happens to the behavior of the solution of this system?
Therefore, the main purpose of this article is to give some sufficient conditions of global existence and finite time blow-up of the solutions for system (1). However, we have to face the following difficulties:
How to handle the aforementioned nonlinear source terms, whether the existence of local solutions of system (1) can be obtained?
Under this type of nonlinear source terms , how to obtain the invariant set of the solution of system (1)?
How to apply potential well theory, concave function and energy estimation method, respectively, to obtain a sufficient condition which is the existence of global solution and explosion of local solution under low initial energy and critical initial energy .
This article is organized as follows. In Section 2, some preliminary results are given. In Section 3, we apply the Galerkin method to prove the existence of the local solution. In Sections 4 and 5, we prove global existence and finite time blow-up of the solution at low initial energy and critical initial energy . In Section 6, we give an application that shows a lower estimate of the solution’s blow-up time for system (1).
2 Notations and set up
In order to simplify the notation, we introduce the following abbreviations:
Moreover, for a weight function , denotes the space of measurable functions so that with norm
In this article, first, we introduce Nehari functional and potential functional, respectively, as follows:
Next, we introduce a potential well depth, which has the following form:
We define Nehari manifold
It is easy to know that the manifold can be divided into two unbounded sets:
Moreover, we also define the sublevels of
We introduce the stable set and the unstable set defined by
In addition, by Theorem 4.2 of , it can also be characterized as
We consider the energy functional about system (1)
Finally, is a generic constant that can change from one line to another.
3 Local existence of solutions
such that and
about all and a.e. .
First, we give the result of the local existence and uniqueness for the solutions of system (1).
Assume hold. Then there exists and a unique local solution of system (1) over [ ]. Furthermore, if
If , we say that the solution of system (1) is explosive and is called the blow-up time. If , we say the solution is global.
Now, we introduce a space endow the norm with
For every , every , and every initial data , there exists a unique
which satisfies the following equation:
We consider using the Galerkin method and combining prior estimation to prove the local existence of the weak solution of system (1). For every , let be the standard orthogonal basis of space .
Suppose that space is spanned by
Taking , s.t, ,
Construct approximation solution of system (1)
So that, we can obtain that
Based on the theory of linear ordinary differential equation (15), there exists a unique global solution . Then we obtain a unique defined by (13) that satisfies (14). Multiplying (14) and summing, we have
Next, (18) from integrating once, for every , we obtain
Now, we estimate the last two terms in the right-hand side of (19).
where , , , , .
for all . In addition, is independent of . By this uniform estimate and Grönwalląŕs inequality, we obtain
Hence, up to a subsequence, we can derive a weak solution of (10) with the above regularity by taking the limit in (13). Due to , we have . Furthermore, since and , we obtain . Finally, from (10), we can easily obtain . Therefore, there exists a , which not only satisfies (10), but also (9). The existence of is obtained.
Next, the proof by contradiction is used to prove uniqueness of this proposition. If and were two solutions of (10), which satisfy the same initial data, by subtracting the equations and testing with , instead of (19), we obtain that
Obviously, it immediately yields . We finish the proof of this lemma.
Let and , for any , we consider
Now, through Lemma 3.1, we define for any , where is the unique solution for (10). Next, we shall show that is a contractive operator which satisfies for a suitable . Obviously, given , the corresponding solution satisfies the following energy identity for all ,
For the last two terms, we apply the same method to estimate (although slightly differently) as for (20) and we obtain
Choosing sufficiently small, we obtain , which shows that . Next, we take . Let , , and substitute them into (10), respectively, setting , we obtain for all and a.e.
Be similar to the discussion above, we obtain
for some and assume is sufficiently small. Therefore, by the contraction mapping principle, there exists a unique (weak) solution for system (1) defined on . The main statement of Theorem 3.1 is proved. Next, we concern the remain assertion. By the construction and analysis above, we know that the local existence time of merely depends (through ) on the norms of the initial data, so that, once continues to be bounded, the solution may be continued, also see , for a similar argument. Therefore, if , we obtain
By multiplying the first equality of (1) with and integrating with respect to , we have
for all . In this case, is nonincreasing, so that
for all . Together with (25), we must obtain
The proof of Theorem 3.1 is now completed.□
4 Global existence and finite time blow up when
Now, let us turn to the global existence of solutions starting with suitable initial data and low initial energy, that depends on Theorem 3.1.
If hold, will be the unique local solution. Moreover, suppose that , for . Then . Namely, system (1) admits a global weak solution .
Through (26) we infer that the energy map is decreasing. From the above condition, we have
In fact, if the above situation is not true, there exists such that . According to the variational characterization (7) of , we have
Through (26), we obtain
This implies and . Hence .
Note that not all local solutions of system (1) are global in time. Particularly, in low initial energy and , the local solutions usually blow up. In the next theorem, we applied concavity method which was introduced by Levine in [21,22] to show the finite time blow up of some solutions of system (1) under .□
In addition, we obtain a lower estimation of the blow-up time of the following solution:
We assume that there exists such that . Without loss of generality, we assume that , and by (26), we can know that for all , so that . This shows that for all .
Assume by contradiction that the solution is global. Then, for any , we consider defined by
Since is continuous and for all , there exists such that
Note that satisfies the first equality of system (1)
Next, multiplying the above equation by and integrating with respect to over , we obtain
can be expressed as
Next, we estimate . From (27), we obtain
Since , we have
Through the Schwarz inequality, we have
Moreover, we introduce , by computing, we have
Integrating the second and the first expression of (37), we can obtain
We can deduce
which contradicts . We completed the main proof of this theorem.□
Next, we give a lower estimation of the blow-up time . First, we introduce the definition
where is a positive constant. Moreover, .
Calculating , we can obtain
Applying the Hölder inequality and the Sobolev embedding inequality, we have
Integrating (40) with respect to form 0 to , we have
Due to , , we have that
5 Global existence and finite time blow up when
In this section, we will use potential well method to prove global existence and finite time blow-up of the solution to system (1) at critical initial energy level .
Let . If , , then system (1) admits a global weak solution .
First, let , . Let us consider the initial conditions
and the corresponding system (1). Due to , we have or . Next, we prove this theorem considering two cases (i) and (ii).
Case i: If , it implies . Furthermore,
Case ii: If , it implies and . Moreover,
Next, to prove finite time blow up of the solution of system (1) under , we first present the following lemma.
Let , and , then all the weak solutions of system (1) belong to .
Let be any weak solution under critical initial energy of system (1), which satisfies . is the maximum existence time of the solution .
Next, we apply reduction to absurdity to prove . Assume this was not the case, then there exists a time , that is, . Moreover, for any , . Based on the value of potential well depth, we can obtain . Combine that with the energy (26) and (30), we have
Therefore, we can derive
From the aforementioned equation, for , we can obtain . Obviously, , so that, . It is in contradiction to the previous assumption. Then the set is invariant under the flow of system (1).□
Let . If and , , then the local solution of system (1) blows up in finite time.