Sharp Condition for Global Existence and Blow-Up on Klein-Gordon Equation
We study the initial boundary value problem of the nonlinear Klein-Gordon equation. First we introduce a family of potential wells. By using them, we obtain a new existence theorem of global solutions and show the blow-up in finite time of solutions. Especially the relation between the above two phenomena is derived as a sharp condition.
Klein-Gordon equation is one of the famous evolution equations arising in relativistic quantum mechanics. There are a lot of literature giving the outline of its study trace. For the following type nonlinear Klein-Gordon (NLKG) equation: a lot of papers show the global and local well-posedness and blow-up properties for the Cauchy problem of the above NLKG equation, which can be found in [1–5]. Especially Zhang derived a sharp condition for the global existence of the Cauchy problem of the above NLKG equation in . By introducing a so-called ground state solution, which is the positive solution of the nonlinear Euclidean scalar field equation , he applied a host of very useful properties of the ground state solution to show the sharp condition for this Cauchy problem. In the present paper, we try to make use of the classical potential wells argument , which is different from that in , to clarify the sharp condition for initial boundary value problem (IBVP) of the same NLKG equation.
2. Potential Wells and Their Properties
In this paper, we study the initial boundary value problem of nonlinear Klein-Gordon equation
where for ; for .
For problem (2.1), we define the energy function and some functionals as follows: In aid of the above functionals, we define the potential well as follows: where Then we further give the following definitions
Now, it is ready for us to define a family of potential wells and the outside sets of the corresponding potential wells sets as follows:
The following lemmas are given to show the relations between the functional and .
Lemma 2.1. If then , where and .
Proof. If , then Hence .
Lemma 2.2. If then .
Proof. Note that gives which implies that can be deduced from (2.8).
Lemma 2.3. As the function of is increasing on , decreasing on , and takes the maximum at .
Proof. Now we prove that for any or . Clearly, it is enough to prove that for any or and for any , there exist a and a constant such that . In fact, for the previous we also define by , then , . Let , then
Take , then .
For , we have For we have These give the conclusion.
3. Sharp Condition for Global Existence and Blow-Up
Definition 3.1 (weak solution). The function with is called a weak solution of problem (2.1) for if the following conditions are satisfied:(1),(2)for all .
Theorem 3.2 (global existence). Let satisfy
for ; for .
Let , . Suppose that , , or . Then problem (2.1) admits a global weak solution , with .
Proof. Let be a system of base functions in . Construct the approximate solutions of problem (2.1) as done in 
Multiplying (3.2) by and summing for we can obtain
Integrating with respect to we obtain
For the cases and or , we have
Hence we arrive at then
Hence, there exist a and a subsequence such that in weak star and a.e. in , in weak star, in weakly.
In (3.2) for fixed , letting , we get
Integrating from to , we obtain that , is a global weak solution of problem (2.1).
Next we prove the fact that for . First of all, we will show that . Let be any solution of problem (2.1) with which gives that . If then from the definition of we obtain . If then also. It is easy to see for sufficiently large .
It is enough for us to prove for sufficiently large and . If it is false, then there must exist a for sufficiently large such that , that is, From the energy inequality , we get for sufficiently large , that is, Then we can see that is impossible. On the other hand, if , we obtain . By the definition of , we get , which contradicts (3.11). Hence is true.
Theorem 3.3 (blow-up). Assume that , , , and , then the solution of problem (2.1) must blow up in finite time, that is, there exists a such that .
Proof. Let be any solution of problem (2.1) with and . Set , then
where is the first eigenvalue of problem
Now we will consider the following two cases to finish the proof:(i)if , then ;(ii)if , we should discuss this case in aid of set .
Let be two roots of equation . For any we will prove .
First let us prove . From the energy equality we get and gives for . Thereby we obtain .
Next let us show that for and . If it is false, we can find a as the first time such that , that is, or for some . However from the conservation law we can see that is impossible. If then for . At the same time, Lemma 2.2 yields that and . Hence, by the definition of we get , which contradicts . So we obtain for and . Hence, and . Let , then and . By (3.12) we obtain
For we have Hence there exists a such that which gives By (3.13) for sufficiently large , we obtain By a direct computation we can see that Let , then we get that is, Applying properties of concave function we can get that there exists a bounded such that
From the above two theorems we can easily get a sharp condition for global existence and blow-up of solutions to problem (2.1) like the following.
This work is supported by Natural Science Foundation of Heilongjiang Province (A200702; A200810); Science and Technology Foundation of Education Office of Heilongjiang Province (11541276).
J. M. Ball, “Finite time blow-up in nonlinear problems,” in Nonlinear Evolution Equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), M. G. Grandall, Ed., vol. 40 of Publications of the Mathematics Research Center, University of Wisconsin, pp. 189–205, Academic Press, New York, NY, USA, 1978.View at: Google Scholar | Zentralblatt MATH | MathSciNet