Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations

This paper is concerned with strong blow-up instability (Definition 1.3) for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. In the single case, namely the nonlinear Klein-Gordon equation with power type nonlinearity, stability and instability for standing wave solutions have been extensively studied. On the other hand, in the case of our system, there are no results concerning the stability and instability as far as we know. In this paper, we prove strong blow-up instability for the standing wave to our system. The proof is based on the techniques in Ohta and Todorova [25]. It turns out that we need the mass resonance condition in two or three space dimensions whose cases are the mass-subcritical case.


Introduction
In this paper, we consider the system of the quadratic nonlinear Klein-Gordon equation where 2 N 5, m > 0 and M > 0 denote the mass of the particles, λ and µ are complex constants, c > 0 is the speed of light, and (u, v) is a C 2 -valued unknown function with respect to (t, x). (NLKG) is a reduced model of the Dirac-Klein-Gordon system concerning proton-proton interactions and the Maxwell-Higgs system which appears in the abelian Higgs model (cf. [14,32]). Under the mass resonance condition M = 2m, (NLKG) is regarded as a relativistic version of the system of the quadratic nonlinear Schrödinger equation Indeed, by considering the modulated wave solution to (NLKG) of the form (u c , v c ) = e itmc 2 u, e 2itM c 2 v , (NLKG) is rewritten as (1.1) Taking c → ∞, under M = 2m, (1.1) reaches to (NLS) with λ 1 = −λ/2m and µ 1 = −µ/2M (see [15]). We here assume λ = eµ for some e > 0 and c = 1 in (NLKG). By the scaling, (NLKG) can be reduced to where κ = M/m. We set u = (u 1 , u 2 ). Formally, (1.2) has the conserved energy Re u 2 1 , u 2 L 2 and the conserved charge Q( u, ∂ t u) = Im The purpose of this paper is to investigate instability of the standing wave solution to (1.2) of the form u = e iωt φ 1,ω , e 2iωt φ 2,ω , where ω ∈ R and φ ω = (φ 1,ω , φ 2,ω ) is a R 2 -valued function. In the case of (NLS), Hamano [13] proves strong blow-up instability (see Definition 1.3 below) of standing wave solutions in N = 5 by giving a threshold for scattering or blow-up below the ground state (see also Dinh [9]). In [8], Dinh investigates stability of standing solutions for N 3. On the other hand, In Garrisi [10], and Zhang, Gan and Guo [35], stability of standing wave solutions to the system of nonlinear Klein-Gordon equations has been studied. However the nonlinearity of their system is different from that of (1.2). In terms of (1.2), stability and instability of standing solutions have not been studied as far as we know.

Shatah and Strauss
Otherwise, one calls that e itω φ ω is orbitally unstable. As for subsequent results in N = 1, see Comech and Pelinovsky [7], and Wu [34]. we also refer to Grillakis, Shatah and Strauss [11,12], and Maeda [19] regarding general theory for the orbital stability of solitary wave solutions to an abstract Hamiltonian system.
In view of the previous works, as the first step of the investigation for stability and instability of standing wave solutions to the system such as (1.2), we aim to establish instability for the standing wave solution to (1.2) in the energy-subcritical cases 2 N 5 by applying the technique in [25].
, then φ ω satisfies the system of elliptic equations We here give a definition of solutions to (1.5) as follows: The static energy J ω corresponding to (1.5) is defined by The definition of the ground state in this paper is the following: 15]). We say that a pair of real-valued functions φ ω ∈ We here remark that it is proved in [15] that (1.5) has positive radially symmetric solutions in G ω if N 5 and |ω| < min (1, κ/2).
Before we state the main results, we give definitions of strong blow-up instability for the standing wave solution.
We are in position to state the main results.
In the case |ω| = ω c , we have the following: Theorem 1.5. Let N = 2, 3. Assume κ = 2. Let φ be any nontrivial radially symmetric solution to (1.5) with ω = ω c . Then the standing wave solution (e iωct φ 1 , e 2iωct φ 2 ) is strongly blow-up unstable. The same assertion occurs in ω = −ω c . Remark 1.6. There results are an extension of [25] regarding (1.3) into (1.2). Remark 1.7. Arguing as in [25] due to [5,20], the existence of a global solution in time satisfying can be excluded. Remark 1.8. Compared with (1.3), we need the mass resonance condition κ = 2 when N = 2, 3. We do not know if the condition is essential. Remark 1.9. In order to apply the technique in [25], it is crucial that the ground states are radial. Since (1.5) is the special case of the system of elliptic equations considered in Brezis and Lieb [3], all of least energy solutions to (1.5) are radially symmetric up to a translation in R N if 2 N 5 and |ω| < min (1, κ/2) (see Remark 11 in Byeon, Jeanjean and Mariş [4]).

1.2.
The key of the proofs. A strategy of the proofs of the main results rely on the argument in [25]. This approach is inspired by the technique used to consider some of blow-up problems of solutions to the nonlinear Schrödinger equation, for instance, see [22,23]. In the mass-critical or masssupercritical case N = 4, 5, the key identity of the proof is a local version of a virial type equality Re u 2 1 , u 2 L 2 . In the mass-subcritical case N = 2, 3, the proof is based on a use of a modified virial type equality The left hand side of (1.6) and (1.7) are not well-defined on (H 1 (R N )) 2 × (L 2 (R N )) 2 . Hence we need to approximate the weight function x in (1.6) and (1.7) by the suitable bounded radial weighted function given in section 2. In order to control the error terms generated by the approximation (see Lemma 2.2 below), one exploits techniques given in the proof of Theorem A in [25], which forces to remove the case N = 1 and the ground states are restricted to be radially symmetric.
x ∈ R N } for any Banach spaces X. We often use the following functionals on H 1 : The norms for C 2 -valued functions is defined by for any p 1, all Banach space Y , Z and any interval I ⊂ R, 0 ∈ I.
The rest of the paper is organized as follows: In section 2, we introduce bounded radial weighted functions necessary to approximate virial type identities (1.6) and (1.7), and give key estimates generated by the approximation to show the main results. Section 3 is devoted to the local well-posedness for (1.2). In section 4, we characterize the ground states by using variational argument. Some of the lemmas due to the variational argument will be proven in section 5. We next turn to the proof of the main results in section 6. Finally appendix A provides the proof of the uniform boundedness of solutions to (1.2).

Preliminary
Let us first introduce bounded radial weighted functions which play an important role to prove the main results. Let Φ ∈ C 2 ([0, ∞)) be a nonnegative function such that for all ρ > 0. The following properties are given by [25]: Lemma 7]). For any ρ > 0, It holds that Here is key estimates due to approximating virial type identities (1.6) and (1.7) to show the main results. Lemma 2.2. Let u ∈ C([0, T max ), H 1 ) be a radially symmetric maximal solution of (1.2). Then there exists C 0 > 0 such that for any t ∈ [0, T max ) and all ρ > 0, where Proof. Since u is a radially symmetric with respect to x, we remark that for j = 1, 2. By using the integration by part and (1.2), we have Arguing as in the above, one also obtains Combining these identities with (2.1), (2.2), (2.3) and (2.4), we have (2.5).
Moreover, since a calculation shows one reaches to (2.6).

Local well-posedness in the energy space
In this section, by arguing as in [33], we shall prove the local wellposedness in the energy space We here define the following notations: We give the definition of solutions to (3.1).

Definition 3.1 (Solution). We say a function u(t) is a solution to
in H 1 for any t ∈ I. We call I is a maximal interval of u if u(t) cannot be extended to any interval strictly larger than I. We denote the maximal interval of u by We have the following: In N = 5, the key of the proof of Proposition 3.2 is the Strichartz estimate as follows: Then the following estimates hold: where p ′ is defined by 1 p + 1 p ′ = 1 for any p 2. Proof of Proposition 3.2. Let us introduce the complete metric space Here the constant M will be chosen later. Set We shall prove that Φ is a contraction map in X T,M . Let us first show that Φ maps from X T,M to itself. When N = 2, 3, 4, we estimate . By Sobolev embedding, the last term can be calculated as follows: In similar way, it follows from Sobolev embedding that Combining these above, we see that Combining Sobolev embedding with (3.3), one obtains Let us move on to the case N = 5. We deduce from Proposition 3.3 that This implies that Thus Φ( u) ∈ X T,M . We shall prove that Φ is a contraction map in X T,M . It come from Proposition 3.3 that Hence we see from (3.3) that Hence Φ is a contraction map in X T,M as long as T = T u 0 .7) and (3.8), we have a solution to (3.1) in X T,M . Computing as in (3.6), we also see from Proposition 3 . When 2 N 4, we easily see from Sobolev embedding that In N = 5, arguing as in (3.5), making a use of Proposition 3.3, one has . Therefore it is concluded that ∂ t u ∈ L ∞ (0, T ; L 2 ). The remainder of the proof is standard, so we omit the proof.

Characterization of the ground states
In this section, we will prove the existence of the ground states characterized by the solution to two constrained minimization problems We further set .
The following holds: The next lemma is very helpful to show Proposition 4.1. (1.5). Then following hold: We divide the proof of Proposition 4.1 into the mass-supercritical case N = 5, the mass-critical case N = 4, and the mass-subcritical case N = 2, 3. Let us begin with the case N = 5. In order to show the case N = 5, we need the following lemma: Then the following properties hold: the proof is the same as in that of [ for any λ > 0 and all function f (x). Then, since ψ m is radially symmetric, Lemma 4.3 (i) implies ψ m ∈ K. Further, we see from Lemma 4.3 (vi) that ψ jm = |φ λ 0 ( wm) jm | * . By means of the property of the Schwarz symmetrization (e.g. [17]), one has for any u ∈ H 1 , because of |u 1 |, |u 2 | 0. Combining the above with Lemma 4.3 (iii) and (iv), it holds that Therefore, { ψ m } is a nonnegative, radially symmetric, non-increasing minimizing sequence of (4.1). Also, since { ψ m } converges in H 1 , { ψ m } is bounded in H 1 . Combining this fact with K( ψ m ) = 0 and Sobolev embedding, together with the Young inequality for any a, b 0, we have This implies that L( ψ m ) CL( ψ m ) N 4 . We see from N = 5 that there exists C 0 > 0 such that for any m ∈ N. By using K( ψ m ) = 0 again, together with the above, one has for all m ∈ N. By means of the Strauss compact embedding H 1 rad (R N ) ֒→ L q rad (R N ) for any q ∈ (2, 2+4/(N −2)) (see [31]), there exist Ψ ∈ (H 1 rad (R N )) 2 and subsequences { ψ m } ∞ m=1 (we still use the same notation) such that ψ m → Ψ as m → ∞ weakly in (H 1 rad (R N )) 2 and strongly in (L 3 rad (R N )) 2 , respectively. Also, (4.5) gives us Ψ = 0. Set Ψ 0 = Ψ λ 0 ( Ψ) . By Since ψ 1 = ψ ∈ M ω , we have f ′ (1) = 0. By using K( ψ) = 0, one sees that Combining these above, we see that On the other hand, one has It follows from K( ψ) = 0 that Since ψ ∈ M ω , there exists a Lagrange multiplier λ ∈ R such that Unifying (4.6) and (4.7), together with the above, we have and take ψ ∈ G ω . Then J ω ( ψ) = l. Thanks to Lemma 4.2 (iv) and ψ ∈ G ω , ψ ∈ K is valid. From J ω ( ψ) = l and ψ ∈ K, we have l d 1 ω . In order to show d 1 ω l, let us take φ ω ∈ M ω . Since M ω ⊂ C ω , taking the definition of l into account, we see from J ω ( φ ω ) = d 1 ω that d 1 ω l. Hence d 1 ω = l is concluded. The equivalence of the two problems is immediate. This completes the proof.
Let us prove the case N = 4. We first remark the following: There exists a pair of non-negative, radially symmetric function φ 0 ∈ P such that We here employ the following result which is so-called linear profile decomposition: C for some C > 0. Then there exist v ∈ H 1 (R 4 ) and a sequence {y n } ⊂ R N satisfying the following: There exists a sub-sequence of { u n } (we denote it by the same notation) such that v n := u n (· + y n ) → v ≡ 0 weakly in H 1 (R 4 ), (4.10) We postpone the proof of Proposition 4.5 and continue to prove Proposition 4.1. In order to employ Proposition 4.5, we shall construct a bounded sequence in H 1 (R 4 ). Let us define the scaled function Note that (4.9) gives us P ( φ n ) > 0. Then we have This implies that { ψ n } is L 2 × L 2 -bounded and L( ψ n ) 1/2. Thus, { ψ n } is the H 1 -bounded minimizing sequence for (4.8). By using the Young inequality, we see that for any n ∈ N, Therefore, applying Proposition 4.5 to { ψ n }, there exist ψ 1 ∈ H 1 and a subsequence of { ψ n } still denoted by the same notation such that ψ 1 n := ψ n (· + y 1 n ) → ψ 1 ≡ 0 weakly in H 1 (R 4 ), (4.13) for some {y 1 n } ⊂ R 4 . Let us assume K( ψ 1 ) > 0 to show K( ψ 1 ) 0. By using (4.14) and K( ψ 1 n ) 0, we deduce that K( ψ 1 n − ψ 1 ) K( ψ 1 n ) − K( ψ 1 ) 0 for sufficiently large n. Therefore, it follows from the definition of d 2 ω that for sufficiently large n. Hence, combining the above with (4.15), one sees that for sufficiently large n. From lim n→∞ J 2 ω ( ψ 1 n ) = d 2 ω , taking n → ∞, this tells us that J 2 ω ( ψ 1 ) 0, which contradicts ψ 1 ≡ 0. Thus we have K( ψ 1 ) 0. for each j = 1, 2, which yields Thus ψ 1 is a solution of (4.8). Set Ψ 1 = |ψ 1 1 | * , |ψ 1 2 | * . Together with (4.17), it follows from (4.16) and (4.3) that Let us next show Ψ 1 is a solution of (4.1). Set This yields s = 1. Therefore it is concluded that K( Ψ 1 ) = 0. We shall prove d 1 ω = d 2 ω . Remark that J ω ( u) = J 2 ω ( u) as long as K( u) = 0. By the definition of d 1 ω and d 2 ω , it holds that d 2 It is then established that Ψ 1 is a solution of (4.1). Hence Ψ 1 ∈ M ω , that is, M ω is nonempty.
Let us finally prove Proposition 4.5. We need the following lemma to show Proposition 4.5.
Proof of Proposition 4.5. Firstly, we have (4.10) immediately from Lemma 4.6. Also, by (4.10), it is easy to show (4.12) because v jn as n → ∞ for j = 1, 2. Let us prove (4.11). Arguing as in the proof of (4.12), it holds that as n → ∞ for j = 1, 2. Next, we shall show where q ∈ [2, 2 + 4/(N − 2)) and Q y is defined by . [1, p168]), we have v n → v a.e in R N as n → ∞. Hence, by using the boundedness of { v n } in H 1 (R N ) and Sobolev embedding, together with the Lebesgue convergence theorem, (4.23) is obtained. Thus (4.11) holds. This completes the proof.
We finish this section by proving Proposition 4.1 in N = 2, 3.
Proof of Proposition 4.1 in N = 2, 3. Let us first show that M ω is nonempty. We here remark that for any m ∈ Z + . This implies that { φ m } is bounded in H 0 (R N ). Hence, combining K 0 ω ( φ m ) = 0 with the Gagliardo-Nirenberg inequality, together with the Young inequality, we see that Since N = 2, 3, this allows us to exists C > 0 such that for any m ∈ Z + . Therefore, { φ m } is bounded in H 1 (R N ). Hence, by means of the Strauss compact embedding H 1 rad (R N ) ֒→ L q rad (R N ) for any q ∈ (2, 2+ 4/(N − 2)) (see [31]), there exist a subsequence { φ m } (we still use the same notation) and w ∈ H 1 (R N ) such that φ m → w weakly in H 1 (R N ) and strongly in (L 3 (R N )) 2 .
Let us show w = 0. Suppose that w = 0. By using K 0 ω ( φ m ) = 0 and φ m → 0 strongly in (L 3 (R N )) 2 , we deduce that φ m → 0 strongly in H 1 (R N ). On the other hand, together with the Young inequality, one sees from K 0 ω ( φ m ) = 0 and Sobolev embedding that Here, in the above last line, we employ the inequality due to the Young inequality for any ε > 0 and j = 1, 2. Unifying the above and φ m = 0, there exists C > 0 such that for any m ∈ Z + . Hence, we have φ 1m C for any m ∈ Z + . However this contradicts w = 0, that is, w = 0. In particular, w ∈ (H 1 (R N )) 2 \ 0. Collecting (4.24), (4.25) and K 0 ω ( φ m ) = 0, we reach to Here, if K 0 ω ( w) < 0, then Lemma 5.3 (i) (will be shown in section 5) leads to This yields K 0 ω ( w) = 0. Hence we conclude that w attains (4.2), that is M ω is nonempty.
We shall prove M ω ⊂ C ω . Let w ∈ M ω . Then there exists a Lagrange multiplier η ∈ R such that .

Some of the variational lemmas
In this section, we will prove some of the lemmas due to the variational results in section 4. We here remark the identity (E − ωQ)( u, ∂ t u) = J ω ( u) (5.1)
Proof. Let us begin with the proof of the fact K( u(t)) < 0 for any t ∈ [0, T max ) by a contradiction. Suppose that there exists t 1 ∈ (0, T max ) such that K( u(t 1 )) = 0 and K( u(t)) < 0 for any t ∈ [0, t 1 ). Lemma 5.1 (i) gives us for any t ∈ [0, t 1 ), Taking t → t 1 , we have u(t 1 ) = 0. By (4.1), one sees J ω ( u(t 1 )) d 1 ω . On the other hand, it follows from (5.1), ( ϕ, ψ) ∈ R 1 ω and the fact E and Q are conserved for all t that ω . This yields a contradiction. Thus, K( u(t)) < 0 for any t ∈ [0, T max ). Together with the above and the fact E and Q are conserved for all t, combining Lemma 5.1 (i) with (5.1), it turns out that for any t ∈ [0, T max ), which completes the proof.

Proof of the main results
This section consists of the proof of the main results. Before going to the proof of the main results, we shall handle the following lemma concerning the uniform boundedness of global solutions to (1.2): Lemma 6.1. Let u ∈ C([0, ∞), H 1 ) be a solution to (1.2). Then Proof. The proof is carried out as in Proposition 3.1 and Lemma 3.5 in [5]. In detail, we refer readers to the Appendix A.
6.1. Proof of Theorem 1.4. Let us start to prove the mass-supercritical and critical case N = 4, 5.
Proof of Theorem 1.4 in N = 4, 5. The strategy of the proof is based on [25,Theorem A]. For any ε > 0, we can take λ = λ( φ ω , ω) > 1 satisfying Lemma 5.1 (ii) yields δ > 0. Here, (6.2) allows us to exist J > 0 such that for j = 1, 2 and all T > 0. we deduce from the mean value theorem that for Let us prove by a contradiction. Assume that there exists a global solution u ∈ C([0, ∞), H 1 ) of (1.2) with u(0) = λ φ ω and ∂ t u(0) = λ(iωφ 1,ω , 2iωφ 2,ω ). Since u is radially symmetric with respect to x for any t 0, we can define I 1 ρ ( u, ∂ t u) by (2.7), so that one has (2.5). By using (2.5), Lemma 5.1 and Lemma 5.2, we have d dt for any t 0 and all ρ > 0, where I 1 ρ (t) = I 1 ρ ( u(t), ∂ t u(t)) is defined by (2.7) and Integrating the both side of (6.4) for t, one sees from T i+2 − T i 1 that for all i ∈ Z + . Let us here show that there exists a constant C 1 > 0 not depending on i such that To this end, let χ(t, r) ∈ C ∞ 0 (R 2 ) satisfy χ(t, r) = 1 if |t| 2 and |r| 1, χ(t, r) = 0 if |t| 4 or |r| 1/2, and 0 χ(t, r) 1. Set v(t, r) = χ(t − T, r/(2 k ρ)) u(t, |r|) for any ρ > 1, all T > 4 and k ∈ Z 0 . Since u is a radially symmetric function, by means of (6.3), we estimate for any ρ > 1, all T > 4 and k ∈ Z 0 . Together with the above, it follows from the Young inequality and Sobolev embedding that which yields (6.7). In view of (6.1), this allows us to exist ρ 0 > 0 such that for any i 4. We see from the above and (6.6) that δ for any i 4, which contradicts that for any i 1, Hence, the solution blows up at finite time. This completes the proof in N = 4, 5.

Application of the uniform boundedness of global solutions.
In the mass-subcritical or mass-critical case 2 N 4, we establish the following: is a global solution of (1.2) satisfying P ( u(t)) > 0 for any t 0, then Proof of Proposition 6.2. By the Young inequality ab εa p + ε − 1 p−1 b q with p −1 + q −1 = 1, it holds that Combining P ( u) > 0 with (6.2) and the conservation of the energy, we have Hereafter, we follow the argument by [20]. In the same way as in [25, Lemma 2.1], it turns out that Further, the Gagliardo-Nirenberg inequality gives us for j = 1, 2, where θ = 2N 3(10−N ) . Together with (6.11) and (6.12), by using the Young inequality again, for any l > 0, we establish 1 l P ( u) 1 2 L( u) + C 1 (6.13) for some constant C 1 depending on A and L. Here set b = min(1, κ). When b < 1, by the conservation of the energy, one obtains . we see from (6.13) that , ∂ t u(0)) + 2C 1 , which implies sup t 0 ( u, ∂ t u)(t) H 1 ×L 2 < ∞. The case m 1 is easier. Thus, the proof is completed.
Combining Proposition 6.2 with the Strauss decay estimate f L ∞ (|x| ρ) Cρ − N−1 2 f H 1 rad (6.14) for any ρ > 0 (see [31]), we establish an alternative proof of Theorem 1.4 in N 4 and Theorem 1.5 which is similar to that of Theorem 1 and Theorem 4 in [25], respectively. For self-containedness, we only give the proof of Theorem 1.4 in N = 2, 3.  (0)). Let us show by the contradiction. Namely, assume that there exists t 1 ∈ [0, ∞) such that d dt