WELL-POSEDNESS ISSUES FOR SOME CRITICAL COUPLED NON-LINEAR KLEIN-GORDON EQUATIONS

. The initial value problem for some coupled non-linear wave equations is investigated. In the defocusing case, global well-posedness and ill- posedness results are obtained. In the focusing sign, the existence of global and non global solutions are discussed via the potential-well theory. Finally, strong instability of standing waves are established.


1.
Introduction. This manuscript is concerned with the initial value problem for a coupled wave system with power-type nonlinearities a jk |u k | p |u j | p−2 u j = 0; u j (0, .) = ψ j , ∂ t u j (0, .) = φ j . :=E 0 (u(t), ∂ t u(t)) + µ 2p m j,k=1 where the kinetic energy defined by If µ = 1, the energy is always positive and the problem (1) is said to be defocusing, otherwise a control of the (H 1 ) m norm with the energy is no longer possible and (1) is focusing.

RADHIA GHANMI AND TAREK SAANOUNI
In order to motivate this note, let us present some related works. Concerning the non-linear coupled wave system, in [26], Segal proposed the following problem where α, β, g and h are non zero real constants. This system arises in quantum fields theory which describes the motion of charged mesons in an electromagnetic field. Later on, many authors treated this problem and several results concerning global existence and blow-up of solutions were established. Medeiros and Menzala [19] studied such a system and found the existence of weak solutions for a mixed problem in a bounded domain. This result was generalized by Miranda and Medeiros in [15,16] to a non-linear terms of the form |v| p+2 |u| p u. Decay of solutions to the system (2) was established in [5] by Ferreira and Menzala. The existence of global/nonglobal solutions for a non-linear coupled wave system was discussed by many authors [30,31,32]. Recently, a coupled non-linear Klein-Gordon equation with damping term was treated by Pişkin and Korpusov [12,23,24].
Before going further, let us recall some existing results about the Cauchy problem for the scalar wave equation which is a particular case of (1) for m = 1. The defocusing semilinear wave equation with power p > 1 reads (N LW ) p ∂ 2 t u − ∆u + u|u| p−1 = 0, u : R × R N → R. This problem has been widely investigated and there is a large literature dealing with the well-posedness theory of (N LW ) p in the scale of Sobolev spaces H s . For the global solvability in the energy space H 1 × L 2 , there are mainly three cases. In the subcritical case p < p c := N +2 N −2 , Ginibre and Velo [6] proved that problem (N LW ) p has a unique solution in the energy space. In the critical case p = p c , the global existence was first proved by Struwe in the radially symmetric case [27], then by Grillakis [7] in the general case and later on by Shatah-Struwe [29] in other dimensions. In the supercritical case p > p c , local well-posedness was recently solved by Kenig-Merle [10] for initial data in the homogeneous Sobolev spacesḢ sp ×Ḣ sp−1 with 1 < s p < 3 2 . Except for some partial results about weak ill-posedness [4,13,14], well-posedness in the energy space is still an open problem.
In two space dimensions, any polynomial nonlinearity is subcritical with respect to H 1 norm and the critical one is of exponential growth. Nakamura and Ozawa [20,21] showed the global well posedness in the defocusing case for small data. Atallah [3] treated the radial case, then Ibrahim, Majdoub and Masmoudi [8] gave a more clear critical threshold. Struwe [28] constructed global solutions for smooth data. Similar results without any smallness condition hold [17,18].
The purpose of this manuscript is two-fold. First, using some Strichartz type estimates, global well-posedness of (1) in the subcritical defocusing and instability in the energy supercritical case are obtained. Second, in the focusing case, using the potential well method, global and non global existence of solutions are discussed via the existence of ground state. Moreover, strong instability of standing waves is proved.
The rest of the paper is organized as follows. The next section contains the main results and some technical tools needed in the sequel. The third and fourth sections are devoted to prove well-posedness of (1). In section five, an ill-posedness result is proved. The section six, existence of critical ground state is established. The seventh section is devoted to discuss global and non global existence of solutions via the potential-well theory. The last section is devoted to obtaining strong instability of standing waves.
Define the product space where H 1 (R N ) is the usual Sobolev space endowed with the complete norm Denote the real numbers We mention that C (respectively C T ) will denote a constant (respectively a constant depending on T ) which may vary from line to line and if A and B are nonnegative real numbers, A B means that A ≤ CB. For 1 ≤ r ≤ ∞ and (s, T ) ∈ [1, ∞) × (0, ∞), we denote the Lebesgue space L r := L r (R N ) with the usual norm . r := . L r , . := . 2 and For simplicity, denote the usual Sobolev Space W s,p := W s,p (R N ) and H s := W s,2 . If X is an abstract space C T (X) := C([0, T ], X) stands for the set of continuous functions valued in X and X rd is the set of radial elements in X, moreover for an eventual solution to (1), we denote T * > 0 it's lifespan.

Main results and background.
In what follows, the main results and some estimates needed in the sequel are collected.
2.1. Main results. First, we deal with local well-posedness of the wave problem (1).
Remark 2.1. The unnatural condition p ≥ 2 seems to be technical and gives some restriction on the dimension.
In the critical case, global existence and scattering hold in the energy space, for small data. In the defocusing case, we obtain an ill-posedness result, precisely we prove that the flow map is discontinuous. Theorem 2.3. Assume that N ≥ 3 and p > p * . There exist a sequence ϕ k := (ϕ 1 k , . . . , ϕ m k ) in (Ḣ 1 ) (m) and a sequence (t k ) satisfying and such that any weak solution u k to (1) with initial data (ϕ k , 0), satisfies In particular, the Cauchy problem Now, we are interested on the focusing sign in (1). Let us recall few results about the existence of a ground state solution to the stationary problem associated to (1). Define for u := (u 1 , . . . , u m ) ∈ H, the action If α, β ∈ R, we call constraint Take the minimization problem Definition 2.2. We call a ground state any solution to −∆ψ j +δ * ψ j = m k=1 a jk |ψ k | p |ψ j | p−2 ψ j , 0 = Ψ := (ψ 1 , . . . , ψ m ) ∈ H, m δ * α,β = S δ * (Ψ).
In the critical case, the situation is as follows.
Remark 2.4. We don't prove that Ψ is a solution to (4) because uniqueness of such a solution is not clear in general. Despite, we will call Ψ as ground state.
The next sets are invariant under the flow of (1).
The last result concerns the instability by blow up for standing waves.
In the next subsection, we give some standard estimates needed in the paper.
The following Gagliardo-Nirenberg inequality will be useful.
Write some chain rule for fractional derivatives [11].
Let us give a classical absorption result.
1−θ , we conclude the proof by a continuity argument.
The following is a classical result about ordinary differential equations [2]. Lemma 2.8. Let F : R → R be a smooth function and consider the following ODË The ODE has a periodic non constant solution if and only if the function G : y → 2 F (x 0 ) − F (y) has two distinct simple zeros α and β with α ≤ x 0 ≤ β and G has no zero in the interval ]α, β[. The period is then given by In addition, x is decreasing on [0, T 4 ] and x( T 4 ) = 0.
We close this subsection with some result about ordinary differential equations.
Proof. Assume with contradiction, the existence of such a function. Then This is a Riccati inequality with blow up time T < 1 G(0) G (0) . This contradiction achieves the proof.
3. Local well-posedness. This section is devoted to prove Theorem 2.1. The proof contains three steps. First we prove the existence of a local solution to (1), second we show uniqueness and finally we establish global existence in subcritical case.
3.1. Local existence. We use a standard fixed point argument. For T > 0, we denote the space endowed with the complete norm where (q, r) is the admissible pair given by q = . Let M > 0 be a positive real number small enough to fix later and B T (M ) be the ball in E T with center zero and radius M. Let the function be given by With the mean value Theorem . . , ψ m )). We prove the existence of some small T, δ > 0 such that φ is a contraction of the ball B T (M ), with center zero and radius M. Take u, v ∈ E T (M ), applying Strichartz estimate (6), we get It remains to prove that φ maps B T (M ) into itself. Taking in the last inequality v = 0, yields for M := 2C E 0 (u 0 , u 1 ), Thanks to a Picard fixed point theorem, φ has a fixed point in E T . The existence is proved.

3.2.
Uniqueness. In what follows, we prove the uniqueness of solutions to the equation (1) in the energy space. Let T > 0 u, v two solutions of (1) in (C T (H 1 ) ∩ C 1 T (L 2 )) m and w := u − v. Then , ∂ t w j (0, .)) = (0, 0). Taking T > 0 small enough, with a continuity argument, we may assume that max j=1,...,m Taking account of the precedent computation, we have The following Lemma concludes the uniqueness proof.
a solution to (1). Then, there exist T δ > 0 such that Proof. Let T (t)(u 0 , u 1 ) to be the solution to the free problem associated to (1). By Strichartz estimate, we get for small T > 0, . The proof of this Lemma is achieved taking account of Lemma 2.7 via the fact that 3.3. Global existence in the subcritical case. We prove that the maximal solution of (1) is global in the subcritical defocusing case. The global existence is a consequence of energy conservation and previous calculations. Let u ∈ (C([0, T * ), H 1 )∩ C([0, T * ), L 2 )) (m) be the unique maximal solution of (1). By contradiction, suppose that T * < ∞. Consider for 0 < s < T * , the problem By the same arguments used in the local existence proof, we can find a real number τ > 0 and a solution v = (v 1 , . . . , v m ) to (P s ) on C [s, s + τ ], H). Using the conservation of energy we see that τ does not depend on s. Thus, letting s be close to T * such that T * < s + τ, this contradicts the maximality of T * and finishes the proof.   Proof. The proposition follows from a contraction mapping argument, we let the function φ(u) given by .

CNLW 613
Then, using the fractional chain rule 2.6, it follows that Similarly, using Strichartz estimate with the couple (s, ρ) = (1, ρ 1 ), it follows that For small a = 2δ, b > 0, we obtain With a classical Picard argument, there exist u ∈ X a,b a solution to (1). Finally, in order to prove that the solution belongs to the energy space, it is sufficient to apply Strichartz estimate via previous computations.
We are ready to prove Theorem 2.2.
Proof of Theorem 2.2. Thanks to Strichartz estimate, So, taking account of the previous Proposition, it suffices to prove that ξ(u, ∂ t u) remains small on the whole interval of existence of u. By conservation of energy and Sobolev's inequality, write Global existence in the defocusing case follows from the last inequality since In the focusing case, with conservation of the energy and Sobolev's inequality, we have So by Lemma 2.7, if ξ(u 0 , u 1 ) is sufficiently small, then ξ(u, ∂ t u) stays small.

RADHIA GHANMI AND TAREK SAANOUNI
Now, we prove scattering. Using classical arguments [6], it is sufficient to prove that u ∈ S(R). Arguing as previously, The proof is achieved taking account of Lemma 2.7.

5.
Ill-posedness. In this section we prove Theorem 2.3. The proof uses the finite speed of propagation and a quantitative study of the associated ODE. Construction of ϕ k . For k ≥ 1 and k := k 1− p p * > 0, consider the following sequence, ϕ k := (ϕ 1 k , 0, . . . , 0) such that k is continuous. By an easy computation, we have Thus, Then, Construction of t k . Consider the ordinary differential equation associated to (1).
Using Lemma 2.8, the previous ODE has a unique global periodic solution with period Because p > p * , Now we are in position to construct the sequence (t k ). Recall that by finite speed of propagation, any weak solution u k to (1) with data (ϕ k , 0) satisfies Hence .
Take t k = T k 4 , then φ 1 k (t k ) = 0 and for |x| < k k − t k , we have The proof is closed.
6. Existence of critical ground state. In this section we prove the existence of a ground state solution to (5) in the critical case. Precisely, we establish Proposition Let the real number Claim. d 0 α,β = m 0 α,β . Since K 0 α,β = 0 implies that S 0 = H 0 α,β , it follows that m 0 α,β ≥ d 0 α,β . Conversely, take 0 = φ ∈ H such that K 0 α,β (φ) < 0. Thus, when 0 < λ −→ 0, we get So, there exists λ ∈ (0, 1) satisfying K 0 α,β (λΨ) = 0 and The claim is proved. Because of the definitions of K 0 α,β and H 0 α,β , it is clear that m 0 α,β is independent of (α, β) and Taking the scaling λφ, yields Here, C * denotes the best constant of the Sobolev injection m j,k=1 7. Invariant sets and applications. This section is devoted to obtaining global and non global solutions to the system (1) as claimed in Theorem 2.4. Let us start with a classical result about stable sets under the flow of (1).
The next auxiliary result reads.
the maximal solution to (1). Then, there exist δ > 0 such that Proof. With contradiction. Assume that K 1,− 2 N (u(t n )) → 0 for some sequence of positive real numbers t n ∈ (0, T * ). The next contradiction closes the proof.
Let us prove the main result of this section.
The proof is complete using Lemma 2.2. 2. By Lemma 7.1 u(t) ∈ A + 1,1 for any t ∈ [0, T * ). So Thus, u is bounded in (Ḣ 1 ) (m) . Preciesly The proof of Theorem 2.5 is based on the following Lemmas.
Let us give an auxiliary result.