Klein-Gordon Equations on Modulation Spaces

and Applied Analysis 3 Theorem 6 (persistence of regularity). Let 1 ≤ p 1 , p 2 ≤ ∞, 0 ≤ s 1 , s 2 < ∞, and k ∈ N, and letu be a strongMs p2,1 solution to (1)with itsmaximal existence interval I. If u(t 0 ) ∈ M s1 p1,1 and u t (t 0 ) ∈ M s1−1 p1,1 for some t 0 ∈ I, then u is also a strong Ms p1,1 solution with the same maximal interval. Remark 7. Combining the above theorem with global result in M 2,1 × M −1 2,1 in [2], one can easily get the global wellposedness inM p,1 ×M s−1 p,1 with small initial data for p ∈ [1, 2] and s ≥ 0. Thus, it is not possible to develop a singularity which causes theMs p1 ,1 × M s1−1 p1 ,1 norm to blow up while theMs p2 ,1 × M s2−1 p2 ,1 norm remains bounded.We also see that the regularity is stable, because if a solution lies in C(I,Ms p2 ,1 ) and is not in Ms p1,1 × M s1−1 p1,1 at some initial time t 0 , it never belongs to M s1 p1 ,1 ×M s1−1 p1 ,1 at any later (or earlier) time. As an application (see also Lemmas 31 and 32), we have the following stronger version of blowup criterion. Corollary 8 (stronger blowup criterion). Let p ∈ [1,∞], k ∈ N, and a strong solution of Cauchy problem (1) blows up in a finite time Tmax < ∞ (Tmin > −∞) if and only if ‖u(t)‖ L k t ([t0 ,Tmax],M 0 ∞,1 ) = ∞(‖u(t)‖ L k t ([Tmin ,t0],M 0 ∞,1 ) = ∞) . (15) Remark 9. From another point of view, the above blowup criterion implies that ‖u(t)‖ M s p,1 cannot blow up too slowly when t tends to a finite blowup time T; that is, lim t→T ‖u(t)‖ M s p,1 |T − t| −1/k+ε = +∞ (16) for every ε > 0. We also obtain a scattering theorem for these equations provided a bounded Lr t (R,Ms p,1 ) norm. Theorem 10 (L t (R,Ms p,1 ) bounds imply scattering). Let u 0 ∈


Introduction
In this paper, we consider the Cauchy problem for the following nonlinear Klein-Gordon equation in the space R × R  = R  × R   : + ( − Δ)  +  () = 0,  ( 0 , ) =  0 ,   ( 0 , ) =  1 , where (, ) is a complex-valued function in  × R  for some time interval  containing  0 , the initial data ( 0 ,  1 ) lies in the product of modulation spaces   ,1 ×  −1 ,1 (1 ≤  ≤ ∞,  ≥ 0), and the nonlinear term () = ( +1 ) is any ( + 1)-time product of  and ,  ∈ N. To understand this research problem and its historical developments, the reader may see Ruzhansky et al. [1] for a brief survey of nonlinear evolution equations on the modulation spaces.Concerning the well-posedness of solution to the Schrödinger equation in the modulation space, readers can refer to [2,3].
We give some remarks about our results.The known study of the Klein-Gordan equations (or other dispersive equations) on modulation spaces must be based on the assumption that the nonlinear term () is a polynomial.This assumption is also necessary in this paper; in fact, this holds for any positive real constant .
We recall that   = /2 − 2/ is the critical index for (1).Up to now, we cannot solve (1) in   for the case that  <   (the sup-critical case).On the other hand, we notice that the modulation space  ,1 has low regularity property.More precisely, for sufficiently large , we have the following embedding: In other words, the modulation space  , has lower regularity than   for large .So, for large   (high dimension for instance), one can solve (1) in   ,1 which contains supcritical initial dates in   for  <   .
Compared with (,   ,1 ) (used in [4]), the space    (,   ,1 ) seems more suitable for applying continuity argument, which is the key point for obtaining the perturbation theorem, especially the long-time version.So we choose    (,   ,1 ) as our work space and establish the nonlinear estimate associated with this work space in Section 2.
In Section 3, we will establish the local theory.We first use the fixed point theorem to construct a local-in-time solution  ∈    (,   ,1 ) to (1).Then, we verify that such solution is a strong   ,1 solution in the sense that  ∈ (,   ,1 ) ∩  1 (,  −1 ,1 ) and is unique in the category of strong solution.Finally, we study the regularity of solutions and deduce a stronger blowup criterion which implies the high rate of blowup.We will develop the scattering results in Section 4. In Section 5, we establish a stability theory for (1) and obtain the continuous dependence as a corollary.
Denote the operator  = Id − Δ, and for any function .Using this notation, we define and the Klein-Gordon semigroup: We now state our main results.Unless otherwise specified, we assume that the letters ,  are integers such that 1/( + 2) + 1/( + 1) < 1/2 and (, ) is -admissible (Definition 25).First, we have the following local theorem.

Theorem 1 (local well-posedness). Let I be a compact time interval that contains 𝑡
for some 0 <  ≤  0 , where  0 is a small constant (depending only on n, k).Then, there exists a unique solution  ∈    (,   ,1 ) to (1).Moreover, u is a   ,1 strong solution to (1) in the sense that  ∈ (,   ,1 ) ∩  1 (,  −1 ,1 ), and one also has From Lemma 19, we can verify the condition (8) by choosing  sufficiently small.So this theorem already gives local existence for large ‖ 0 ‖   ,1 , ‖ 1 ‖  −1 ,1 data.On the other hand, by inequality (63), we have the following global result as an application.
More precisely, we have the following global wellposedness result which gives the decay rate of solutions.Corollary 3 (another form of global well-posedness for small fine data).Assume that ( 0 ,  1 ) ∈ for some small constant  > 0.Then, there exists a unique global solution : ∈  := { : sup In the proof of Theorem 1, uniqueness is an immediate conclusion by the fixed point theorem.But, in fact, we have the following stronger result.

Theorem 4 (unconditional uniqueness in 𝑀 𝑠
,1 ).Let I be a time interval containing  0 ,  ∈ [1, ∞],  ∈ N, and let , V ∈ (,   ,1 ) be two strong solutions to (1) in the sense of (45) with the same initial data By combining the above uniqueness result with the local theorem, one can define the maximal interval  of the strong solution; thus, we have the following standard blowup criterion.

Theorem 5 (blowup criterion). Let 𝑢
, and let  = ( min ,  max ) be the maximal interval.If  max < ∞ ( min > −∞), then one has The above blowup criterion will be improved soon as an application of Lemmas 31 and 32.For completeness, we also give a proof for this weak version.Then, we give a regularity result.
Theorem 6 (persistence of regularity).Let 1 ≤  1 ,  2 ≤ ∞, 0 ≤  1 ,  2 < ∞, and  ∈ N, and let be a strong solution with the same maximal interval.Remark 7. Combining the above theorem with global result in  2,1 ×  −1 2,1 in [2], one can easily get the global wellposedness in   ,1 ×  −1 ,1 with small initial data for  ∈ [1, 2] and  ≥ 0. Thus, it is not possible to develop a singularity which causes the norm remains bounded.We also see that the regularity is stable, because if a solution lies in (, Remark 9. From another point of view, the above blowup criterion implies that ‖()‖   ,1 cannot blow up too slowly when  tends to a finite blowup time ; that is, lim for every  > 0.
We also obtain a scattering theorem for these equations provided a bounded    (R, for some constant  > 0.Then, there exist as  → ±∞. Finally, we will discuss the stability theory.The stability theory for (1) means that given an approximate solution to (1), with  and  − ũ,   − ũ small in a suitable space, is it possible to show that the genuine solution  to (1) stays very close to ũ in some sense (for instance, in the   ,1 )?Note that the question of continuous dependence of the data corresponds to the case  = 0 and the uniqueness theory to the case  = 0, ( 0 ) = ũ( 0 ).We have the following shorttime perturbations and long-time perturbations.Theorem 11 (short-time perturbations).Let I be a compact time interval, and let ũ be an approximate solution to (1)  ) ≲  () , (33) As applications of the above stability theorems, we have the following corollaries.
Also, one can deduce continuous dependence for  ∈ [1, ∞],  ∈ N directly without using perturbation theorem, and the proof is not difficult, so we omit the details.

Preliminaries
If  and  are two quantities (typically nonnegative), we will often use the notation  ≲  to denote the statement that  ≤  for some absolute constant  > 0, where  can depend on , , , , but it might be different from line to line.Given  = ( 1 ,  2 , . . .,   ) ∈ Z  , we write The norm ‖‖  ∞  (R  ) is defined with the usual modification.We also abbreviate ‖ ⋅ ‖    (R  ) for ‖ ⋅ ‖   , or ‖ ⋅ ‖  , when there is no confusion.We use    (, ) to denote the space-time norm: with the usual modifications when , , or  is infinite.For the operator  = Id − Δ, the operator and the Klein-Gordon semigroup have been defined in Section 1.Thus, we may recall Duhamel's formula: Also, we recall the integral form of Gronwall's inequality.
Lemma 15 (Gronwall inequality and integral form [5]). Let  : [ 0 ,  1 ] → R + be continuous and nonnegative, and suppose that A obeys the integral inequality for all  ∈ [ 0 ,  1 ], where  ≥ 0 and  : [ 0 ,  1 ] → R + is continuous and nonnegative.Then, we have Let S := S(R  ) be the Schwartz space and S  := S  (R  ) the tempered distribution space.We introduce the definition of modulation space, which was introduced by Feichtinger [6] in 1983 by short-time Fourier transform.We will also display some basic properties of this function space.
Applying the frequency-uniform localization techniques, one can get an equivalent definition of modulation spaces (see [7] for details) as follows.Let   be the unit cube with the center at , so {  } ∈Z  constitutes a decomposition of R  .First, we construct a smooth cut-off function.Let  ∈ S(R  ) and let  : R  → [0, 1] be a smooth function satisfying We see that Then, {  ()} ∈Z  satisfies the following properties: In fact, {  ()} ∈Z  constitutes a smooth decomposition of R  and   () = ( − ), in which The frequency-uniform decomposition operators can be exactly defined by for  ∈ Z  .
Definition 16 (modulation space).Let  ∈ R, 0 < ,  ≤ ∞, and one defines the modulation space Below, we list some basic properties for the space   , (R  ).
where C is independent of   .
We establish the following nonlinear estimate.
For , by Lemma 19, we have Accordingly, for  ∈   ,1 , For the nonlinear part,             ∫ If (, ) ∈ (,   ,1 ), then we have ()(, ) ∈ (,  −1 ,1 ).In fact, taking advantage of (92) and the Lebesgue dominated convergence theorem, we can get Recall that () ∈ (,   ,1 ) and apply (92) and the Lebesgue dominated convergence theorem to the first term of (93); we have Abstract and Applied Analysis Consequently, Next, the proof of time continuity of   is similar to .It only needs to take care of the difference of smoothness and the action of the Bessel potential.Finally, we obtain  ∈ (,   ,1 )∩ 1 (,  −1 ,1 ); that is, it is the strong   ,1 solution.

Global
Well-Posedness for Small Fine Data.Let us construct a time decaying norm as This idea for the NLS can be traced back to the work of Strauss [12] and Wang and Hudzik [2].We consider the following mapping: in the metric space D = { : Γ() ≤ } with (, V) = Γ( − V).For any  ∈ D, in view of Lemma 20, we get that and (recall the notation that ⟨⟩ = 1 + ||) Owing to (1/2 − 1/)( + 1) > 1, it follows that and that Combining the above four inequalities, we have Hence, if we choose  ≲ /2 and  small enough, then using the standard contraction mapping argument the Cauchy problem (1) has a unique solution satisfying (13).
for  ∈ [ 0 , ] and obtain the conclusion immediately.
Secondly, we reduce another regularity index from  to 0. By the Leibniz rules for modulation space Lemma 22, we can deduce the following lemma.The proof of this lemma is very similar to the proof of Lemma 31, so we omit the details.
, they have the same maximal existence interval.

Scattering Theorems
The goal of this section is to derive scattering results.

Proof of Scattering
Theorem.Without loss of generality, we assume  0 = 0, and let By the fact we used in Section 3.2.1 that is from Definition 25 for -admissible pair (, ), there exists γ ≥ 1 such that Take advantage of the nonlinear estimate and Hölder inequality: Since ‖‖    (R,  ,1 ) ≤ , we have as ,  → ∞.This implies that V() is Cauchy in   ,1 as  → ∞.Denote V + 1 to be the limit: In a similar way, we obtain Recall that V + = ()V + 1 + (−)V + 2 ; so taking advantage of the nonlinear estimates, we get      () − V +    as  → +∞.So is V − , respectively.In fact, in our proof, we also have V + 1 ∈   ,1 .One may ask whether there exists a global   ,1 solution to Cauchy problem (1) which is scattering associated with The following remark gives us a partial answer.
To construct , we only need to solve the Cauchy problem from  = +∞ to some finite time  =  >  1 with initial data at  = +∞, that is, to solve the following equation:

Stability Theory
In this section, we will derive the stability theory including short-time result and long-time result and deduce some corollaries.
Lemma 32.Let  be a bounded time interval containing  0 ,  ≥ 0, and let  ∈ (,   ∞,1 ) be a strong   ∞,1 solution to (1). ) remains bounded.This implies that the blowup phenomenon is independent of exact  and , so if an initial data (( 0 ),