Cauchy problem for NLKG in modulation spaces with noninteger powers
aa r X i v : . [ m a t h . A P ] N ov CAUCHY PROBLEM FOR NLKG IN MODULATION SPACESWITH NONINTEGER POWERS
HUANG QIANG, FAN DASHAN, AND CHEN JIECHENG
Abstract.
In this paper, we consider the Cauchy problem for the nonlinearKlein-Gordon equation whose nonlinearity is | u | k u in the modulation space,where k is not an integer. Our method can be applied to other equations whosenonlinear parts have regularity estimates. We also study the global solutionwith small initial value for the Klein-Gordon-Hartree equation. By this we canshow some advantages of modulation spaces both in high and low regularitycases. Introduction and main results
In this paper, we study the Cauchy problem for the following nonlinearKlein-Gordon equation (NLKG): u tt + ( I − △ ) u = ±| u | k u, u (0) = u , u t (0) = u , (1 . k ∈ (0 , + ∞ ) \ Z + , u tt = ∂ /∂ t, and △ = ∂ /∂ x + ... + ∂ /∂ x n is the Laplace operator. It is well known that the NLKG has the following equivalentintegral form: u ( t ) = K ′ ( t ) u + K ( t ) u − Z t K ( t − τ ) | u | k udτ, (1 . ω = ( I − ∆) and K ( t ) = sin tω ω , K ′ ( t ) = cos tω . The aim of this paper is to study the local and global well posedness of NLKGin modulation spaces. The modulation space M sp,q was originally introduced byFeichtinger in [4] , where its definition is based on the short-time Fourier transformand the window function. Feichtinger’s initial motivation was to use the modulationspace to measure smoothness for some function or distribution spaces. Since then,this space was received an extensive study on its analysis/topological construc-ture and algebraic properties. See, for example, [6, 10, 11, 13] and the referencestherein for more details. Later, people found that this space is a good working Date : November 11, 2014.2000
Mathematics Subject Classification.
Key words and phrases.
Modulation spaces, Nonlinear Klein-Gordon equation, Cauchy prob-lem, noninteger power. frame to study certain Cauchy problems of nonlinear partial differential equations.To this end, Wang and Hudzik gave another equivalent definition in [22] usingthe frequency-uniform-decomposition operators. With this discrete definition, theywas able to consider the global solutions for nonlinear Schr¨odinger equation andnonlinear Klein-Gordon equation in the space M sp,q . Following their pioneer work,we can find a lot of research papers in the literature that address various harmonicanalysis and PDE problems on the modulation spaces. In the following, we lista few of these results, among numerious of papers. Gr¨obner in his PH.D. the-sis [7] introduced the α modulation spaces that reveals the essential connectionbetween the modulation spaces and the Besov spaces; Han and Wang[8] followedGr¨obner’s idea to give a discrete version of the α modulation space based onthe frequency-uniform-decomposition, and they obtained more properties relatedto this space; Feichtinger, Huang and Wang [5] studied the trace operator in modu-lation, α -modulation and Besov spaces. Also, for PDE problems, Wang, Zhao, andGuo [23] studied the local solution for nonlinear Schr¨odinger equation and Navier-Stokes equations; Wang and Huang [21] obtained the local and global solutionsfor generalized KdV equations, Benjamin-Ono and Schr¨odinger equations; TsukasaIwabuchi studied the local and global solutions for Navier-Stokes equations, as wellas the heat equations (see [9]). However, we observe that the nonlinear parts ofabove mentioned equations is either | u | k u with k ∈ Z + or a multi-linear function F ( u, .., u ). The reason of such a restriction is that all the estimates in [23], [21],[9] are based on the algebra property (cid:13)(cid:13) u (cid:13)(cid:13) M sp, (cid:22) k u k M sp, , that causes that the exponent k must be a positive integer in the nonlinear term | u | k u. In a recent article [16], Ruzhansky, Sugimoto and Wang stated some newprogresses on the modulation spaces. In the same article they posed three openquestions. One of these questions is to study nonlinear PDE whose nonlinear term | u | k u has a non-integer k ∈ (0 , + ∞ ).Motivated by the above question, in this paper, we will give a partial answerto the above problem. Before we attack the problem, let us briefly describe someobvious difficulties in handling this problem. Unlike the Lebesgue spaces or theBesov spaces, we do not have the Littlewood-Paley theory or any of its analog onthe modulation spaces. So we can only use its algebra property and local analysison its windows in the frequency spaces. However, both of these two tools handleonly the case of integer k . These bring the main difficulty in our problem. Toachieve our aim, fortunately we observe that modulation spaces and Besov spacescan be embedded each other. So, we can use this property to deduce the prob-lem on the modulation spaces to those on the Besov spaces and then transfer theobtained estimates back to the modulation spaces. Of course, during this embed-ding process, we might loss some regularity. So this method might be applied insome equations whose nonlinear parts have regularity estimates. These equationsinclude the Klein-Gordon equation, the heat equation and some other equationswith regularity estimates.Now we state our main theorems in the following. AUCHY PROBLEM FOR NLKG IN MODULATION SPACES WITH NONINTEGER POWERS3
Theorem 1.
Let s ∈ R , k ≥ [ s ] , < p < ∞ , and ≤ q < ∞ . Assume that q satisfies the following conditions: max { n − nq , n − k [1 − ( 1 q −
12 ) n ] } < s < n . when q < and n ( q − ) < ;and max { , nq ′ − k [1 − n ( 12 − q )] } < s < nq ′ (1 . when q > , n ( − q ) < .For any ( u , u ) ∈ M s ,q × , M s − ,q , there exists a T > such that the equation(1.1) has a unique solution in the space L ρ (0 , T ; M s − βp,q ) \ L ∞ (0 , T ; M s ,q ) , (1 . where ρ and β are any real numbers satisfying ρβ = n +1 n − and β ∈ (0 , n +12 n − ) . Since β > L ρ (0 , T ; M s − βp,q ) might be closed to the space L ∞ (0 , T ; M sp,q ) , which has more regularity and better integrality . Remark 1.
When consider local well-posedness of NLKG in Besov space B sp,q , thedomain of p, q is ≤ q ≤ ∞ and ( − p ) n ∈ [0 , . Comparing this to Theorem1, we can see that the domain of p in the modulation space is similar to thedomain of q in the Besov space, and q behaves in modulation spaces quite like theperformance of p in the Besov space. One can easily understand this nature, whenone compares the embedding relations between the modulation spaces (see 2.1) tothe Sobolev embedding in Besov spaces. In Theorem 1, we use an auxiliary space L ∞ (0 , T ; M s ,q ). The purpose is thatwe want to expand the domain of p, q by the Stricharz estimate. Actually, wecan also prove the unconditional local well posedness in L γ (0 , T ; M sp,q ) without theauxiliary space. As we all know, the unimodular semigroup e it | ∆ | α/ is boundedon the L p space or on the Besov space B sp,q if and only if p = 2 (or α = 1 at n = 1) . So if we want to obtain the local well posedness in B sp,q when p = 2, weshould use the Sobolev space H s as an auxiliary space. On the other hand, for theSobolev space H s alone, it is difficult to estimate nonlinear part when s < n , so italso needs the Besov space as an auxiliary space. But in modulation spaces, we areable to obtain the M sp,q − boundedness for the unimodular semigroup and we canalso estimate the nonlinear part in the low regularity case. As these advantages,the next corollary shows that we can obtain the following unconditional local wellposedness in modulation spaces. This feature is not available either on the Besovspaces, or on the Sobolev spaces, when one studies the problem in the low regularitycase. Corollary 1.
Let ≤ q < ∞ , ≤ p < ∞ , s ∈ R and k ≥ [ s ] . Assume that theysatisfy q ∈ [ p ′ , p ] , (1 − p ) n < and max { − n ( 1 q − p ) , nq ′ − k } < s < nq ′ − k (1 − p ) n. (1 . Then for any initial value ( u , u ) ∈ M sp,q × , M s − p,q , there exists a T > such thatthe equation (1.1) has an unique solution in L γ (0 , T ; M sp,q ) for any γ ≥ k + 1 . HUANG QIANG, FAN DASHAN, AND CHEN JIECHENG
Remark 2.
Throughout the proof, we will use the Stricharz estimate and a nonlin-ear estimate in Besov spaces. Both of these two tools requires < p < ∞ . So, forthe case ≤ p ≤ , we are not able to obtain the well-posedness in the Besov spaces.However, such a restriction can be removed when we study the same problem onthe modulation spaces. In the next theorem, we indeed obtain the solution in M sp,q when ≤ p ≤ . Recall that in the Besov spaces B sp,q , if we want to use Sobolevembedding to control the norm B sp,q by the norm B s p ,q with p < p , it needsmore regularity s > s . But in modulation spaces M sp,q , we have uniformly estimatefor the index p (see (2.1)) which has no influence to the regularity index s . Hencewe can use this embedding property and boundedness of unimodular semigroup tosolve the problem in the case ≤ p ≤ . Specifically, we have the following theorem: Theorem 2.
Let < p ≤ , ≤ q < ∞ s ∈ R and k ≥ [ s ] . Assume that theysatisfy that q ∈ [ p, p ′ ] , ( p − n < , and max { − n ( 1 q − p ′ ) , nq ′ − k } < s < nq ′ − k (1 − p ′ ) n. (1 . Then the same conclusion as Corollary 1 holds .
Remark 3.
In the case p = 2 , the condition q ∈ [ p ′ , p ] in the above theorem means q = 2 . In [22] , Wang and Hudzuk proved that the space M s , is equivalent to theSobolev space H s , so this result is not interesting. Actually, when p = 2 , it is notnecessary to choose q = 2 . The range of q for p = 2 can be wider. We will addressthis special case in Remark 6. Now we turn to state the global solution of NLKG in modulation spaces.
Theorem 3.
Let ≤ q < ∞ , ≤ p < ∞ s ∈ R and k ≥ [ s ] . Assume that theysatisfy q ∈ [ p ′ , p ] , (1 − p ) n < − α and max { − α − n ( 1 q − p ) , nq ′ − k (1 − α ) + α } < s < nq ′ − k (1 − p ) n + α . where α = θ ( n + 1)( 12 − p ) , δ = θ ( n − − p ) . (1 . for some θ ∈ [0 , . In addition, assume q ∈ [ γ ′ .γ ] , γ ≥ δ , and k > θ + n . Thenthere exists a small ν > such that for any k u k M sp,q + k u k M s − p,q ≤ ν , equation(1.1) has an unique global solution u ∈ L ∞ ( R ; M sp,q ) \ L k +2 ( R ; M s − α p,q ) . (1 . Remark 4.
When we choose θ = 1 in Theorem 3, we can find the global solutionwith small initial value for k > n . In [22] , Wang and Hudzik proved that if k isan integer, the global existence interval is k ≥ n which is wider then ours. That isbecause during the embedding between Modulation spaces and Besov spaces we lostsome regularity. Hence we need more power of u to guarantee more regularity tomake up this lost. By far, we find the local solution for 1 < p < ∞ and the global solution for2 < p < ∞ when k is not integer. In all cases, nonlinear estimate in modulationspaces relies heavily on the corresponding nonlinear estimate in Besov spaces andsome regularity is lost. However, we obtain two advantages in the modulation spaces AUCHY PROBLEM FOR NLKG IN MODULATION SPACES WITH NONINTEGER POWERS5 and they are quite unique comparing to the results on the Besov space. First, inCorollary 1, we obtain the unconditional local well posedness with low regularity.Second. in Theorem 2, we solve the problem in the case 1 < p <
2. Moreover, if thenonlinearity is a multi-linear function, we can find that the modulation spaces havemany other advantages. Below, we will use the nonlinear Klein-Gordon-Hartreeequation(NLKGH): u tt + ( I − △ ) u + ( | x | − µ ∗ u ) u = 0 , u (0) = u , u t (0) = u (1 . R × R , and compare the result with the same solution in Besov spaces obtainedin [12]. As we mentioned before, the role of the index q in the modulation spaceis significant. The regularity index s might depend on q in some estimates, forinstance see (2.2). In order to obtain a good time-space estimate, in [22] Wangand Hudzik gave up a traditional method of dual estimate. They introduced thespace l s,q (cid:3) ( L r (0 , T ; L p ( R n ))) (see Definition 1 ) to replace the standard modulationspace. Then, for q ∈ [ γ ′ .γ ], they were able to invoke the Minkowski inequalityto obtain some time-space estimates in L r (0 , T ; M sp,q ). Look back to Theorem 3,in the proof we need the embedding between modulation spaces and Besov spacesto solve the case k / ∈ Z , so we work on the space L r (0 , T ; M sp,q ) whose Stricharzestimate needs to restrict to q ∈ [ γ ′ .γ ]. But for the equation (1.11), we want touse the relation between s and q to extend the domain of µ . So we hope that therestrictions on q are as less as impossible. To achieve this target, we choose thespace l s,q (cid:3) ( L r ( R ; L p ( R n ))) to find the global solution and establish the followingresult. Theorem 4.
Suppose that ( u , u ) ∈ M s ,q × M s − ,q , where < q < ∞ , s ≥ , α and δ are defined in (1.9). Assume that the domain of p satisfies (1 − p ) ∈ [ 12 θ ( n − , θ ( n −
1) ) . For n (1 − p ) ≤ µ ≤ s + nq ) + 1 − α − n , we can find a constant ε > for whichif k u k M s ,q + k u k M s − ,q ≤ ε then, for equation (1.3), there exists a unique global u ∈ l s,q (cid:3) ( L ∞ ( R ; L p ( R n ))) \ l s − α ,q (cid:3) ( L ( R ; L p ( R n )) . (1 . Remark 5. In [12] , Miao and Zhang studied equation (1.11) on the Besov spaces.In the case n = 3 , they showed that the exponent µ must satisfy µ = 6( s + 1)3 + η and µ ≥
62 + η where η ∈ [0 , . These requirements imply s ≥ η . So the minimum regularityshould be s = when we choose η = 1 . But on the modulation space in Theorem1.9, if we let θ = 1 , we can choose p such that n (1 − p ) = when n = 3 . Then,if we choose q closed to 1, it is not difficult to find that the minimum value of s can HUANG QIANG, FAN DASHAN, AND CHEN JIECHENG be . We observe that in [22] , Wang and Hudzik proved that M p,q has no derivativeregularity for any < p, q ≤ ∞ . Hence, our result is another form of low regularityfor global solution of NLKGH. Second, If one wants to obtain some high regularityestimate for this equation, in [12] the domain of µ is ( s + 1) ≤ µ ≤ s + 1) whenthe authors take the Besov space as the working space. Checking the domain of µ on modulation spaces in Theorem 4, clearly it is larger, since the low bound is fixedwhich is independent on s . So, both in low regularity and high regularity cases,modulation spaces seems better than Besov spaces. We are not surprising that the modulation space have these advantages compar-ing to the Besov space. In the Besov spaces, many estimates, such as the admissiblepairs, H¨older’s inequality, boundedness of fractional integral operator, the Sobolevembedding, etc., rely all on the exponent p , while the index q is dummy. But in themodulation spaces, the admissible pair relies on p , the Sobolev embedding relies on q (see (2.2)), H¨older’s inequality and boundedness of fractional integral operatorrely on both p and q . Moreover, we have the uniformly estimate for the index p (see (2.1)). In other words, working in the Besov spaces, one needs to give toomuch restrictions of p , and q plays no role. In the modulation spaces, p and q sharethese restrictions together, and both p and q have uniformly estimates.The proofs of theorems will be represented in the third section.2. Preliminaries
In this section we recall the definitions and some properties of the modulationspace and Besov space. Also, we will prove several lemmas, particularly a key lemmato estimate | u | k u in the modulation space when k is not an integer. Definition 1. (Modulation spaces) Let { ϕ k } ⊂ C ∞ ( R n ) be a partition of the unitysatisfying the following conditions: suppϕ ⊂ { ξ ∈ R n | | ξ |≤ √ n } , X k ∈ Z n ϕ k ( ξ ) = 1 , for any ξ ∈ R n , where ϕ k ( ξ ) := ϕ ( ξ − k ) . For each k ∈ Z n , denote a local squareprojection (cid:3) k on the frequency space by (cid:3) k := F − ϕ k F , where F and F − denote the Fourier transform and its inverse, respectively. Bythis frequency-uniform decomposition operator, we define two kinds of modulationspaces, for < p, q ≤ ∞ and s ∈ R , by M sp,q ( R n ) := { f ∈ S ′ ( R n ) : k f k M sp,q ( R n ) = ( X k ∈ Z n h k i sq k (cid:3) k f k qp ) q < ∞} and l s,q (cid:3) ( L r (0 , T ; L p ( R n ))) := { f ( t, · ) ∈ S ′ ( R n ) : k f k l s,q (cid:3) ( L r (0 ,T ; L p ( R n )) ) < ∞} , where k f k l s,q (cid:3) ( L r (0 ,T ; L p ( R n ))) = ( X k ∈ Z n h k i sq k (cid:3) k f k qL r (0 ,T ; L p ( R n )) ) q and h k i := (1 + | k | ) (See [22] for details). If the domain of t is ( −∞ , + ∞ ) ,we donate l s,q (cid:3) ( L r L p ) for convenience. The space l s,q (cid:3) ( L r (0 , T ; L p ( R n ))) was firstintroduced by Planchon [14] , [15] when he studied the nonlinear Schr¨odinger equation AUCHY PROBLEM FOR NLKG IN MODULATION SPACES WITH NONINTEGER POWERS7 and the nonlinear wave equation. In the definition, the order of L r norm and l q norm is changed. This change seems important in modulation spaces. As we know, q is a very important index in modulation spaces which can impact the regularity.So, in many cases, we should deal with q carefully and choose l q norm in the laststep. Moreover, we will recall some properties of modulation spaces which will beuseful in this paper. More details can be found in [22] . In the following content, if no special explanation, we always assume that s, s i ∈ R, ≤ p, p i , q, q i ≤ ∞ . Proposition 1. (Isomorphism [22] ).Let < p, q ≤ ∞ , s, σ ∈ R . For the Bessel potential J σ = ( I − △ ) σ , the mappings J σ : M sp,q → M s − σp,q and J σ : l s,q (cid:3) ( L r (0 , T ; L p ( R n ))) → l s − σ,q (cid:3) ( L r (0 , T ; L p ( R n ))) are isomorphic mappings. Proposition 2. (Embedding, [22] ). M s p ,q ⊂ M s p ,q and l s ,q (cid:3) ( L r (0 , T ; L p )) ⊂ l s ,q (cid:3) ( L r (0 , T ; L p )) if .(i) s ≥ s , < p ≤ p , < q ≤ q (2.1)(ii) q > q , s > s , s − s > n/q − n/q (2.2) In (2.1), we can see that both p and q have uniform estimates, and from (2.2)we can find that the condition on q is similar to the Sobolev embedding. Now weneed the relationship between modulation spaces and Besov spaces. Lemma 1. (Embedding with Besov spaces, [22] )Assume B sp,q is the Besov spaces, and ≤ p, q ≤ ∞ , s ∈ R . We have the followingembedding: M s + σ ( p,q ) p,q ⊂ B sp,q , σ ( p, q ) = max (0 , n ( 1 p ∧ p ′ − q )) (2 . B s + τ ( p,q ) p,q ⊂ M sp,q , τ ( p, q ) = max (0 , n ( 1 q − p ∨ p ′ )) . (2 . u k . Recall that such estimate in Besov spaces B sp,q has a longhistory. Cazenave obtained the case 0 < s < < s < N in [3]. Later, Wang proved a general case in [19]. Ourproof will be based on Wang’s result. Since all results on the Besov space B sp,q arestated for the case q = 2, we need the following embedding to obtain informationfor all 1 ≤ q < ∞ , as we will handle all q in the space M sp,q . Proposition 3. ( [18] ) Let ǫ > , for any ≤ p, q , q ≤ ∞ , then we have B s + ǫp,q ⊂ B sp,q (2 . HUANG QIANG, FAN DASHAN, AND CHEN JIECHENG
Lemma 2. (Nonlinear estimate in Besov space)Suppose ≤ p < ∞ , ≤ q ≤ ∞ and ≤ δ < s < s < ∞ , [ s − δ ] ≤ k If they satisfy k ( 1 p − sn ) + 1 p − δn = 1 p ′ , p − sn > . then we have k| u | k u k B s − δp ′ ,q (cid:22) k u k k +1 B s p,q (2 . Proof:
When q = 2, in [20] we can find the following inequality in the givencondition: k| u | k u k B s − δp ′ , (cid:22) k u k k +1 B sp, . (2 . s is an open set, we may choose s ǫ and δ ǫ so that s < s ǫ < s and δ ǫ < δ, and require them satisfy (2.5) and (2.6). So (2.7) gives the inequality k| u | k u k B sǫ − δǫp ′ , (cid:22) k u k k +1 B sǫp, . Finally, by Proposition 3, we obtain the desired estimate k| u | k u k B s − δp ′ ,q (cid:22) k| u | k u k B sǫ − δǫp ′ , (cid:22) k u k k +1 B sǫp, (cid:22) k u k k +1 B s p,q . Now, with Lemma 1 we can embed the modulation space into the Besov spaceand invoke Lemma 2 to obtain the nonlinear estimates on the Besov spaces. Thenuse Proportion 3 to transfer the estimate back to the modulation spaces. This isthe following lemma, which is crucial in this paper.
Lemma 3. (Nonlinear estimate in modulation spaces)Let ≤ q < ∞ , ≤ p < ∞ , s ∈ R [ s − r ] ≤ k . Assume that q ∈ [ p ′ , p ] , (1 − p ) n < r ,and max { r − n ( 1 q − p ) , nq ′ − rk } < s < nq ′ − k (1 − p ) n. (2 . Then we have k u k +1 k M s − rp ′ ,q (cid:22) k u k k +1 M sp,q . (2 . Proof:
By Lemma 1 we have k u k +1 k M s − rp ′ ,q (cid:22) k u k +1 k B s − r + τ ( p ′ ,q ) p ′ ,q . (2 . r − n ( q − p ) < s , we have s − r + τ ( p ′ , q ) >
0. Using Lemma 2, we obtain k u k +1 k B s − r +( 1 q − p ) np ′ ,q (cid:22) k u k k +1 B s − ( 1 p ′ − q ) np,q . (2 . s = s + ε in (2.6), then s satisfies k ( 1 p − n ( s − ( 1 p ′ − q ) n ) + 1 p − n ( r − n (1 − p ) − ε ) = 1 p ′ . (2 . s = nq ′ − rk + εk (2 . τ ( p ′ , q )+ σ ( p, q ) = n (1 − p ) < r , where it is easy to find 0 < ε < r − n (1 − p ).Combining this with (2.14) and the condition of Lemma 2, we easily see that thedomain of s is max { r − n ( 1 q − p ) , nq ′ − rk } < s < nq ′ − nk (1 − p ) . AUCHY PROBLEM FOR NLKG IN MODULATION SPACES WITH NONINTEGER POWERS9
Finally, we use Lemma 1 again to obtain k u k +1 k B s − r +( 1 q − p ) np ′ ,q (cid:22) k u k k +1 B s − ( 1 p ′ − q ) np,q (cid:22) k u k k +1 M sp,q . The lemma is proved.
Remark 6.
The condition q ∈ [ p ′ , p ] is not necessary, but only for continence inthe calculation. So it does not mean that q must be equal to when p = 2 . Infact, we can find a larger domain of q when p = 2 . More precisely, with the samemethod as above, we may obtain the estimate k u k +1 k M s − r ,q (cid:22) k u k k +1 M s ,q , for q < and max { r + n − nq , n − k [ r − ( 1 q −
12 ) n ] } < s < n , n ( 1 q −
12 ) < r (2 . or for q > and max { r, nq ′ − k [ r − n ( 12 − q )] } < s < nq ′ , n ( 12 − q ) < r. (2 . Remark 7.
For ≤ p < , if we switch p and p ′ in the condition of Lemma 3, thesimilar conclusion will be obtained, that is k| u | k u k B s − δp,q (cid:22) k u k k +1 B s p ′ ,q . (2 . k / ∈ Z . When k ∈ Z + , wecan find the following result in [9] , which will be useful in the proof of Theorem 4 Lemma 4.
Let s ≥ , ≤ p, q, p i , q i ≤ ∞ ( i = 1 , , , satisfy p = 1 p + 1 p = 1 p + 1 p , q + 1 = 1 q + 1 q = 1 q + 1 q , (2 . We have k uv k M sp,q (cid:22) k u k M sp ,q k v k M p ,q + k u k M p ,q k v k M sp ,q (2 . This conclusion also holds for l s,q (cid:3) ( L r L p ) . That is, for ≤ r, r i ≤ ∞ ( i =1 , , , satisfying r = 1 r + 1 r = 1 r + 1 r , we have k uv k l s,q (cid:3) ( L r L p ) (cid:22) k u k l s,q (cid:3) ( L r L p ) k v k l ,q (cid:3) ( L r L p ) + k u k l ,q (cid:3) ( L r L p ) k v k l s,q (cid:3) ( L r L p ) (2 . I α ( f )( x ) = Z R n f ( x − y ) | y | n − α dy. Lemma 5. (Boundedness of fractional integral operator in modulation space) [17]
Let < α < n and < p , p , q , q < ∞ . The fractional integral operator I α isbounded from M sp ,q ( R n ) to M sp ,q ( R n ) or from l s,q (cid:3) ( L r L p ) to l s,q (cid:3) ( L r L p ) ifand only if p ≤ p − αn and q ≤ q + αn . (2 . Proof of the main Theorems
Before we present the proofs, we need to state the Stricharz estimates of NLKGin modulation spaces. This estimate on the modulation spaces was obtained in [22].
Lemma 6. (Strichart estimate of NLKG in modulation spaces [22] ).Let ≤ p < ∞ , ≤ q < ∞ , γ ≥ ∨ (2 /δ ) , where α and δ are defined in (1.9). Wehave following the estimates: k K ′ ( t ) f k M − αp,q (cid:22) (1 + t ) − δ k f k M p ′ ,q , (3 . k K ′ ( t ) f k l − α/ ,q (cid:3) ( L γ ( R,L p )) (cid:22) k f k M ,q , (3 . k Z t K ( t − τ ) f dτ k l − α/ ,q (cid:3) ( L γ ( R,L p )) (cid:22) k f k l − ,q (cid:3) ( L ( R,L )) , (3 . k Z t K ( t − τ ) f dτ k l − α/ ,q (cid:3) ( L γ ( R,L p )) (cid:22) k f k l α/ − ,q (cid:3) ( L γ ′ ( R,L p ′ )) . (3 . In addition, if q ∈ [ γ, γ ′ ] , then we have k K ′ ( t ) f k L γ ( R,M − α/ p,q ) (cid:22) k f k M ,q , (3 . k Z t K ( t − τ ) f dτ k L γ ( R,M − α/ p,q ) (cid:22) k f k L ( R,M − ,q ) , (3 . k Z t K ( t − τ ) f dτ k L γ ( R,M − α/ p,q ) (cid:22) k f k L γ ′ ( R,M α/ − p,q ) , (3 . K and K ′ on the modulationspaces. Lemma 7. [1] . Let ≤ p, q < ∞ , s ∈ R . We have the following inequalities: k K ′ ( t ) f k M sp,q (cid:22) (1 + t ) n | − p | k f k M sp,q , (3 . k K ( t ) f k M sp,q (cid:22) (1 + t ) n | − p | k f k M s − p,q , (3 . where K ( t ) = sin t ( I − ∆) ( I − ∆) , K ′ ( t ) = cos( I − ∆) . Proof of Theorem 1.
Let δ, α and θ be defined in (1.9). Consider themapping Φ : u → K ′ ( t ) u + K ( t ) u − Z t K ( t − τ ) | u | k udτ on the Banach space X = L ∞ (0 , T ; M s ,q ) \ L ρ (0 , T ; M s − β ,q ) (3 . AUCHY PROBLEM FOR NLKG IN MODULATION SPACES WITH NONINTEGER POWERS11 where ρ = δ and β = α . For all 2 < p ≤ ∞ , we can choose θ such that δ ( p ) < k Φ ( u ) k X (cid:22) k u k M sp,q + k u k M s − p,q + k| u | k u k L (0 ,T ; M s − ,q ) . (3 . r = 1 in Remark 6 and using the H¨older inequality, we obtain that thenonlinear term k| u | k u k L (0 ,T ; M s − ,q ) (cid:22) k u k k +1 L k +1 (0 ,T ; M s ,q ) (cid:22) T k u k k +1 L ∞ (0 ,T ; M s ,q ) . (3 . k Φ ( u ) k X (cid:22) k u k M sp,q + k u k M s − p,q + T k| u | k u k X . (3 . k Φ ( u ) − Φ ( v ) k X (cid:22) T (cid:16) k u k kX + k v k kX (cid:17) k u − v k X Denote B M = { u ∈ X : k u k X ≤ M } . We choose
M, T > B M → B M is an onto mapping and k Φ ( u ) − Φ ( v ) k X ≤ k u − v k X . Thus, we complete the proof of theorem by the standard method of contractionmapping.In the proofs of Corollary 1 and Theorem 2, we can not use the Stricharz estimatein the space X = L γ (0 , T ; M sp,q ) . Hence we will invoke the boundedness of Klein-Gordon semigroup in modulationspaces to estimate the linear part.In the proof of corollary 1, we first choose θ = 0 in (1.9) such that α = δ = 0 in(3.1). So we obtain k K ( t ) f k M sp,q (cid:22) k f k M sp ′ ,q . Now by Lemma 7, we obtain k Φ ( u ) k X (cid:22) (1 + T ) n | − p | ( k u k M sp,q + k u k M s − p,q ) + k Z t | u | k udτ k L γ (0 ,T ; M s − p ′ ,q ) . Choosing r = 1 in Lemma 3 and using H¨older’s inequality, we have k Φ ( u ) k X (cid:22) (1 + T ) n | − p | ( k u k M sp,q + k u k M s − p,q ) + T − kγ k u k k +1 X . (3 . k Φ( u ) k X (cid:22) k t ( − p ) n k L γ (0 ,T ) ( k u k M sp,q + k u k M s − p,q ) + k Z t K ( t − τ ) | u | k udτ k X (cid:22) ( T γ + T ( − p ) n + γ )( k u k M sp,q + k u k M s − p,q ) + k Z t K ( t − τ ) | u | k udτ k X . By choosing r = 1 in Remark 6 and Remark 7, we use H¨older’s inequality to obtainthe following estimate for the nonlinear term: k Z t K ( t − τ ) | u | k udτ k X (cid:22) k Z t [1 + ( t − τ )] n ( p − ) k u k +1 k M s − p,q dτ k L γ (0 ,T ) (cid:22) k Z t [1 + ( t − τ )] n ( p − ) k u k k +1 M sp,q dτ k L γ (0 ,T ) (cid:22) k u k k +1 L γ (0 ,T ; M sp,q ) · k t γ − k − γ (1 + t n ( p − ) ) k L γ (0 ,T ) (cid:22) (1 + T n ( p − ) ) T γ − kγ k u k k +1 X . If we first assume
T <
1, from the above estimates we obtain that k Φ( u ) k X ≤ C T [( k u k M sp,q + k u k M s − p,q )] + k u k k +1 X . (3 . k Φ( u ) − Φ( v ) k X ≤ C T ( k u k kX + k v k kX ) k u − v k X , (3 . C T → , as T → . Now, by (3.15) and (3.16), the contractionmapping yields the conclusion of Theorem 2.
Proof of Theorem 3.
We denote the space X = L ∞ ( R ; M s ,q ) \ L k +2 ( R ; M s − α p,q ) , where α and δ are defined in (1.9), and k + 2 ≥ ∨ ( δ ). Since δ < , q ∈ [ p ′ , p ],we can choose q such that ( k + 2) ′ ≤ q ≤ δ . So we may assume k + 2 ≥ δ forconvenience. By Lemma 6, we have k Φ( u ) k X (cid:22) k u k M s ,q + k u k M s − ,q + k Z t K ( t − τ ) | u | k udτ k L k +2 ( R ; M s − α p,q ) . The last term above can be estimated in the following by using Lemma 3 andchoosing r = 1 − α , s = s − α in the lemma. An easy computation gives k Z t K ( t − τ ) | u | k udτ k L k +2 ( R ; M s − α p,q ) (cid:22) k u k +1 k L k +2 k +1 ( R ; M s + α − p ′ ,q ) (cid:22) k u k k +1 L k +2 ( R ; M s − α p ′ ,q ) . So, we have k Φ( u ) k X (cid:22) k u k M s ,q + k u k M s − ,q + k u k k +1 X . By the standard method used in the proofs of Theorem 1 and Theorem 2, we canobtain the existence and uniqueness of global solution if the initial value k u k M s ,q + k u k M s − ,q is small enough.Finally, we find the domain of k . We notice that when we use Lemma 3, theindex p should satisfy n (1 − p ) < − α. This gives 2 n ( 12 − p ) < − ( n + 1) θ ( 12 − p ) AUCHY PROBLEM FOR NLKG IN MODULATION SPACES WITH NONINTEGER POWERS13 by a simply calculation. So we find that the domain of p is ( − p ) > n +( n +1) θ .The condition k + 2 ≥ δ generates k ≥ δ − > θ + 2 n . Proof of Theorem 4.
Checking the above proof, we know that the globalexistence theory for small initial data is a straightforward result of the nonlinearestimate. Thus the main issue is to obtain an estimate for the nonlinear part. Weuse Y to denote the space Y = l s,q (cid:3) ( L ∞ ( R ; L p ( R n ))) \ l s − α ,q (cid:3) ( L ( R ; L p ( R n )) . By Lemma 4, we obtain the following estimate for the nonlinear part: k ( | x | − µ ∗ | u | ) u k l s + α − ,q (cid:3) ( L / T L p ′ ) (cid:22) k| x | − µ ∗ | u | k l s + α − ,q (cid:3) ( L T L p ) k u k l ,q (cid:3) ( L T L p ) + k| x | − µ ∗ | u | k l ,q (cid:3) ( L T L p ) k u k l s + α − ,q (cid:3) ( L T L p ) = I + II.
We will only estimate term I , since the second term II can be estimated in thesame way. Using Proposition 2, Lemma 4 and Lemma 5, we have I (cid:22) k u k l s + α − ,q (cid:3) ( L T L p ) k u k l ,q (cid:3) ( L T L p ) (cid:22) k u k l s + α − ,q (cid:3) ( L T L p ) k u k l ,q (cid:3) ( L T L p ) k u k l ,q (cid:3) ( L T L p ) (cid:22) k u k l s − α ,q (cid:3) ( L T L p ) . Similarly, we can obtain II (cid:22) k u k l s − α ,q (cid:3) ( L T L p ) . Therefore, we have k ( | x | − µ ∗ | u | u ) k l s + α − ,q (cid:3) ( L / T L p ′ ) (cid:22) k u k l s − α ,q (cid:3) ( L T L p ) (cid:22) k u k Y . Again, by the standard method of contraction mapping, we prove the conclusion ofthe theorem.Finally we check the range of µ. In the above proof, we notice that the conditionsin the estimate of I (the same condition in the estimate of II ) imply that p shouldsatisfy 1 p + 1 p = 1 p ′ , p + 1 p = 1 p , p ≤ p − n − µn , and p , p , p ≤ p. These clearly yield that 2 n (1 − p ) ≤ µ. Also, we note that q should satisfy1 q + 1 q = 1 q + 1 , q + 1 q = 1 q + 1 , q ≤ q + n − µn ,nq , nq < s − α nq , and s + α − nq < s − α nq . Thus, a direction computation gives that µ ≤ s + 2 nq + 1 − α − n. So, the domain of µ is2 n (1 − p ) ≤ µ ≤ s + 2 nq + 1 − α − n. To compare the low bound of µ to that in [12] when n = 3 , we choose θ = 1in (1.9). The condition 4 ≥ δ means that 4(1 − p ) ≥ . Since n = 3, the value2 n (1 − p ) is at least . References [1] A.B´enyi, K.Gr¨ocheing, K.A. Okoudjou, L.G.Rogres,
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Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China
E-mail address , Huang Qiang: [email protected] (Fan Dashan)
Department of Mathematics, University of Wisconsin-Milwaukee, Mil-waukee, WI 53201, USA
E-mail address , Fan Dashan: [email protected] (Chen Jiecheng)
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China
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