ANALYTIC SMOOTHING EFFECT FOR THE CUBIC HYPERBOLIC SCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS

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1 Electronic Journal of Differential Equations, Vol ), No. 34, pp ISSN: URL: ttp://ejde.mat.txstate.edu or ttp://ejde.mat.unt.edu ftp ejde.mat.txstate.edu ANALYTIC SMOOTHING EFFECT FOR THE CUBIC HYPERBOLIC SCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS GAKU HOSHINO, TOHRU OZAWA Abstract. We study te Caucy problem for te cubic yperbolic Scrödinger equation in two space dimensions. We prove existence of analytic global solutions for sufficiently small and exponential decaying data. Te metod of proof depends on te generalized Leibniz rule for te generator of pseudo-conformal transform acting on pseudo-conformally invariant nonlinearity. 1. Introduction We study te Caucy problem for te yperbolic Scrödinger equations in two space dimensions i t u + u = λ u 2 u, t, x) R R 2, 1.1) were i = 1, u : R R 2 t, x) ut, x) C, t = / t, = 1 2 2, 2 j = / x j, x = x 1, x 2 ), and λ C. Te two dimensional cubic yperbolic Scrödinger equation describes te gravity waves on liquid surface and ion-cyclotron waves in plasma see for instance [1, 29, 30] and references terein). We refer te reader to [4, 20, 21, 33] and reference terein for recent study on te Caucy problem for non elliptic nonlinear Scrödinger equations. Especially, analyticity of solutions to non elliptic nonlinear Scrödinger equations is studied in [4]. Space-time analytic smooting effect for te local solutions to te nonlinear Scrödinger equations in n space dimensions is studied in [6, 25]. Space-time analyticity is caracterized by te Galilei generator Jt) = x + it and te pseudoconformal generator Kt) = x 2 +nit+2itt t +x ). Since te following equality olds [7]) i t + 1 ) 2 K = K + 4it) i t + 1 ) 2, te following inequality plays an important role in construct analytic solutions a l K + µt) l f; L p 0, T ; X) 1 a l C l! 1 abt l! Kl f; L p 0, T ; X), l 0 l Matematics Subject Classification. 35Q55. Key words and prases. Nonlinear Scrödinger equation; non elliptic Scrödinger equation; analytic smooting effect; global solution. c 2016 Texas State University. Submitted April 4, Publised January 25,

2 2 G. HOSHINO, T. OZAWA EJDE-2016/34 were µ C, abt < 1, a, T > 0, b > 2, 1 p and X is an appropriate Banac space of functions on R n. In [14, 15, 16], we prove te space-time analytic smooting effect for te global solutions to te nonlinear Scrödinger equations wit sufficiently small data by using te Leibniz rule for te pseudo-conformal generator suc as: ) K + 4it u 4/n u = ) u 4/n Ku 2 n n u2 u 4/n 2 Ku. Te yperbolic Galilei transform G v and te pseudo-conformal transform P θ are defined by G v u)t, x) = e iv x itv2 1 v2 2 ) ut, x 1 + 2v 1 t, x 2 2v 2 t), v R 2, P θ u)t, x) = 1 θt) 1 e i θ 41 θt) x2 1 x2 2 ) t u 1 θt, x 1 1 θt, x ) 2, θ R, 1 θt respectively. We see tat 1.1) is invariant under te transforms G v and P θ see [29, 30]). In tis paper, we consider te analyticity in bot space-time variables of solutions to 1.1). We prove te Leibniz rule for te pseudo-conformal generator K t) = x 2 1 x itt t + x ) + 4it olds even for 1.1) see Lemma 2.3 below). For stating our main result precisely, we introduce te following notation. L p denotes te usual Lebesgue space L p R 2 ), 1 p. Te Fourier transform F is defined by F[ϕ]ξ) = 2π) 1 R 2 e iξ x ϕx)dx and F 1 is its inverse. We denote te linear part of 1.1) by L = i t +. Te free propagator of yperbolic Scrödinger equation is defined by U t) = e it, t R. We use te notation suc as U φ ) t) = U t)φ. We put U 1 t) = e it 2 1, U 2 t) = e it 2 2. Ten U t) = U 1 t) I)I U 2 t)) = I U 2 t))u 1 t) I), were I is te identity in L 2 R) and denotes te tensor product. Te relations are abbreviated as U t) = U 1 t)u 2 t) = U 2 t)u 1 t). Te Galilei generators are defined by J t) = J 1 t), J 2 t)) = x 1, x 2 ) + 2it 1, 2 ) = U t)x 1, x 2 )U t), t R. For t 0, we put M t) = e i x 2 1 x2 2 4t and we use te notation suc as M 1 )t) = M t). For t 0, J is represented as: J t) = M t)2it 1, 2 )M 1 t). According to [8, 13, 22], we define A δ t) = U t)e δ x U t), t R, δ R 2, were δ x = δ 1 x 1 + δ 2 x 2. For t 0, A δ is represented as: A δ t) = M t)e 2itδ 1, 2) M 1 t), were e 2itδ 1, 2) = F 1 e 2tδ ξ1, ξ2) F. We define te generator of dilations by We define P t) = t t + x, t R. K t) = x 2 1 x itP t), t R.

3 EJDE-2016/34 ANALYTIC SMOOTHING EFFECT 3 For t 0, K is represented as: K t) = 4itM t)p t)m 1 t). We define te pseudo-conformal generator by K t) = K t) + 4it = U t) x 2 1 x it 2 t ) U t), t R. We introduce te following basic function space: wit te norm X = L R; L 2 ) L 4 R; L 4 ), u; X = u; L R; L 2 ) + u; L 4 R; L 4 ). Let D R n, a > 0 and w be a real-valued function on R 2. We define te following function spaces: G D x; L 2 ) φ L 2 ; φ; G D x; L 2 ) <, φ; G D x; L 2 ) sup e δ x φ; L 2, G D J ; X ) u X ; u; G D J ; X ) <, u; G D J ; X ) sup A δ u; X, G D,a x, w; L 2 ) φ G D x; L 2 ); φ; G D,a x, w; L 2 ) <, φ; G D,a x, w; L 2 ) l 0 G D,a J, K ; X ) a l l! wl φ; G D x; L 2 ), u G D J; X ); u; G D,a J, K ; X ) <, u; G D,a J, K ; X ) l 0 a l l! Kl u; G D J ; X ). For any r > 0 and any Banac space X, we put B r X ) = u X ; u; X r. We consider te following integral equation associated wit te Caucy problem 1.1) wit data φ: u = U φ iλf u 2 u ) were F f ) t) = t 0 U t s)fs)ds, t R. Since te free propagator U is written as a product of one dimensional free Scrödinger propagators, U as te same properties as tose of te free Scrödinger propagator see Lemma 2.1 below). Especially, we ave by te representation U t)φ = 4πt) 1 e x 1 y 1 x 2 y 2 4it φy 1, y 2 )dy 1 dy 2, t 0, R R U t)φ; L 2 = φ; L 2, t R, U t)φ; L C t 1 φ; L 1, t 0,

4 4 G. HOSHINO, T. OZAWA EJDE-2016/34 and U t)φ; L p C t 1+ 2 p φ; L p, t 0 for 2 p. Tere are many papers on te Caucy problem for te nonlinear Scrödinger equations and on te analyticity of solutions to te nonlinear evolution equations we refer te reader to [2, 3, 5, 18, 30, 32] for te former and to [4, 6, 11, 12, 13, 17, 19, 22, 23, 24, 25, 26, 27, 28, 31] for te latter. We say tat a domain D R 2 is symmetric if D satisfies te following conditions: 0 D and for any δ = δ 1, δ 2 ) D we ave δ 1, δ 2 ), δ 1, δ 2 ) D. We state our main result: Teorem 1.1. Tere exists an ε > 0 suc tat for any a > 0, any symmetric domain D R 2 and any φ B ε G D,a x, x 2 1 x 2 2; L 2 )), 1.1) as a unique solution u G D,a J, K ; X ). Remark 1.2. Since x 2 1 x 2 2 x x 2 2, te following inequality olds: φ; G D,a x, x 2 1 x 2 2; L 2 ) φ; G D,a x, x x 2 2; L 2 ). Remark 1.3. Regarding analyticity, te operators J and K correspond analyticity in space and in time, respectively. Remark 1.4. As stated in te teorem, a and D may be taken independent of ε. 2. Preliminaries In tis section, we introduce te some basic lemmas. Lemma 2.1 [2, 30, 32]). For any r j, q j ) satisfying 2/r j = 1 2/q j, wit q j [2, ), j = 1, 2, te following inequalities old: U φ; L r1 R; L q1 ) C φ; L 2, F f; L r1 R; L q1 ) C f; L r2 R; L q2 ), were p denotes te Hölder conjugate of p defined by 1/p + 1/p = 1. Te following result is similar to te previous results in [7, 13], were we can find commutation relation between pseudo-conformal generator Kt) not yperbolic K t)) and te linear operator i t not L = i t + ). Lemma 2.2. Let t R. We ave [K t), L] = 8itL, [A δ t), L] = 0, were [A, B] = AB BA is te commutator. Proof. Since L = U t)i t U t), we ave and [K t), L] = U t) [ x 2 1 x it 2 t, i t ] U t) = U t) [ 4it 2 t, i t ] U t) = 8itL [A δ t), L] = U t) [ e δ x, i t ] U t) = 0. Lemma 2.3. Let t R. We ave K t) + 8it ) u 1 u 2 u 3 = K t)u 1 )u 2 u 3 u 1 K t)u 2 )u 3 + u 1 u 2 K t)u 3 ).

5 EJDE-2016/34 ANALYTIC SMOOTHING EFFECT 5 Proof. By te Leibniz rule for P t) = t t + x, we ave K t) + 8it ) u 1 u 2 u 3 = K t) + 12it ) u 1 u 2 u 3 = M t) 4itP t) + 12it ) M 1 t) ) u 1 u 2 u 3 = M t) 4itP t) + 12it ) M 1 t)u 1M 1 t)u 2M 1 t)u 3 = K t) + 4it)u 1 )u 2 u 3 u 1 K t) + 4it ) ) u 2 u 3 + u 1 u 2 K t) + 4it ) ) u 3 = K t)u 1 )u 2 u 3 u 1 K t)u 2 )u 3 + u 1 u 2 K t)u 3 ). Lemma 2.4. Let t R. We ave K t) + 8it ) l u1 u 2 u 3 ) = for all l Z 0. Te above lemma follows immediately by Lemma Proof of Teorem 1.1 1) l2 l! l 1!l 2!l 3! Kl1 t)u 1K l2 t)u 2K l3 t)u 3, Let φ B ε G D,a x, x 2 1 x 2 2; L 2 )), u G D,a J, K ; X ). We define Φ : u Φu by Φu = U φ iλf u 2 u ). Let r > 0, we define a metric space Xr), d) by Xr) = B r G D,a J, K ; X )), du, v) = u v; G D,a J, K ; X ). We see tat Xr), d) is a complete metric space. We sow tat Φ is a contraction mapping in Xr), d). By Lemma 2.1, we ave Φu; X C φ; L 2 + C u 3 ; L 4/3 R; L 4/3 ) C φ; L 2 + C u; L 4 R; L 4 ) 3. Hence Φ is a mapping in X. Since M 1 A δ gives an analytic continuation M 1 A δψ)t, x) = M 1 t, x 1 + 2itδ 1, x 2 2itδ 2 )ψt, x 1 + 2itδ 1, x 2 2itδ 2 ), for t 0, x + 2itδ R 2 + 2itD and ψ G D J ; L R; L 2 )), by Lemmas 2.2 and 2.4, we ave A δ KΦu l = U e δ x x 2 1 x 2 2) l φ iλf A δ K + 8is ) ) l u 2 u = U e δ x x 2 1 x 2 2) l φ iλf A δ = U e δ x x 2 1 x 2 2) l φ iλ 1) l2 l! l 1!l 2!l 3! F 1) l2 l! ) l 1!l 2!l 3! Kl1 ukl2 ukl3 u ) A δ K l1 ua δk l2 ua δk l3 u

6 6 G. HOSHINO, T. OZAWA EJDE-2016/34 for all l Z 0. In te same way as above, we ave A δ K l Φu; X C e δ x x 2 1 x 2 2) l φ; L 2 l! + C A δ K l1 l 1!l 2!l 3! C e δ x x 2 1 x 2 2) l φ; L 2 + C ua δk l2 A δ K l2 u; L4 R; L 4 ) A δ K l3 u; L4 R; L 4 ). By te assumption D = D, we ave a l l! sup A δ KΦu; l X C al l! sup e δ x x 2 1 x 2 2) l φ; L 2 + C Terefore, we obtain ua δk l3 u; L4/3 R; L 4/3 ) l! l 1!l 2!l 3! A δk l1 u; L4 R; L 4 ) 3 j=1 a lj l j! sup A δ K lj u; X. Φu; G D,a J, K ; X ) C φ; G D,a z, x 2 1 x 2 2; L 2 ) + C u; G D,a J, K ; X ) 3. Similarly, we ave Φu Φv; G D,a J, K ; X ) C u; G D,a J, K ; X ) 2 + v; G D,a J, K ; X ) 2) u v; G D,a J, K ; X ) for all u, v G D,a J, K ; X ). Hence, Φ is a mapping in G D,a J, K ; X ). We take ε, r > 0 satisfying Cε + Cr 3 r, Cr 2 < 1. Ten Φ is a contraction mapping in Xr), d). Tis completes te proof of Teorem 1.1. Acknowledgment. Te autors would like to tank te anonymous referees for teir important comments. Tis work was supported by Grant-in-Aid for JSPS Fellows. References [1] D. R. Crawford, P. G. Saffman, H. C. Yuen; Evolution of a random inomogeneous field of nonlinear deep-wave gravity waves, Wave Motion, ), [2] T. Cazenave; Semilinear Scrödinger Equations, Courant Lecture Notes in Mat., 10, Amer. Mat. Soc., [3] T. Cazenave, F. B. Weissler; Some remarks on te nonlinear Scrödinger equation in te critical case, Lecture notes in Mat. Springer, Berlin, ), [4] A. DeBouard; Analytic solution to non elliptic non linear Scrödinger equations, J. Differential Equations., 104, 1993), [5] J. Ginibre; Introduction aux équations de Scrödinger non linéaires, Paris Onze Edition, L161. Université Paris-Sud, [6] N. Hayasi, K. Kato; Analyticity in time and smooting effect of solutions to nonlinear Scrödinger equations, Commun. Mat. Pys., ),

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8 8 G. HOSHINO, T. OZAWA EJDE-2016/34 Gaku Hosino Department of Applied Pysics, Waseda University, Tokyo , Japan address: Toru Ozawa Department of Applied Pysics, Waseda University, Tokyo , Japan address: