LOCAL EXISTENCE OF SOLUTIONS TO THE CAUCHY PROBLEM FOR NONLINEAR SCHRÖDINGER EQUATIONS
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(2) 112. N. HAYASHI AND E.I. KAIKINA. where n 2 and. X. Yn j Yn k 1 2 N =N (u w u w ) = u u (wj ) (w k ) j =1 k=1 l0 jj+jj+j jl1. with w = (wj )1j n , 2 C, l1 l0 2 N, l0 2. Our main purpose in this paper is to consider (1.1) in lower order Sobolev spaces compared with the previous works 2, 12, 13]. Our function spaces are similar to ones used in 10, 13-16] in which the condition every component of @w N is pure imaginary. (1.2). is assumed. Condition (1.2) is sucient for application of the classical energy method. When the nonlinear terms do not satisfy the condition (1.2) in order to treat the derivatives of unknown function in nonlinear terms we need some smoothing property of solutions to the linear Schrodinger equation 3, 4, 11, 17, 19] or some multiplication factor associated with nonlinear structure (see 1, 8, 9, 18] ). However, the smoothing properties obtained in 3, 11, 17, 19] require some smallness condition on the data (see 12]). An application of the gauge transformation method to the Cauchy problem (1.1) is useful only for the one dimensional case 1, 8] and general space dimensions in the case of some special nonlinearities (see 9, 18]). There are only a few results for the Cauchy problem (1.1) with general nonlinearities in the case of large initial data. In papers 2], 13] the existence of local solutions in higher order Sobolev spaces was proved by using smoothing eects obtained in 4] and are based on the theory on pseudo-dierential operators of order 0. More precisely in 2] it is assumed that the initial data u0 2 H m+l0 \ H m1 \ H m;12 , where m n2 ] + 6 and l is a suciently large integer. Here and below we denote the weighted Sobolev space by. H ms = ff 2 L2 kf kms = k(1 + jxj2 )s=2 (1 ; )m=2f kL2 < 1g: We use the notation s] denoting the largest integer less than or equals to s. To treat the Cauchy problem (1.1) in lower order Sobolev spaces we avoid in this paper to use the well known results of pseudo-dierential operators (L2 boundedness theorem, Sharp Garding inequality, see 2]) which need higher order Sobolev spaces. We now state our main result in this paper.. Theorem 1.1. We assume that u0 2 H m0 \ H m;32 m n2 ]+6. Then there exists a unique solution of (1.1) and a positive constant T > 0 such that. u 2 C ( ;T T ] H m0 \ H m;32). T = T () = O. . 1. el1+2. . as ! 1.
(3) ON NONLINEAR SCHRO DINGER EQUATIONS. 113. where p = ku0 km0 + ku0 km;32 , and l1 is as in N . Furthermore u satises the following smoothing property. X Z T (1 + x2j );s=2 (1 ; @x2j )1=4 u(t)k2m0 dt < 1. 1j n. ;T. where s > 1=2. To prove Theorem 1.1 we use the following Notation and function spaces.. tor of order 0. We let for > 0. We introduce the pseudo-di erential opera-. Z xj. Kxj = exp A. 0. xj hx0j i;1; dx0j hD Dx i. . j. where the constant A > 0 will be chosen below to be su ciently large, Dxj = i@xj = i@j , hxj i = (1 + jxj j2 )1=2 and hDxj i = (1 ; @j2 )1=2 . We de ne the operator. K=. n Y. X n Z xj. Dxj hx0j i;1; dx0j hD xi. Kxj = exp A 0 j =1 j =1 k Z 1 Ak X n xj X 0 ; 1; 0 Dxj = hxj i dxj hD i xj k=0 k! j =1 0. . j. which was rst used in 4]. It is easy to see that K is a bounded operator from L2 into itself and there exists a inverse operator K ;1 which is also a bounded operator from L2 into itself. Moreover, we have. kKuk kK ;1 uk eC1 nA kuk C1 =. Z1 0. hxj i;1; dxj K ;1 = K. where kk = kkL2 is the norm of the usual L2 space, Lp is the Lebesgue space with the norm k kp = k kLp . The operator K is useful to obtain smoothing properties of solutions to (1.1). To prove Theorem 1.1 we use the following function space. XTmfksg = ff 2 C (0 T ] L2 ) kf kXTmfksg < 1g where. kf k2XTmfksg = kf k2Tm;1fksg + kf k2YTm + k@t f k2Tm;3fk;2sg + k@t f k2YTm;2.
(4) 114. N. HAYASHI AND E.I. KAIKINA. and. ;. kf k2Tlfksg = sup kf (t)k2l0 + kf (t)k2ks t20T ]. kf kYTm 2. = e;2C1 nA. . n ZT X A X ; hDx i1=2 KD f ( )k2 d kh x i j j jj=m 8 j =1 0 1+ 2. + sup kKD f (t)k2 t20T ]. Q. P. where D = nj=1 (i@j )j j j = nj=1 j and A depends on the size of the data. It is su cient to choose A = l1 +1 for Theorem 1.1. We also de ne a closed ball in XTmfksg as follows. . XTmfksg = f 2 XTmfksg kf kTm;1fksg . k@t f kTm;3fk;2sg l1+1 kf kYTm. l1+1 . k@t f kYTm;2 . 2(l1 +1). :. In this paper we only treat the case T > 0 because the case T < 0 can be treated similarly.. x2. Preliminaries In this section we formulate Lemma 2.1 which states the well-known Sobolev embedding inequality. Then we prepare Lemma 2.2 which is needed to estimate the nonlinear terms. In Lemma 2.3 we give some smoothing e ects of solutions to the linear Schrodinger equation.. Lemma 2.1(The Gagliardo- Nirenberg-Sobolev inequality). Let 1 q r 1. Let j m 2 N f0g satisfy 0 j < m. Let p and a satisfy 1=p = j=n + a(1=r ; m=n) + (1 ; a)=q j=m a < 1 if m ; j ; n=r 2 N f0g, a = j=m otherwise. Then. X. jj=j. k@ kp C (n m j q r). X jj=m. provided that the right hand side is nite. For Lemma 2.1, see, e.g., A.Friedman 5].. k@ kar k k1q;a .
(5) ON NONLINEAR SCHRO DINGER EQUATIONS. 115. Lemma 2.2. We have. . K (f@j g) Kh. CeCnAkf k n ]+22kgkkhk 2. + C kf k n2 ]+12 khxj i;s hDxj i1=2 Kgkkhxj i;s hDxj i1=2 Khk. provided that the right hand side is nite, where s 2 (1=2 1]. Proof.. By a simple calculation we get. K (f@j g) = K hxj i;s @j (hxj is fg) ; sK hxj i;1 xj fg ; K (@j f )g = hxj i;s @j K (hxj is fg) + F (f g). (2.1). where. F (f g) = K hxj i;s @j ](hxj is fg) ; sK hxj i;1 xj fg ; K (@j f )g: In the same way as in the proof of Lemma A.4, (a.31), 6] we have. kK hxj i;s @j ]f k CeCnA kf k: Hence by Lemma 2.1 we obtain. kF (f g)k CeCnA. . . kf k n2 ]+11 + kf k n2 ]+20 kgk:. (2.2). By virtue of the Schwarz inequality we get. . hxj i;s @j K (hxj is fg) Kh. . =. @j hDxj i1=2 K (hxj is fg) hDxj i1=2 hxj i;s Kh. hDxj i khDxj i1=2 K (hxj is fg)kkhDxj i1=2 hxj i;s Khk:. (2.3). We have via Lemma A.1, (a.1), 6]. khDxj i1=2 hxj i;s Khk CeCnAkhk + khxj i;s hDxj i1=2 Khk. (2.4). and by Lemma A.2, 6]. khDxj i1=2 K (hxj is fg)k CeCnAkhxj is f k1kgk + kK hDxj i1=2 (hxj is fg)k:. (2.5).
(6) 116. N. HAYASHI AND E.I. KAIKINA. We represent the second term of the right hand side of (2.5) as follows. K hDxj i1=2 (hxj is fg) = K hDxj i1=2 hxj i2s f ]hxj i;s g + hxj i2s fK hDxj i1=2 hxj i;s g By Lemma 3.2, 7] we get. kK hDxj i1=2 hxj i2s f ]hxj i;s gk CeCnA khxj i2s f k n2 ]+20khxj i;s gk (2.6) and. khxj i2s fK hDxj i1=2 hxj i;s gk khxj i2s f K hDxj i1=2 hxj i;s ]gk + khxj i2s f hxj i;s K hDxj i1=2 ]gk + khxj i2s f hxj i;s hDxj i1=2 Kgk CeCnA kf k n2 ]+12kgk + kf k n2 ]+12khxj i;s hDxj i1=2 Kgk: (2.7) Therefore by virtue of estimates (2.1)-(2.7) we have the result of the lemma. We next consider the Cauchy problem for the linear Schrodinger equation. i@t u + u = f (t x) 2 R Rn u(x 0) = u0 (x) x 2 Rn :. (2.8). Lemma 2.3. We have the following inequality for the solution u of the Cauchy problem (2.8) with > 0. n Zt X A khxj i; 1+2 hDxj i1=2 Ku( )k2 d kKu(t)k + 4 2. j =1. 2C1 nA. e. 0. ku0 k + C (1 + A 2. 2. )eCnA. . Z t. . + Im Kf ( ) Ku( ). d 0. Zt 0. ku( )k2 d. provided that the right hand side is bounded. Proof.. We put. Pxj =. Z xj 0. Dxj 1 + : hx0j i;2s dxj 0 hD s = i 4 xj.
(7) ON NONLINEAR SCHRO DINGER EQUATIONS. 117. Then the operator Kxj de ned in Introduction is written as 1 Am X. Pxmj A > 0: m ! m=0 We have Dx2j Dx2j 2 ; 2s i@j Pxj ] = 2hxj i hD i + i(@j hxj i ) hD i xj xj therefore by a simple calculation we get for m 1 Kxj =. i@j2 Pxmj ] = = where. mX ;1. m1 =0 mX ;1 m1 =0. Pxmj ;m1 ;1i@j2 Pxj ]Pxmj 1 Pxmj ;m1 ;1Qxj Pxmj 1 +. mX ;1 m1 =0. Pxmj ;m1;1 Rxj Pxmj 1 . (2.9). D2 D Qxj = 2hxj i;2s hDxj i Rxj = i(@j hxj i;2s ) hDxj i :. xj xj m ; m ; 1 m m m 1 1 1 Then using the identity Pxj Qxj Pxj = Qxj Pxj + Pxj ;m1;1 Qxj ] and. (2.9) we nd i@j2 K ] = i@j2 Kxj ]. Y. 1ln l6=j. = AQxj K +. Kxl. X ;2 1 Am mX. m;m1 ;1 Qx ]P m1 j xj. m! m1=0Pxj. m=2 1 m m X A X;1 m;m1;1 m1 Y + Pxj Rxj Pxj Kxl : m=1 m! m1 =0 1ln l= 6 j. Applying the operator K to both sides of (2.8), we obtain X 2 i@t Ku + Ku + ii@j K ]u = Kf: 1 j n. (2.10) (2.11). Via estimates (2.9) - (2.11) we obtain X i@t Ku + Ku + i AQxj Ku 1j n. +. ;2 1 Am mX X X P m;m ;1 Qx ]P m u i. 1j n. m! m1=0 xj. 1. j. xj 1. m=2 1 X Am mX;1 m;m1;1 m1 Y Kxl u = Kf: +i Pxj Rxj Pxj m=1 m! m1 =0 1ln l= 6 j. (2.12).
(8) 118. N. HAYASHI AND E.I. KAIKINA. We multiply both sides of (2.12) by Ku, integrate over Rn and take the imaginary part to get 1 d kKu(t)k2 + A X Re(Q Ku(t) Ku(t)) xj 2 dt 1j n + Im(Wu(t) Ku(t)) = Im(Kf (t) Ku(t)) (2.13) where. W=. 1 Am mX ;2 X X (i Pxmj ;m ;1 Qxj ]Pxmj u m! 1. 1. 1j n. m=2 m1 =0 1 m ; 1 X Am X m;m1;1 m1 Y +i Pxj Rxj Pxj ) Kxl : m=1 m! m1 =0 1ln l= 6 j. We consider the second term of the left hand side of (2.13). Using the representation Dx2j =hDxj i = hDxj i ; hDxj i;1 we have (Qxj Ku Ku) ; ; = hxj i;s hDxj iKu hxj i;s Ku ; hxj i;s hDxj i;1 Ku hxj i;s Ku.
(9) = hxj i;s hDxj i1=2 ]hDxj i1=2 Ku hxj i;s Ku.
(10) ; ;hxj i;s hDxj i;1 Ku hxj i;s Ku 2 ;CeC nA hxj i;s hDxj i1=2 Ku kuk + hxj i;s hDxj i1=2 Ku ;s 1=2 2 1 CnA 2 ;Ce kuk + hxj i hDxj i Ku :. 2. + hxj i;s hDxj i1=2 Ku hDxj i1=2 hxj i;s ]Ku + hxj i;s hDxj i1=2 Ku 1. 2 By virtue of (2.13) and (2.14) we get. 1 d kKu(t)k2 + A hx i;s hD i1=2 Ku2 xj 2 dt 2 j jIm(Wu(t) Ku(t))j + jIm(Kf (t) Ku(t))j + C kuk2 eCnA :. (2.14). (2.15). We now consider the rst and second terms of the right hand side of (2.15). By the estimates. kPxj uk C kuk and kRxj uk C kuk.
(11) ON NONLINEAR SCHRO DINGER EQUATIONS. 119. we have. X 1 m ;1 Am X Pxmj ;m ;1 Rxj Pxmj Y Kxl u m! m =0 1j n m=1 1ln X. 1. 1. 1. . l6=j. ;1 1 Am mX X X m;m ;1 Rx P m Y Kx uk k P xj j xj l m! 1. 1j n m=1. 1. 1ln l6=j. m1 =0. 1 Am mX ;1 X X m;m ;1 kRx P m Y Kx uk C j xj l m! 1. 1j n m=1. 1. 1ln l6=j. m1 =0. ;1 1 Am mX X X m k Y Kx uk CnAeCnA kuk: C l m! 1j n m=1. 1ln l6=j. m1 =0. (2.16). We now consider the term. ;2 1 Am mX Y X X m ; m ; 1 m Pxj Qxj ]Pxj Kxl u(t) m ! m =0 1j n m=2 1ln 1. 1. 1. . X. 1 Am mX ;2 X. m! m1 =0 1j n m=2. l6=j. kPxmj ;m1;1 Qxj ]Pxmj 1. Since Pxl j Qxj ] =. l;1 X l1 =0. Y. 1ln l6=j. Kxl u(t)k:. (2.17). Pxl;j l1 ;1 Pxj Qxj ]Pxl1j. the right hand side of (2.17) is bounded from above by m ;2 1 Am mX ;2 m;X X X m;m ;l ;2 kPx Qx ]P l +m Y Kx u(t)k: C j j xj l m! 1. 1j n m=2. m1 =0 l1 =0. 1. 1. 1. 1. 1ln l6=j. (2.18).
(12) 120. N. HAYASHI AND E.I. KAIKINA. We next consider the term kPxj Qxj ]f k. We easily nd that Pxj Qxj ] =. Z xj. D2. xj hx i;2s xj hx0j i;2s dxj 0 hD D i j hD i. xj Z x j D Dxj ; 2hxj i;2s hDxj i hx0j i;2s dxj 0 hD xj 0 xj i Z xj Dx2j Dxj 0 2. xj. hx0j i;2s dxj 0 hD i ]hxj i;2s hD i xj xj Z xj Dxj ;2s ) Dxj ; hx0j i;2s dxj 0 hD ( D h x i x j j i hD i =. 0. 0. 2. Z xxjj. xj. 0. D. ; hxj i;2s hDxj i ]. xj. Dxj hx0j i;2s dxj 0 hD i:. (2.19). xj. Hence by Lemma A.1, 6] we get. kPxj Qxj ]f k C kf k:. (2.20). From (2.18), (2.19) and (2.20) it follows that. X 1 m;2 Am X Pxmj ;m ;1 Qxj ]Pxmj u(t) CA2 eCnA kuk: m=2 m! m =0 1. 1. 1. (2.21). We apply (2.16) and (2.21) to (2.15) to obtain the result of Lemma 2.3. . x3. Proof of Theorem 1.1 We di erentiate (1.1) with respect to xj to obtain. . . n X i@t uxj + 21 uxj = @uN uxj + @u
(13) N uxj + @@l u N @l uxj + @@lu N @l uxj : l=1. Let us consider the linearized version of the Cauchy problem (1.1). 8 i@t u + 12 u = N (v v1 vn v v1 vn ) (t x) 2 R Rn > < P n 1 i@t uj + 2 uj = @v N vj + @v
(14) N vj + l=1 @vl N @l uj + @v
(15) l N @l uj > : n u(0 x) = u0 (x) uj (0 x) = @j u0 (x) x 2 R . (3.1) p m 2 N, =. where 1 j n, v vj 2 XTm;1fm;42g, , m 2 ku0 km0 + ku0 km;32 and T = O(1=el1 +2 ). We de ne the mapping (u uj ) = n ] + 6.
(16) ON NONLINEAR SCHRO DINGER EQUATIONS. 121. M (v vj ) and show that M is a mapping from (XTm;1fm;42g)n+1 into itself for some time T if m n2 ] + 6. Without loss of generality we can assume that is su ciently large because we are interested in the case of large initial data. We rst consider a priori estimates of solutions in the norms. kukTm;2fm;42g kuj kTm;2fm;42g k@t ukTm;4fm;62g and k@t uj kTm;4fm;62g: By Holder's inequality, Lemma 2.1 and the classical energy method we obtain sup kuj (t)km;20 ku0 km;10. t20T ]. + Cl1. Z T; 0. + kuj (t)km;20 + eC2 A. X. . jj=m;1 1j n. kKDuj k dt. here and in what follows C2 depends on n and C1 is de ned in Notation and function spaces . Hence we have. X. sup kuj (t)km;20 2p + Cl1 TeC2 A sup. t20T ]. t20T ] jj=m;1 1j n. kKD uj k. (3.2). for small time T = O(1=l1 +1 ). Multiplying both sides of the second equation of (3.1) by x2 , and applying a classical energy method to the resulting equation we get sup kx uj (t)km;40. t20T ]. 2. ZT; p +C kxru (t)k j. m;40 + kuj (t)km;40. n ; X 2 + x @v N vj + @v
(17) N vj + @vl N @l uj + @v
(18) l N @l uj m;40 l=1 Z T p +C. 0. 0. (kx2 uj (t)km;40 + kuj (t)km;20 )dt. + Cl1 T (1 + kuj kTm;2fm;42g) p + Cl1 T (1 + kuj kTm;2fm;42g): From (3.2) and (3.3) it follows that. (3.3). kuj kTm;2fm;42g C p + Cl1 TeC2A sup. X. t20T ] jj=m;1 1j n. kKD uj k:. (3.4).
(19) 122. N. HAYASHI AND E.I. KAIKINA. In the same way as in the proof of (3.4) we get. k@t uj kTm;4fm;62g X C (p + pl1 ) + C2l1 TeC2A sup kKD @t uj k t20T ] jj=m;3 1j n. (3.5). for small time T = O(1=2(l1 +1) ). We easily see that the following estimates kukTm;2fm;42g C p (3.7) and.
(20) k@t ukTm;4fm;62g C p + pl1 (3.8) hold. We next consider a priori estimates of solutions to (1.1) in the norms. kukYTm;1 kuj kYTm;1 k@t ukYTm;3 k@t uj kYTm;3 :. This is our main point in the proof of the result of Theorem 1.1. In order to obtain the desired estimates we use smoothing properties of solutions to the linear Schrodinger equation. We make use of the operator K to get a smoothing e ect. However we can not use K directly to the nonlinear Schrodinger equations with nonlinear terms involving @l u because K@l u 6= K@l u. We now remove such terms from the original equations by a diagonalization technique (see 2]). We de ne the pseudo-di erential operators. A=. n X l=1. @ vl N @ l B =. P. n X l=1. @v
(21) l N @l :. In order to eliminate the term nl=1 @v
(22) l N @l uj from (3.1) we rewrite the second equation of system (3.1) in the matrix form u 1 ; 2A ;2B u @ N v + @ N v v j v
(23) j j i@t uj + 2 2B ; + 2A uj = ;@v N vj ; @v
(24) N vj : j (3.9) We de ne the 2 2 matrix operators and 0 as follows 1 1 ;2 ;2 2 Bh D i ; 2 Bh D i 0 = 2Bh = ;2Bh Di;2 Di;2 1 1 where hDi = (1 ; )1=2 . Applying the operator to both sides of (3.9), we obtain 2A ;2B u j i@t uuj + 12 ; 2B ; + 2A uj j @ N v + @ N v u v j v
(25) j (3.10) = ;@ N v ; @ N v + i@t ] uj : j v j v
(26) j.
(27) ON NONLINEAR SCHRO DINGER EQUATIONS. By a direct calculation we nd 0 =. . 1. ;BhDi;2. Di;2 ; Bh 1 ; BhDi;2 Bh D1i;2 =. . 1. Di;2 Bh. BhDi;2 1. 0. Di;2 BhDi;2 1 0 0 BhDi;2Bh D1 ;i;Bh 2 0 = ; ;2 0 1. . . . . Di BhDi;2 : Bh. 0. We substitute the identity into (3.10) to get. u . 123. . . 2A ;2B 0 uj + 21 ; 2B ; + 2A uj = ;@@v NN vvj +;@@v
(28) NN vvj + i@t ] uuj j v j v
(29) j u ; 2 Di;2 1 ; 2 A ; 2 B Bh D i Bh 0 j ; 2 2B ; + 2A Di;2 BhDi;2 uj : 0 Bh (3.11). i@t uj j. It is easy to see that. ; 2A. ;2B. . 0 2B ; + 2A 1 BhDi;2 ; 2A ;2B 1 ;2 ;Bh D i = Bh ;2 i;2 1 D;i 2A + 2Bh1 Di;2 B 2;B2B + ;BhD+i;22A(; +;2BhA)D ;2 1 ;Bh D i = 2B + Bh ;2 ;2 Di;2 ;Bh 1 ; 2A Di (0 ; 2A) ; + 2A ; 2BhDi B = 0 ;0+ 2A ;2 ; BhDi;2 ; 2B ; Bh D i + Bh Di;2 + Bh Di;2 + 2B 0. . A 11 +. ;A12. . A12 ;A11 . where. A11 = 2BhDi;2 B +. . . D i ;2 ) ; 2B + BhDi;2 (; + 2A) (;Bh. A12 = 2BhDi;2 A + (2A ; 2BhDi;2 B)BhDi;2:. (3.12).
(30) 124. Since. N. HAYASHI AND E.I. KAIKINA. ;BhDi;2 ; BhDi;2 ; 2B = ; B]hDi;2 ; 2BhDi;2 ; 2B = ; B]hDi;2 ; 2BhDi;2. we have 2A ;2B ; 2A A B 0 11 12 0= ; 2B ; + 2A 0 ; + 2A + ;B12 ;A11 where B12 = 2BhDi;2 A+(2A; 2BhDi;2B)BhDi;2 ; B]hDi;2 ; 2BhDi;2 (3.13) and B]hDi;2 = 2. n X l=1. (r@v
(31) l N ) r@l hDi;2 +. n X l=1. (@v
(32) l N )@l hDi;2 :. Therefore we obtain u 1 ; 2A u 0 j i@t u + 2 uj 0 ; + 2 A @j N v + @ N v u j 1 A B u j = ;@v N vj ; @v
(33) N vj + i@t ] uj ; 2 ;B11 ;A12 11 uj 12 j v j v
(34) j ; 2A ;2B BhDi;2 Bh Di;2 u 1 0 j ; 2 2B ; + 2A Di;2 BhDi;2 uj 0 Bh (3.14) where ;2 0 2( i@ B ) h D i t i@t ] = 2(i@ B)hDi;2 : 0 t Now di erentiating (3.9) with respect to x we get analogously to (3.14) D u 1 ; 2A D u 0 j j i@t Du + 2 0 ; + 2A D uj j = DD ((;@@v NN vvj +;@@v
(35) NN vvj )) D Pnv (@ Nj @ v
(36) u + @j N @ u ) ; Pn (@ N @ D u + @ N @ D u ) v l j v
(37) l j v l j v
(38) l j + ;D Pl=1 n (@ N @ D u + @ N @ D u ) n (@ N @ u + @ N @ u ) + Pl=1 v l j v
(39) l j v l j v
(40) l j l=1 l=1 ; 2 D uj ; 1 A11 B12 D uj + 2(i@ B0)hDi;2 2(i@t B0):hDi D uj D uj 2 ;B12 ;A11 t ;2 Bh uj Di;2 1 ; 2 A ; 2 B Bh D i 0 D ; 2 2B ; + 2A Di;2 BhDi;2 D uj 0 Bh (3.15).
(41) ON NONLINEAR SCHRO DINGER EQUATIONS. 125. where j j = m ; 1. From the de nitions of the operators B A11 A12 B12 , we see that every term of the right hand side of (3.15) is harmless in our problem. We will prove that the L2 norm of every term of the right hand side of (3.15) is estimated from above by the Sobolev spaces of order m ; 1. Using Lemma 2.1 and Holder inequality we will prove the following inequality . Yl Yl D fj C kfj k11;j kD fj kj j=1 j=1 P where
(42) j 2 0 1] are such that lj =1
(43) j = 1: First let us consider the case l = P 2: By the Leibniz rule we get kD f f kC jj kD;0 f D0 f k and then. 1 2 j00j=0 0 ; by the Holder inequality we have kD f1 D f2 k C kD;0 f1 kp kD0 f2 kq 1 2. with p1 + 1q = 12 : Then applying Lemma 2.1 we obtain 0. if p1 = j;n j +
(44) 1. 1. kD;0 f1kp C kf1k11;1 kD f1k1 kD0 f2kq C kf2k11;2 kD f2k2. 0.
(45). jj and in the same way 2 ; n.
(46). if q1 = jn j +
(47) 2 12 ; jnj : Now the condition
(48) 1 +
(49) 2 = 1 follows from p1 + 1q = 1 . Therefore we get 2. 2 D Y f C Y2 kf k1;j kD f kj j j=1 j j=1 j 1 P. where
(50) j 2 0 1] are such that 2j =1
(51) j = 1: Then using this particular result Q and arguing by induction with respect to l 2 we have with g = lj =1 fj. kD. lY +1. fj k = kD gfl+1 k. j =1 l Y. 1; C fj j=1 1 Yl 1; C. j =1. 0. l D Y f kf k1;l kD f kl l+1 j=1 j l+1 1 0. +1. +1. kfj k1 0 kfj k(11;j )0 kDfj kj 0 kfl+1k11;l+1 kD fl+1kl+1 .
(52) 126. N. HAYASHI AND E.I. KAIKINA. P. where
(53) 0 +
(54) l+1 =1: By virtue of the equality lj =1
(55) 0
(56) j +
(57) l+1 =1 we get the desired estimate. Using this estimate, we consider estimates of nonlinear terms. We may assume that N is a polynomial of degree l1 Q2 since weQconsider the largeP solutions. Therefore we get with @v N =v1 ;1 v2 nk=1 vkk nk=1 vkk , P 1 + 2 + 1kn
(58) k + 1kn k =l1. X jjm;1 Yn. kD @v Nk C. X. kD vk 1 kvk11 ; 1;1kD vk 2 kvk12 ; 2. jjm;1 Yn kvk k1k ; k kD vk k k kvk k1k ; k kD vk k k k=1 k=1 n Y C kvkm1;;110kvkm2;10 kvk kmk;10 kvk kmk;10 Cl1;1 : k=1. In the same way we obtain for j j m ; 1. kD @v
(59) Nk + kD @vj Nk + kD @v
(60) j Nk + k@v Nk1 + k@v
(61) Nk1 Cl1;1 : Hence we have with
(62) 1 +
(63) 2 = 1 and
(64) 3 +
(65) 4 = 1. D(@ N v + @ N v ) v j v
(66) j D (;@v N vj ; @v
(67) N vj ) L L 2. 2. C kD(@v N vj ; @v
(68) N vj )k + C kB hDi;2 D (;@v N vj ; @v
(69) N vj )k C k@v Nk11;1 kD @v Nk1 kvj k11;2 kD vj k2 + C k@v Nk11;3 kD;1 @v Nk3 kvj k11;4 kD vj k4 Cl1;1 kvj km;10 + Cl1;1 l1;1 kvj km;10 C (1 + l1 )l1 for 0 t T (3.16). Note that we do not have the derivatives of the highest order m in the form.
(70) ON NONLINEAR SCHRO DINGER EQUATIONS. 127. D (@v N @l uj ) ; @v N @l D uj when j j = m ; 1. Therefore we have. Pn (D(@ N @ u + @ N @ u ) ; @ N @ Du ; @ N @ Du ) Pnl=1 ; v l j v
(71) l j v l j v
(72) l j l=1 @v N @l D uj + @v
(73) N @l D uj ; D (@v N @l uj + @v
(74) N @l uj ) L L X 1; 1; +1 C. k@v Nk1 1 kD @v Nk 1 k@l uj k1 2 kD @l uj k 2 + C k@v NkkD. jj=m;2 + C k@v Nk1 kD;1 @v Nk3 k@v Nk1;3 k@l uj k1;4 kD;1 @l uj k4 X Cl1;1 kuj km;20 + Cl1;1 eC2 A kKD uj k jj=m;1. 0 1 X + C2l @kuj km;20 + eC A kKD uj kA 0 jj=m;1 1 X C (1 + l )l eC A @kuj (t)km;20 + kKDuj (t)kA 1. 2. 2. uj k. 2. 1. 1. (3.17). 2. jj=m;1. for all 0 t T . Since by Lemma 2.1 k@t @v
(75) Nk1 Cl1 we have. 2(i@ B0)hDi;2 t. . . 2(i@t B)hDi;2 D uj 0 Duj L2 L2 C k@t @v
(76) Nk1 kD;1 uj k Cl1 kuj km;20:. (3.18). In the same manner we get the following estimates. 1 A B Du ;B11 ;A12 Duj 2 12 11 j L L X l 3l CA 2. C (1 + )e 1. 1. 2. kuj (t)km;20 +. . 2. jj=m;1. kKD uj (t)k. (3.19). and. 1 ; 2A ;2B BhDi;2Bh Di;2 D u 0 j 2B ; + 2A Di;2 BhDi;2 D uj L L 0 Bh 2 X 2. Cl1 (1 + 3l1 )eC2 A kuj (t)km;20 +. jj=m;1. kKD uj (t)k. for all 0 t T . Thus estimates (3.16)-(3.20) show that the right hand side of (3.15) is not di cult to treat. We now prove a priori estimates of solutions. 2. (3.20).
(77) 128. N. HAYASHI AND E.I. KAIKINA. to (3.15) by using (3.16)-(3.20). In order to get the desired a priori estimates we rewrite (3.15) as a single equation i@t wj + 1 wj = Pn @vl N @l wj + F1 l=1 2 (3.21) wj = D uj + 2BhDi;2 D uj where j j = m ; 1. From (3.16)-(3.20) we see that. kF1 k Cl1 (1 + 3(l1 +1) )eC2 A. . kuj (t)km;20 +. + C (1 + l1 )l1 +1 eC2 A for 0 t T:. X. jj=m;1. . kKD uj (t)k. (3.22) In the same way as in the derivation of (3.21) we have the equation connected with D @t uj ( i@ w~ + 1 w~ = Pn @ N @ w~ + F l j 2 t j 2 j l=1 vl (3.23) ; 2 w~j = D @t uj + 2BhDi D @t uj where jj = m ; 3. Here F2 has the estimate. kF2 k C. 2l1. . (1 + 6l1 )eC2 A (k@t uj (t)km;40 +. + kuj (t)km;20 +. X. kKDuj (t)k. . X. kKD @t uj (t)k. jj=m;3 + C (1 + 2l1 )2l1 eC2 A. (3.24) for all 0 t T . We apply Lemma 2.3 to the equation (3.21) and use (3.22) to obtain jj=m;1. n Zt X A 2 kKwj (t)k + 4 khxk i; 1+2 hDxk i1=2 Kwj ( )k2 d k=1 0 Zt 2C1 nA 2 2 C2 A e kwj (0)k + C (1 + A )e kwj ( )k2 d 0 . Zt Z t. X n . + 2 kF1 ( )kkKwj ( )kd + 4 Im K @vl N @l wj ( ) Kwj ( ). d 0 0 l=1 2C1 nA 1=2 l = 2 C A 1 2 e ( + ) + e T sup kwj (t)k2 t20T ] ZT X + Cl1 (1 + 3l1 )eC2 A (kuj (t)km;20 + kKD uj (t)k)2 dt 0 jj=m;1 ZT X + C (1 + l1 )l1 eC2 A kKD uj (t)kdt 0 jj=m;1 . Z T. X n . + 4 Im K @vl N @l wj ( ) Kwj ( ). d: (3.25) 0 l=1.
(78) ON NONLINEAR SCHRO DINGER EQUATIONS. 129. Since j j = m ; 1 we have by (3.2). X. kKD uj (t)k kKwj (t)k + 2. kK BhDi;2 D uj (t)k. jj=m;1 C A l kKwj (t)k + Ce 2 1 sup kuj (t)km;20 t20pT ] X C A l l C A 1 2 2 1 kKwj (t)k + Ce 2 + C Te sup kKD uj (t)k : t20T ] jj=m;1 1j n. Hence sup. X. t20T ] jj=m;1 1j n. X. kKD uj (t)k 2 sup. t20T ] jj=m;1 1j n. kKwj (t)k + CeC2 Al1 p. (3.26). for small time T = O(1=el1 +2 ), if we take A = l1 +1 . Similarly, we get sup. X. t20T ] jj=m;1 1j n. kKwj (t)k 2 sup. X. t20T ] jj=m;1 1 j n. kKD uj (t)k + CeC2Al1 p:. (3.27). From estimates (3.25) - (3.27) it follows that l. (1 ; CeC2 1 T ) sup +1. X. kKwj (t)k2. t20T ] jj=m;1 1j n Z n t X X khxk i; 1+2 hDxk i1=2 Kwj ( )k2 d + A4 jj=m;1 k=1 0 1j n . n X Z t. X 2C1 nA l1 e +4. Im K l=1 @vl N @l wj ( ) Kwj ( ). d: 0 jj=m;1 1j n. We apply Lemma 2.2 to the second term of the right hand side of the above inequality to get (1 ; CeC1 nA 4(l1 +1) T ) sup. X. kKwj (t)k2. t20T ] jj=m;1 1j n A X X n ZT + 4 ; Cl1 khxk i; 1+2 hDxk i1=2 Kwj ( )k2 d jj=m;1 k=1 0 1j n 2C1 nA l1 Ce :.
(79) 130. N. HAYASHI AND E.I. KAIKINA. If we take A = l1 +1 and T = O(1=el1 +2 ) we obtain. X. sup. t20T ] jj=m;1 1j n. kKwj (t)k2. n ZT X X A + khxk i; 1+2 hDxk i1=2 Kwj (t)k2 dt. 6 jj=m;1 k=1. Ce. 1j n 2C1 nA l1. 0. e2C1 nA l1 +1:. (3.28). By (3.26) and (3.28) we nd that. X. sup. t20T ] jj=m;1 1j n. kKD uj (t)k. n ZT X X 1+ A ; 1=2 2 2 +8 khxk i hDxk i KD uj (t)k dt jj=m;1 k=1 1j n 2C1 nA l1 +1 e :. 0. (3.29). In the same way as in the proof of (3.29) we have sup. X. t20T ] jj=m;3 1j n. kKD@t uj (t)k. n ZT X X A +8 khxk i; 1+2 hDxk i1=2 KD@t uj (t)k2 dt e2C1 nA2(l1 +1) :. jj=m;3 k=1 1j n. 0. (3.30). Thus we see that there exists a time T = O(1=el1 +2 ) such that. fu u1 un g 2 (XTm;1fm;42g)n+1 :. (3.31). We now let a sequence fu(k) u(1k) u(nk) g satisfy the equation (3.1) with u = u(k) , uj = u(jk) , v = u(k;1) ,vj = u(jk;1) and with the same initial data, (0) where fu(0) u(0) odinger equation. 1 un g is a solution of the linear Schr We prove that the sequence fu(k) u(1k) u(nk) g is a Cauchy sequence in (XTm;1fm;42g)n+1 . From (3.31) we already know that. fu(k) u(1k) u(nk) g 2 (XTm;1fm;42g)n+1 for any k 2 N:.
(80) ON NONLINEAR SCHRO DINGER EQUATIONS. 131. In the same way as in the proof of (3.4)-(3.8) we see that. kU (k) kTm;2fm;42g + kUj(k) kTm;2fm;42g + k@t U (k) kTm;4fm;62g + k@t Uj(k) kTm;4fm;62g 2l1 C A 2 C Te kU (k;1) kTm;2fm;42g + kUj(k;1) kTm;2fm;42g + k@t U (k;1) kTm;4fm;62g + k@t Uj(k;1) kTm;4fm;62g X ; (k;1) + kKD U k + kKDUj(k;1) k jj=m;1 jj=m;3 1j n. . + kKD @t U (k;1) k + kKD @t Uj(k;1) k. (3.32). where U (k) = u(k) ; u(k;1) , Uj(k) = u(jk) ; u(jk;1) . As in the proof of (3.21) we get. i@t Wj(k) + Wj(k) =2. n X. (k;1). @vl N (u. l=0. (k ). (k;2). )@l wj ; @vl N (u. (k;1). )@l wj. . + F1 (u(k;1) ) ; F1 (u(k;2) ) N~ . (3.33). where Wj(k) = wj(k) ; wj(k;1) , wj(k) = D u(jk) + 2B(u(k;1) )hDi;2 D u(jk) . We apply Lemma 2.3 to (3.33) to obtain. X Z t ; 1+2 1=2 (k) 2 A kKWj (t)k + 4 khxl i hDxl i KWj ( )k d (k). CA. 2. 2. eC2 A. Zt 0. 1ln 0. . Z t. (k ). ~ kWj ( )k d + Im K N ( ) KWj ( ). d: 0 (3.34) (k). 2. By Holder's inequality and Lemma 2.1 we get. ZT 0. kF1 (u(k;1) ) ; F1 (u(k;2) )kkKWj(k) (t)kdt. Cl1 e2C2 AT +. . kU (k;1) kTm;2fm;42g + kUj(k;1) kTm;2fm;42g. X ; (k;1) (k;1) kKD U k + kKD Uj k. jj=m;1. sup kKWj(k) (t)k:. t20T ]. (3.35).
(81) 132. N. HAYASHI AND E.I. KAIKINA. We also have by Lemma 2.2 , the Schwarz inequality and (3.31). Z T. X n. Im K @vl N (u(k;1) ) @l Wj(k)(t) 0 l=1 n X (k;1) (k;2) +. l=1. (@vl N (u. ZT n X l. C. 1. ) ; @vl N (u. (k;1). ))@l wj. . (t) KWj (t). dt (k). khxj i; 1+2 hDxj i1=2 KWj(k)(t)k2 dt. 0 j =1 + Cl1 ;1 sup kU (k;1) (t)km;42 + sup kUj(k;1) (t)km;42 t20T ] t20T ] Z n T X khxl i; 1+2 hDxl i1=2 KWj(k)(t)kkhxj i; 1+2 hDxj i1=2 Kwj(k;1)(t)kdt: l=1 0. We again apply Lemma 2.2 , the Schwarz inequality and (3.31) to see that the right hand side of the above inequality is estimated by the value. C. ZT n X l j =1. 1. 0. khxj i; 1+2 hDxj i1=2 KWj(k)(t)k2 dt. + Cl1 ;1 eC2 A. . n ZT X l=1. C. 0. sup kU. t20T ]. (k;1). (k;1). (t)km;42 + sup kUj t20T ]. (t)km;42. . khxl i; 1+2 hDxl i1=2 KWj(k)(t)kdt. n X. 1ln 1. +8. . l1 (1 + Cl1 eC2 A T ). sup kU. t20T ]. (k;1). ZT 0. khxl i; 1+2 hDxl i1=2 KWj(k)(t)k2 dt (k;1). (t)km;42 + sup kUj t20T ]. 2. (t)km;42 :. (3.36). From (3.34), (3.35) and (3.36) it follows that there exists a time T =O(1=el1 +2 ).
(82) ON NONLINEAR SCHRO DINGER EQUATIONS. 133. such that sup. X. t20T ] jj=m;1. + A8. +. kKWj(k) (t)k2. X ZT. khxl i; 1+2 hDxl i1=2 KWj(k)( )k2 d. jj=m;1 0 1ln 1 (k;1) kTm;2fm;42g + kUj(k;1) kTm;2fm;42g 4 kU. 2 X ; (k;1) (k;1) kKD U k + kKD Uj k :. jj=m;1. (3.37). By the de nition of Wj(k) we have. Wj(k) = D u(jk) + 2B(u(k;1) )hDi;2 D u(jk) ; D u(jk;1) ; 2B(u(k;2) )hDi;2 D u(jk;1) = D Uj(k) + 2B(u(k;1) )hDi;2 D Uj(k). . (k;1). + 2 B(u. (k;2). ) ; B (u. . ) hDi;2 D u(jk;1) :. Hence by (3.31) and Lemma 2.1 we obtain. X jj=m;1. . X Z T ; 1+2 1=2 (k) 2 A kKWj (t)k + 8 khxl i hDxl i KWj ( )k d. X. jj=m;1. (k). 2. kKD Uj(k) (t)k2. jj=m;1 1ln. 0. X Z T ; 1+2 1=2 (k) 2 A +8 khxl i hDxl i KD Uj ( )k d. jj=m;1 0 1ln 2 (k;1) (k;2) C A l 2 1 + Ce kUj kTm;2fm;42g + kUj kTm;2fm;42g. (3.38).
(83) 134. N. HAYASHI AND E.I. KAIKINA. and. X. jj=m;1. kKD Uj(k) (t)k2. X Z T ; 1+2 1=2 (k) 2 A +8 khxl i hDxl i KD Uj ( )k d jj=m;1 1ln. . X. jj=m;1. 0. kKWj(k) (t)k2 + A8. + CeC2 A l1. . (k;1). kUj. X ZT jj=m;1 1ln. 0. khxl i; 1+2 hDxl i1=2 KWj(k)( )k2 d (k;2). kTm;2fm;42g + kUj. 2. kTm;2fm;42g :. (3.39). In the same way as in the proofs of (3.37), (3.38) and (3.39) we have sup. X. t20T ] jj=m;3. kK@t Wj(k)(t)k2. X Z T ; 1+2 1=2 A +8 khxl i hDxl i K@t Wj(k)( )k2 d . +. X jj=m;3. . jj=m;3 0 1ln 1 kU (k;1) k (k;1) kTm;2fm;42g Tm;2fm;42g + kUj 4. 2 X ; (k;1) (k;1) kKD U k + kKD Uj k . jj=m;1. kK@t Wj(k)(t)k2 + A8. X. jj=m;3. kKD @t Uj(k) (t)k2. X ZT jj=m;3 1ln. 0. (3.40). khxl i; 1+2 hDxl i1=2 K@t Wj(k)( )k2 d. X Z T ; 1+2 1=2 (k) 2 A khxl i hDxl i KD @t Uj ( )k d +8. jj=m;3 0 1ln (k;1) (k;2) C A l 2 1 + Ce kUj kTm;2fm;42g + kUj kTm;2fm;42g. (3.41).
(84) ON NONLINEAR SCHRO DINGER EQUATIONS. and. X. jj=m;1. . kKD Uj(k) (t)k2 + A8. X. jj=m;1. kKWj(k)(t)k2 + A8. . X ZT jj=m;1 1ln. 0. X ZT. jj=m;1 1ln. 0. 135. khxl i; 1+2 hDxl i1=2 KD Uj(k) ( )k2 d khxl i; 1+2 hDxl i1=2 KWj(k)( )k2 d. . + CeC2 A l1 kUj(k;1) kTm;2fm;42g + kUj(k;2) kTm;2fm;42g :. (3.42). From the estimates (3.32), (3.37) - (3.42) it follows that there exists a time T = O(1=el1 +2 ) such that. kU (k) kXTm;1fm;42g + kUj(k) kXTm;1fm;42g X (k;l) 1 (k;l) kU kXTm;1fm;42g + kUj kXTm;1fm;42g : 8 1l3. We de ne. Lk =. X. kU. 1l3. (k+1;l). (k+1;l). kXTm;1fm;42g + kUj. Then we have by (3.43). (3.43). . kXTm;1fm;42g :. Lk 21 Lk;3. which implies that fu(k) u(1k) u(nk) g is a Cauchy sequence in (XTm;1fm;42g )n+1 . Hence there exists a unique solution fu u1 un g satisfying 8 i@t u + 12 u = N (u u1 un u u1 un ) (t x) 2 R Rn >. < P n 1 i@t uj + 2 uj =@u N uj + @u
(85) N uj + l=1 @ul N @l uj + @u
(86) l N @l uj > : n u(0 x) = u0 (x) uj (0 x) = @j u0 (x) x 2 R 1 j n:. (3.45) By the uniqueness of solutions we see that uj =@j u. Therefore u 2 C (0 T ] H m0 ) \ C (0 T ] H m;32 ) since K ;1 is a bounded operator in L2 . Finally we prove a smoothing e ect for the solutions. We already know sup ku(t)km0 +. t20T ]. 2. X ZT. jj=m 0 1j n. khxj i;s hDxj i1=2 KD u(t)k2 dt < 1: (3.46).
(87) 136. N. HAYASHI AND E.I. KAIKINA. By a direct calculation we get. X. jj=m. kD hxj i;s hDx i1=2 u(t)k j. . C ku(t)km0 + CeC2 A. . X jj=m. ku(t)km0 +. khxj i;s hDx i1=2 D u(t)k. . j. X. jj=m. . kKhxj i;s hDx i1=2 Du(t)k. for s > 1=2: (3.47). j. We also have. kKhxj i;s hDx i1=2 fk kK hxj i;s hDx i1=2 ]fk + k KhDx i1=2 hxj i;s ]fk + khxj i;s K hDx i1=2 ]fk + khxj i;s hDx i1=2 Kfk: (3.48) j. j. j. j. j. We apply Lemma A.1, 6], Lemma 3.2, 6] and Lemma A.2, 6] to the rst three terms of the right hand side of (3.48), respectively. Then we get. kKhxj i;s hDx i1=2 fk CeC2A j. . kfk + khxj i;s hDx i1=2 Kfk j. . :. (3.49). By inequalities (3.46), (3.47) and (3.49) we get the last estimate of Theorem 1.1. This completes the proof of Theorem 1.1. . Acknowledgement. The authors would like to thank the referee for care-. ful reading and useful comments.. References. 1. H.Chihara, Local existence for the semilinear Schrodinger equations in one space dimension, J. Math. Kyoto Univ. 34 (1994), 353-367. 2. H.Chihara, Local existence for semilinear Schrodinger equations, Math. Japon. 42 (1995), 35-52. 3. P.Constantin and J.C.Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), 413-439. 4. S.Doi, On the Cauchy problem for Schrodinger type equations and regularity of solutions, J. Math. Kyoto Univ. 34 (1994), 319-328. 5. A.Friedman, Partial Dierential Equations, Krieger, 1983. 6. N.Hayashi, Local existence in time of small solutions to the Davey-Stewartson system, Annales de l'I.H.P. Physique Theorique 65 (1996), 313-366. 7. N.Hayashi and H.Hirata, Global existence and asymptotic behavior in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system, Nonlinearity 9 (1996), 1387-1409. 8. N.Hayashi and T.Ozawa, Remarks on nonlinear Schodinger equations in one space dimension, Di. Integral Eqs. 7 (1994), 453-461..
(88) ON NONLINEAR SCHRODINGER EQUATIONS. 137. 9. N.Hayashi and T.Ozawa, Global, small radially symmetric solutions to nonlinear Schrodinger equations and a gauge transformation, Di. Integral Eqs. 8 (1995), 1061-1072. 10. T.Kato, Quasi-linear equation of evolution, with application to partial dierential equations, Lecture Notes in Math., Springer-Verlag, vol. 448, 1975, pp. 25-70. 11. C.E.Kenig, G.Ponce and L.Vega, Oscillatory integral and regularity of dispersive equations, Indiana Univ. J. 40 (1991), 33-69. 12. C.E.Kenig, G.Ponce and L.Vega, Small solutions to nonlinear Schrodinger equations, Annales de l'I.H.P. Non. Lin. 10 (1993), 255-288. 13. C.E.Kenig, G.Ponce and L.Vega, Smoothing eects and local existence theory for the generalized nonlinear Schrodinger equations, preprint (1997). 14. S.Klainerman, Long-time behavior of solutions to nonlinear evolution equations, Arch. Rat. Mech. Anal. 78 (1982), 73-98. 15. S.Klainerman and G.Ponce, Global, small amplitude solutions to nonlinear evolution equations, Commun. Pure Appl. Math. 36 (1983), 133-141. 16. J.Shatah, Global existence of small solutions to nonlinear evolution equations, J. Differential Equations 46 (1982), 409-425. 17. P.Sjolin, Regularity of solutions to the Schrodinger equation, Duke Math. J. 55 (1987), 699-715. 18. A.Soyeur, The Cauchy problem for the Ishimori equations, J. Funct. Anal. 105 (1992), 233-255. 19. L.Vega, Schrodinger equations : pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878. Nakao Hayashi Department of Applied Mathematics, Science University of Tokyo Tokyo 162-8601, JAPAN E-mail : nhayashi@rs.kagu.sut.ac.jp Elena I. Kaikina Facultad de Ciencias Basicas, Instituto Technologico de Morelia CP 58080, Morelia, Michoacan, MEXICO E-mail : kaikina@ifm1.ifm.umich.mx.
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