Global strong solutions for nonlinear Schrödinger equations

0362-546~87 Pergamon

GLOBAL

STRONG SOLUTIONS FOR NONLINEAR EQUATIONS

$3 00 + Journals

00 Ltd.

SCHRODINGER

YOSHIO TSUTSUMI

Faculty of Integrated Arts and Sciences, Hiroshima University, Higashisenda-machi, (Receiued 15 May 1986; received for public&on

Naka-ku, Hiroshima 730. Japan

10 July 1986)

Key words and phrases: Global existence, strong solutions, nonlinear Schriidinger equations,

in higher

dimensions. 1. INTRODUCTION WE

AND

A THEORElM

CONSIDER

Schrodinger

the unique global existence of strong solutions for the following nonlinear equations with power interaction:

au j - = - Au + ]u]P-iu, at u(to x>= u,(x), 7

tER,

XER”,

XER”,

(1.1) (1.2)

where to E R. Let a(n) denote 30 if n = 1 or 2 and (n + 2)/(n - 2) if n 2 3. There are many papers concerning the global existence of solutions for (l.l)-(1.2) and the solutions of (l.l)-(1.2) have been constructed in various function spaces (see, e.g., [l-3, 10, 15-191). In this paper we consider the solution u(t, x) of (l.l)-(1.2) in the following class:

c(w, H*(w)) n cl(R; L*(w)).

(1.3)

The solution in (1.3) of (l.l)-(1.2) is called a global strong solution of (l.l)-(1.2). In [3] Ginibre and Velo show that when n Z 1 and 1 < p < m(n), problem (l.l)-(1.2) has a unique global weak solution in C(R; H’(R”)) for any u,, E H’(R”). Naturally the following question arises: When 1 < p < a(n) and u. E H*(l.IY), does the weak solution constructed in [3] belong to the class (1.3)? In other words, this is the problem of existence of global strong solutions for (l.l)-(1.2). In [l] B ai‘11 on, Cazenave and Figueira show that if 1 5 n ZZ3 and 1 < p < a(n), Problem (l.l)-( 1.2) has a unique global strong solution in (1.3) for any u. E H’(Z). However, there seem to be scarcely any papers concerning the existence of global solutions of (l.l)(1.2) in higher spatial dimensions. The proof in [l] is based on HZ@“) GLX(R”),

lSnS3,

(1.4)

that is, Hz@“) is continuously imbedded into L”(R”) for 15 n Z 3. (1.4) does not hold if n 2 4. Furthermore, since 1 < p < a(n), p is nearly equal to 1 in higher dimensions. Therefore, in higher dimensions we can expect at most that the nonlinear term ]u]P-‘u is Lipschitz continuous with respect to u. These facts make it difficult to consider the existence of global strong solutions of (l.l)-(1.2) in higher dimensions. For example, these difficulties seem to prevent us from the application of the well known abstract local existence theorems (see, e.g., Kato [8,9] and Segal[14]) to problem (l.l)-(1.2) in higher dimensions, while in [2] and [lo] Segal’s theorem [14] is used in order to obtain the local solution of (l.l)-(1.2) for 1 s n s 3. 1143

1144

YOSHIO

Tsursuwr

In this paper we show that when 1 < p < cu(n) and n 2 1, problem (l.l)-( 1.2) has a unique global strong solution in (1.3). In our proof the Strichartz space-time estimate plays an important role (for the Strichartz estimate, see Strichartz [13, corollary in Section 31 and Ginibre and Velo [4, proposition 71). Our main result in this paper is the following. THEOREM1.1. Assume that 1 C p < a(n) and n 2 1. For any ldoE H’(lP) there exists a unique global strong solution u(t) in the class (1.3) of (l.l)-(1.2) satisfying the equation (1.1) in L*(w). Our plan in this paper is as follows. In Section 2 we summarize several lemmas needed for the proof of theorem 1.1. In Section 3 we give proof of theorem 1.1. We conclude this section with some notations given. Let U(r) = eib’ denote the evolution operator associated with the free Schrodinger equation. We abbreviate L4(R”) and H”(W”) to Lq and H”‘, respectively. (e, -) denotes the scalar product in L2. We define the norms of f-P and HZ as follows:

Let h(x) be an even and positive function in CT(Iw”) with ]jh]jL~= 1. We put h,(x) = j”h(jx), j=l,2,... * denotes the convolution with respect to spatial variables. We put f(0) =

(up-‘0,

fj(U) = hj *f(hj *

(1.6)

u>

(1.7)

for any “nice” function u from iw”to @. For a closed interval I in R and a Hilbert space H with the scalar product ( - , -)H, &(I; H) denotes the set of all functionsf(t) from I to H such that for all u E H, (f(f), u)~ is continuous from I to d=. For 1 $p 5 x, an open interval I in R and a Banach space X with the norm I(./Ix, LJ’(Z;X) d enotes the Banach space consisting of strongly measurable X-valued functions f(t) defined on I such that ]/j(t)]], E L”(Z). For a complex number z we denote the real part and the imaginary part of z by Re z and Im z, respectively. In the course of the calculations below various constants will be simply denoted by C. In particular, C = C(*, . . . , *) will denote a constant depending only on the quantities appearing in parentheses. 2. LEMMAS In this section we give several lemmas needed for the proof of theorem 1.1. We first describe the following two lemmas concerning the free evolution operator

U(t).

LEMMA2.1. Let q and r be positive numbers such that l/q + l/r = 1 and 2 5 q 5 I. For any t # 0, U(f) is a bounded operator from L’ to Lq satisfying ](U(t)&g

5 (4n]f])(“‘4)-(“‘*) ](&‘,

u E L’,

and for any t # 0, the map t--, U(t) is strongly continuous. strongly continuous for all t E iw.

tfo,

(2.1)

For q = 2, U(t) is unitary and

1145

Nonlinear Schredinger equations LEMMA 2.2. Let 4 and r be positive numbers such that 1 s 4 - 1< cu(n) and (n/2) - (n/q)r

=

2. Then, lM+&‘(RL9)

(2.2)

VEL?,

5 CIM2,

where C = C(n, 9). Lemma 2.1 is well known (see, e.g., [3, lemma 1.2)). For lemma 2.2, see Stichartz [13, corollary 1 in Section 31 and Ginibre and Velo [4, proposition 71. In the proof of theorem 1.1 we shall use the regularizing technique of Ginibre and Velo [3]. We next summarize their results needed for the proof of theorem 1.1. We consider the following integral equations: K(f) = U(f - t,)u,

- i

t qt J-

- s)f(4@)

(2.3)

h,

(0

uj(t)

=

v(t

-

to)hj

*

UO

-

i

’ I

=

s).fj("j(s))

u(t-

dS

10 t--10

U(t--to)h,*u,-i

U(s)fj(uj(f-S))

i=

&,

1,2, . . .

(2.4)

i

(2.3) is the integral version of the initial v”,lue problem (l.l)-(1.2) equations associated with (2.3). For (2.3) and (2.4) we have the following two lemmas.

and (2.1) are the regularized

LEMMA 2.3. Assume that 1

IM0llL~ = II~OIIP9

(2.5)

(2.6) LEMMA 2.4. Assume that 1 < p < a(n). For any p > 0 there exists a T(p) > 0 depending only on n, p and p (but independent of j) such that for any u. E H’ with ]lr~O]lHlZ p (2.4) have unique solutions j = 1,2, . . . , in C([t, - T(p), to + T(p)]; LP+‘) satisfying the following: Uj(t),

Uj(f>

E

f7

Cl([fO

-

T(P)~

[O +

W))];

(2.7)

Hk),

k=l

IIUj(f)llL~= llhj * uOIILz~

f E

~llvuj(r>ll’lz + & = ~((Vhj * u~(]~z + &

auj_ j--at

AUj +fj(uj)

[to -

to +

nJ)l~

(Ihj * uj(f)llf_Trl

(Jhj * hj * uOIJ~Z’~ 7 in

T(P),

Gw (2.9)

t E [to - T(P)> ‘0 + T(P)],

Hm,m h 1, tE [to - T(p),to + T(p)].

(2.10)

1116

YOSHIO

-h,TSLhlI

Furthermore, let u(t) be the solution of (2.3) given by lemma 2.3 with the initial datum at r = t0 equal to the above uu. Then, u,(f)*

u(t)

in

C([r, - T(p), to + T(p)]: LP+‘) (i+

x).

For lemma 2.3, see [3, theorem 3.11 and for lemma 2.4, see [3, propositions and 3.31. At last we give two fundamental lemmas.

(2.11) 2.4, 3.1, 3.2

LEMMA2.5. The convolution with h,(x) is a contraction in L9 for all q, 15 q S =, and in H” it commutes with U(t) and (I+ * K, u) = for all nonnegative integers N. Furthermore, (U, h, * u) for all K, u E L2. Lemma 2.5 is clear. LEMMA2.6. Assume that 1

and for 1

pO. Proof. We first note that for u E Hz

a2 Iiax,dxi”

Ii L!

5 CllAu(lLz,

1 Zi,jzz’.

Therefore, (2.12) follows directly from the Gagliardo-Nirenberg inequality (for the GagliardoNirenberg inequality, see, e.g., [ll]). Since 1 >p6 and ]]u]]~P+Is C]]U]]~~for 1

0 so that p is not smaller than the right-hand side of (3.1). For this p, let T(p) be defined as in lemma 2.4. By lemma 2.4 we have the solutions

Nonlinear

Schrddinger

u;(t),j= 1,2,. . . , of (2,4) satisfying (2.7)-(2.11). theorem we note that l/u,(0llH1 5 C0 (llh, * U&l

1117

equations

By (2.8), (2.9) and the Sobolev imbedding + 1(/l,* h! * t&P;”

‘)

S CO((ILLO(JHl + //UoI((Hpl+) 5 p, fE

[l” - T(P),to+ T(P)17 j = 1,2,

(3.2)

.. ,

where Co is defined in (3.1). Let 4(P+1) r = n(p - 1) throughout this section. For any T > 0 we denote an open interval (to - T, to + T) and a closed interval [to - T, to + T] by IT and jr, respectively. We first prove the following lemma. LEMMA 3.1. If u. E H?, there exist two positive constants ,1f and T depending only on n, p and p such that 0 < T s T(p) and

u(t) E C(I,; HZ),

(3.3)

Illf(t$/? % M(l + 7-q r/2(1 + l\U&/I).

(3.4)

tE&,

where , =

l

n + 2 - (n - 2)p 2(P + 1)

.

Proof. We divide the proof into two steps. Step 1. We prove (3.4). For that purpose, we first show that there exist two positive constants K and T depending only on n, p and p (but independent of j) such that 0 < T 5 T(p) and a at /I

Uj

5 K hh,~

!IL’(/T:/Y+‘)

t

j= 1,2,.

(3.5)

..

By lemma 2.4 we differentiate (2.4) in f and take the L’(I,; LP+‘) norm for any T with 0 < T 5 T(p). Then we have by (2.1) and (2.2)

a at

/I /I Uj

Ly/r;P+l)

5

IlU( *)(iAhj *

S

CI(Ahj * UO -fi(h,

U-0 -

if,(hj * UO)IIIL~(IT:LP+')

* UO)[[LI

+ C

,

j=l,2

,...)

L’(IT)

(34

1148

Tsursu~~

YOSHIO

for any T with 0 < T 5 T(p). By J, and Jz we denote the first term and the second term at the right-hand side of (3.6), respectively. By (2.13) and the fact that I/u,,/]~I5 p we have J, 5 C(&&~

+ ~~%&zp)

lI~011~~1114hf~)

$ C(IMLf~ +

25cll%ll”~ 9

(3.7)

where C = C(n, p, p). We put

uf(s) = 1

0, d z

S@J-r> (3.8) uj(s>T

s E

IT.

Then, by (3.2), the Soboiev imbedding theorem and the generalized e.g., [12]) we have J, S CpP-’

Young inequality (see,

(ni2)+fni(P+t))IJUf(S)liLp_, & L'W

0 c T s T(p),

(3.9)

Li(17;LP+1)’

where r=

4(P + 1) n + 4 - (n - 4)p

and C = C(n,p, p). At the second inequality the generalized

l/r = l/’ +

Young inequality with

4P - 1) _ 1 2(p+l)

.

Since r > f for 1 < p < a(n), we have by (3.9) and Holder’s inequality >

j=1,2

,.

(3.10)

. .,

L’(l7;LPf’)

where ;=

n + 2 - (n - 2)p 2(P + 1)

and Co = C,,(n,p, p). By (3.7) and (3.10) we obtain S ~lK~ll~2 + Co L'(IT;LP+')

- d TY -

Kj at IiL’(fT;LP-‘) (1

j=

1,2,. . . t

(3.11)

for any T with 0 < T 5 T(p). Since jj > 0 for 1 < p C n(n), we can choose T > 0 so small that C, TV d min(l, T(p)).

(3.12)

1119

Nonlinear Schriidinger equations

Therefore, we have (3.5) by (3.11) and (3.12). We next show (3.4) by using (3.5). Multiplying (2.10) by -A(a/Ltt)u,(r).

+

we have

(.fj(u,(t>),-Atru,(t) i IT(~), j= 3

1.2,.

tE

(3.13)

..

We take the real part of (3.13) to obtain ~((AUj(t)(]~~= allAh, * uOI(~L~ + j’Re(fi(ui(s)),-A~uj(r))dr,r~I,,,,

(3.14)

j=l,2,....

f0

by parts in t

By J we denote the second term at the right-hand side of (3.14). By integration we have

J = ReCfj(uj(t)),

-Au,(f))

- Re(fj(hj * CL,), -Ahj * ~0) + J”’ Re($fj(ui(s)), (0 By Schwarz’s inequality,

(3.15)

ALLj(s)) b, t E jr(p).

(2.12), (3.2) and the Sobolev imbedding theorem we have

I(fi(uj(t))T-Auj(t))I 5 IIfj(Uj(r))llL?llAUj(t)IIL2 S

IIuj(f)lJP,?pIIAUj(t)llL2 ~5 C~(‘-~)PllA~~j(r)ll~~+’ ~C+QIIAuj(t)lli2~ ~EJT((,),j= 1,2,. . . ,

(3.16)

where 8 is defined in lemma 2.6 and C = C(n,p, p). At the last inequality Young’s inequality with 1-pB+pB+l -----=I 2 2

we have used

*

since pB < 1 for 1

j = 1,2,.

where C = C(n, p, p). On the other hand, since we have by integration

=

i(l~(“j(t))llZL2 - Ilfj(hj * u0>llt2>~

.. )

(3.17)

by parts

(3.18)

1150

YOSHIOTSUTSti?dI

we have by lemma 2.5, Hiilder’s inequality,

+ it IlAuollt~

(2.10) and (3.2)

‘E&&j=

1,2,.

.. ,

where B is defined in lemma 2.6 and C = C(n,p, p). Therefore, us

(3.19)

(3.16)-(3.17)

and (3.19) give

(3.20) where C = C(n,p, p). By combining (3.14) and (3.20) we obtain

where C = C(n, p, p). On the other hand, HGlder’s inequality and (3.5) give us

5

K2TYJIUoll~? )

fEIT>

j = 1,2,.

(3.22)

.. ,

where n + 2 - (n - 2)p Y=

2(P + 1)

’

C = C(n,p, p) and K and Tare defined in (3.5). From (3.21) and (3.22) it follows that ~~Aui(f)~~~~ 5 C(1 + Tv)(l + ~~u,,~~~~), fEI,,

j= 1,2,.

. .,

(3.23)

Nonlinear

where C = C(n,p,

p). Combining

Schradinger

1151

equations

(2.5) and (3.23), we have

/U,(f)ll& I C(1 + Ty)‘ql

(3.24)

j= 1,2,. . . ,

Gl,,

+ (IU(){(,y’),

where C = C(n, p, p). Lemma 2.4 and (3.2) imply that K(C)E ti for all f E 7, and ui(f) Therefore,

u(r) weakly in H’(j-

td,.

=).

(3.25)

we have by (3.24) lIU(f)ll/f’ 5 !i: 5

C(1

ll”j(fNH’

+

IE i,.

T’)“2(1 + flu&:),

(3.26)

This shows (3.4). Step 2. We next prove (3.3). We easily see by (3.26) and the strong continuity that u(t) E On the other hand, by (3.5), (3.14)-(3.15),

in H’ of u(t)

c,(i,; ff?).

(3.27)

(3.19) and (3.24) we have

IlAUj(t)llZz ~ (IAhj * u~~I'L~ + 2l(fj(uj(t)), -Auj(r))

- (f,(hj *UO)>

-Ahj

+ IIlfj(uj(m2 - llfi(hj * hdll’L4? tE

*

UO)I + Clt- tol’

Imy,j=

1,2,.

. ,

(3.28)

where n + 2 - (n - 2)p Y=

2(P + 1)

and C = C(n, p, p, T, lluollHz). (2.12), (3.24) and (3.26) give us

Ilfj(“j(f))-f(“(f>)llL2 s IIhj* {f(hj * uj(‘))-f((f)))ll~1 + IIhj*f(u(t>)- f(u(t>)II~z 2 C(]Jhj * uj(t)Ilf.G1 + lIU(t)l]PL%‘)IIhj * uj(t>- u(f)lI~zP + IIhj*f(u(l>)-f(u(t))II~2 S

CJlhj * Uj(r) - u(t)ll~P61llhj * u(t) - ~(t)llk~ +

IIhj*f(u(t)) -f(u(t))lIL~

4 C(llhj

* Uj(t) - hj * u(~)\~LPcI + (Ihj *U(f)

IIhj*f(u(f)) -f(u(t>)II~2 +I f llhj *u(t) 5 c(lluj(r)- U(f)llLP

-

U(~)/\LP+~)‘-’

+

+

IIhj*f(u(t)) -f(u(t))llL~7

- U(t)ll~~+l)~-'

si,,

j = 1,2,.

.. ,

(3.29)

where 8 is defined in lemma 2.6 and C = C(n,p, p, T, ~~uO~jH~). By (3.29), lemma 2.4 and the fact that f(u(t)) E L* we obtain fj(Uj(t))

in L’(j+

+f(U(t>)

z). t E ?,.

(3.30)

(3.25) and (3.30) give us (A(uj(t)>r -Au,(t))

* (f(u(t)),

-Arl(r))

(j+

%).

(3.31)

r~).

(3.32)

In the same way we also have (fi(hj * UO>,-AhI * UO)- (f(uo>T -Alc~)(j+ Therefore,

letting j -+ ‘3~in (3.28), we obtain by (3.25) and (3.31)-(3.32) IIAu(t)llil 5 ,J~mllAUj(r)/l~2 5 IlA~ok

+ 21(f(u(t))T -Au(t))

+ Clt - f0 I’/ +

lllfwN~

- (f(~&

-Au,)1

- lIfbdllt4

fEJT.

(3.33)

In the same way as (3.29) we obtain by (2.12) and (3.24)

llf(W

- f(UO>IIL?

s Cllw - %llL-HI.

TVIT,

(3.34)

where 8 is defined in lemma 2.6 and C = C(n, p, p, T, lluol[H:). (3.34), the Sobolev imbedding theorem and the strong continuity in H’ of u(t) imply that f(u(t)>--+f(u0)

in L*(t+

to).

(3.35)

By (3.27), (3.33), (3.35) and the fact that y > 0 for 1

jjAu(r)ll:2

Z

))Au&:.

(3.36)

(3.37)

(1.5) and the strong continuity in H’ of u(t) give us ;i_ii Il40llff~= Il%A4’~

(3.38)

(3.38) and (3.27) imply the strong continuity in H’ of u(t) at t = to. For each t E iT we can use the same argument as above with the initial time to and the initial datum uo replaced by f and u(t). Therefore, we obtain (3.3). Now we are in a position to prove theorem

1.1.

Proof of theorem 1.1. We first note that in lemma 3.1 the size of T depends only on n, p and [~L~~~~~I but does not depend on JJuOIIH~. since Ilu(t)llH~ is bounded by p for any t E 54, by using lemma 3.1 repeatedly we obtain u(t) E C(Z,; H’),

(3.39)

1153

Nonlinear SchrZjdinger equations

for any R > 0. By lemma 2.3 and (2.3) we easily see that arc/Jr(t) E C(R; H-‘) and dU

i~=-A~+lu/P-~u

in H-*.r~ifJ

(3.40)

Since (3.39) and the Sobolev imbedding theorem imply the strong continuity in L? of the righthand side of (3.40) with respect to the time variable t, it follows that for any R > 0

$ (t) E C(I,; (3.39) and (3.41) complete the proof of theorem 4. CONCLUDING

L?). 1.1.

(3.41)

n

REMARKS

(1) When the sign of the nonlinear term in (1.1) is negative, theorem 1.1 also holds for 1

in proof

After this paper was accepted, T. Kato improved theorem 1.1. That is. he proved that theorem 1.1 holds valid for the nonlinear Schriidinger equation with more general nonlinearity, for example, a sum of several power nonlinearities. For the details, see [20]. His proof is based on the space-time behaviour of the free Schrddinger group and the contraction mapping principle. REFERENCES 1. BAILLONJ. B., CAZENAVET. & FIGUEIRAM., fiquation de Schr(idinger nonlineaire,

C. r. Acad. Sci.. Paris 284.

869-872 ( 1977).

2. BREZIS H. & GALLOUETT., Nonlinear Schredinger equations. Nonlinear Analysis 4. 677-681 (1980). 3. GINIBREJ. & VELO G., On a class of nonlinear SchrBdinger equations. I: the Cauchy problem. /. fnnct. Analysis 32. 1-32 (1979). 4. GINIBREJ. & VELO G., Theorie de la diffusion dans I’espace d’inergie pour une classe d’iquations de Schriidinger non lineaire, C. r. Acad. Sci., Paris 298, 137-140 (1984). 5. GINIBREJ. & VELO G., Scattering theory in the energy space for a class of nonlinear Schradinger equations /. Math. pures appl. 64, 36HOl (1985). 6. HAYASHIN. & TSUTSUMIM., L’(IW”)-decay of classical solutions for nonlinear Schr(idinger equations (to appear).

7. HAYASHI N., NAKAMITSUK. & TSUTSUMIM., On solutions of the initial value problem for the nonlinear Schriidinger equations J. funcl. Analysis 71, 218-245 (1987). 8. KATO T., Quasilinear equations of evolution with applications to partial differential equations, Lecture Notes in Mathematics 448, 25-70, Springer, Berlin (1975). 9. KATO T., Linear and quasi-linear equations of evolution of hyperbolic type, C.I.IM.E. II, 127-191 (1976). 10. LIN J. E. & S-~RAUSSW. A., Decay and scattering of solutions of a nonlinear Schr6dinger equation. J. funcf. Analysis 30, 245-263

(1978).

11. NIRENBERGL., On elliptic partial differential equations, Anna/i Scu. norm. sup. Pisa. 13, 115-162 (1959). 12. REED M. & SIMONB., Methodr of Modern Mathematical Physics, Vol. II, Fourier Analysis and Self-adjoinmess. Academic Press, New York (1975). 13. STRICHARTZR. S., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44, 705-714 (1977). 14. !&GAL I., Non-linear semi-groups, Ann. Mafh. 78, 339-364 (1963). 15. TSUTSUMIM., Weighted Sobolev spaces and rapidly decreasing solutions of some nonlinear dispersive wave equations, J. diff. Eqns 42, 26281 (1981).

1154 16. TWTSGMI

YOSHIOTsursuw~

M. & HAYASHIN.. Classical solutions of nonlinear SchrBdinger in higher dimensions. Mafh. ” eauations . 2. 177. 217-23-I (1981). 17. TSY~SUMIY., L2-solutions for nonlinear Schrddinger equations and nonlinear groups. Funkcialaj Ekuacioj (to appear). 18. GINIBREJ. & VELO G., On the global Cauchy problem for some nonlinear Schriidinger equations, Ann. 1. H. P. (Analyre Non Linttaire) 1. 309-323 (1984). 19. HAYASHIN., Classical solutions of nonlinear Schradinger equations, Manuscripta Muh. 55, 171-190 (1986). 20. KATO T., On nonlinear Schriidinger equations, Ann. I. H. P. (Phys. ThPor.). 46, 113-129 (1987).