The Lax-Mizohata theorem for nonlinear gauge invariant equations

Nonlinear Analysis 49 (2002) 159 – 175

www.elsevier.com/locate/na

The Lax-Mizohata theorem for nonlinear gauge invariant equations Karen Yagdjian Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Received 27 April 1999

Keywords: Cauchy problem; Nonlinear equations; Lax-Mizohata theorem; hyperbolicity

0. Introduction The Lax-Mizohata theorem for linear equations. Consider the partial di1erential equation
Dtm u + aj;  (t; x)Dtj Dx u = f(t; x) (0.1) j+||≤m; j¡m

with coe4cients aj;  ∈ C ∞ ([0; T ] × Rn ), t ∈ [0; T ], T ¿ 0, x ∈ Rn . Here Dt = −i@t = −i@=@t, Dx = (−i)|| @x = (−i@=@x1 )1 · · · (−i@=@x n )n ,  is a multi-index. The characteristics of Eq. (0.1) are de:ned as solutions  = j (t; x; ), j = 1; : : : ; m, of the equation
m + aj;  (t; x) j  = 0: j+||=m; j¡m

Consider the Cauchy problem for Eq. (0.1) with data prescribed at t = 0: @lt u(0; x) =

l (x);

l = 0; : : : ; m − 1:

(0.2)

The problem is said to be well-posed (in the corresponding space of functions) if for every given data from that space there exists a solution, the solution is unique and continuously depends on the data. The classical Hadamard’s example shows that the Cauchy problem for the Laplace equation is not well-posed in the space of C ∞ functions. Lax [11] and Mizohata [13] proved E-mail address: [email protected] (K. Yagdjian). 0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 0 ) 0 0 2 3 6 - 4

160

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

Lax-Mizohata Theorem. If the Cauchy problem is C ∞ well-posed; then all characteristics of the linear equation are real-valued. The equation (0.1) with real characteristics is called hyperbolic equation. Ivrii [5], Komatsu [10], and Nishitani [14] extended this theorem to the Gevrey and analytic classes. Gauge invariant equations. The equation L[u] = F({@kt @x u}k+||≤m−1 ; {@kt @x u}k+||≤m−1 ); where L is the mth order linear partial di1erential operator from (0.1), appears quite frequently in many articles and books on mathematical physics: • Ginzburg–Landau equations: stationary, −G + (| |2 − 1) = 0, non-stationary, i@t = −G + (| |2 − 1) (see [15]), • J8orgens equation utt − Gu = −|u|p−1 u (see [1,6]), • dissipative wave equation u + |@t u|2 @t u = 0 (see [16]), • nonlinear Schr8odinger equation iut + uxx ± u|u|2 = 0, • defocusing nonlinear Schr8odinger equation iut + uxx − u|u|4 = 0 (see [2]), • Hirota’s equation ut + iau + ib(uxx − 2|u|2 u) + cux + d(uxxx − 6|u|2 ux ) = 0. All these equations are invariant with respect to the gauge transformation, H = !F("; ") H F(!"; !H")

for all ! ∈ U (1) (i:e: ! ∈ C; |!| = 1); " ∈ Cn" :

Furthermore, nonlinear terms of these equations possess the following behaviour in the neighbourhood of the origin H |F("; ")|=|"| →0

as " → 0:

In the present paper we prove the Lax-Mizohata theorem for the nonlinear gauge invariant equations.

1. Main results We consider the partial di1erential equation j  j  @m t u + F(t; {@t @x u}j+||≤m; j¡m ; {@t @x u}j+||≤m−1 ) = 0;

(1.1)

where the function F is assumed to be continuous with respect to t ∈ J :=[0; T ], and k  H to vectors ":={@kt @x u}j+||≤m; j¡m ∈ Cn" and ":={@ H j+||≤m; j¡m ∈ Cn" . t @x u} We assume that Eq. (1.1) is invariant with respect to gauge transformation, that is H is said a function F is gauge invariant in the following sense. A function F = F(t; "; ") to be gauge invariant if H = !F(t; "; ") H F(t; !"; !H")

for all ! ∈U (1) (i:e: ! ∈C; |!| = 1); " ∈Cn" ; t ∈[0; T ]:

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

161

Taking into account a feature of the Cauchy problem for the hyperbolic equations we require the existence of the cone of dependence in the problem under consideration. Let K& (x0 ; t 0 ):={(t; x) ∈ [0; T ] × Rn ; |x − x0 | ≤ &(t 0 − t)} denote the cone with vertex (x0 ; t 0 ) and slope & ¿ 0. Then, let D& (x0 ; t 0 ):={x ∈ Rn ; |x − x0 | ≤ &t 0 } be the base of the cone K& (x0 ; t 0 ). In what follows || · ||C s (K& (x0 ;T )) and || · ||C s (D& (x0 ;T )) denote the uniform norms of the spaces C s (K& (x0 ; T )) and C s (D& (x0 ; T )), respectively. First we consider a quasilinear equation. 1.1. Quasilinear equations Now we restrict ourselves to the partial di1erential equation
Dtm u + aj;  (t; {@kt @x u}k+||≤m−1 ; {@kt @x u}k+||≤m−1 )Dtj Dx u j+||=m; j¡m

= F(t; {@kt @x u}k+||≤m−1 ; {@kt @x u}k+||≤m−1 )

(1.2)

whose coe4cients aj;  and nonlinear term F are assumed to be continuous with respect k  to t ∈ [0; T ], and to vectors p:={@kt @x u} ∈ Cnp and p:={@ H H ∈ C np . t @x u} Denition 1.1. The solution u(t; x) ≡ 0 to the problem (1.2), (0.2) is said to be stable solution of the problem (1.2), (0.2) if there are positive numbers T , &, M , and (0 such that for every positive ( ≤ (0 and for every x0 ∈ Rn there exists a positive number h(() such that for every vector K = ( 0 ; 1 ; : : : ; m−1 ) ∈ C0∞ (Rn ) of initial data satisfying ||K||C M (D& (x0 ;T )) ≤ h(() there exists a unique solution u = uK (t; x), uK ∈ C m (K& (x0 ; T )), and that solution satis:es ||uK ||C m−1 (K& (x0 ;T )) ≤ (. Remark 1.1. If u ≡ 0 is a stable solution to (1.2), (0.2), then F(t; 0; 0) = 0 for all t ∈ [0; T ]. To justify our de:nition of stable solution let us recall the following well-known results for hyperbolic equations and systems. First of all we mention that in a local existence theorem for strictly hyperbolic equations (see e.g. [4]) the interval [0; T ] of the existence of the solution u (lifespan of u) depends only on the norms of the Cauchy data. Moreover, for a given T one has a solution of the Cauchy problem on [0; T ] provided that the Sobolev norms || 0 ||(s+1) and || 1 ||(s) are small enough, for some integer s ¿ n=2 + 1. Similar results for the quasilinear symmetrizable hyperbolic system and, in particular, for the quasilinear strictly hyperbolic systems, can be found in many books (see, e.g. Li Ta-tsien [12], Kichenassamy [8]). The Cauchy problem for the quasilinear weakly hyperbolic equations is considered by Kajitani and Yagdjian [7], and for fully nonlinear second-order equations by Dreher and Reissig [3], and with small analytic data for fully nonlinear :rst-order systems by Kinoshita [9]. Spagnolo [17] gives a survey of this problem. One can check that for all above-mentioned equations the solution u(t; x) ≡ 0 to the Cauchy problem is a stable solution in the sense of De:nition 1.1.

162

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

Then we introduce some counterexamples to the stability of the trivial solution. Example 1.1 (Kajitani and Yagdjian [7]). The function u(t; x) ≡ 0 solves the equation utt + t 2l uxx + t 2l (ux )2 + (ut )2 = 0

(1.3)

and takes vanishing initial data. For given initial data the solution is unique in C 2 space. Then for every given time interval [0; T ] and positive number s, and for every (-neighbourhood of the initial data (0; 0) in the space C s , there exist initial data from this neighbourhood such that the solution to (1.3) does not exist in C 2 (K& (0; T )). We conclude that the trivial solution u(t; x) ≡ 0 to the Cauchy problem for (1.3) is not a stable solution. The cause of instability in that example is the nonhyperbolicity. Example 1.2 (Kajitani and Yagdjian [7]). Similar consideration of the weakly hyperbolic equation utt − t 2j uxx − t k ux − t 2j (ux )2 + (ut )2 = 0 with k ¡ j − 1 and j ¿ 1, leads to the same conclusion although in this example the cause of instability is the violated Levi condition [18]. Now we are in position to formulate the Lax-Mizohata theorem for the quasilinear equations: H be a continuous function de>ned for all " ∈ Cn" ; and all Theorem 1.1. Let F(t; "; ") t ∈ [0; T ]; which is invariant with respect to gauge transformation; H = !F(t; "; ") H for all ! ∈ U (1); " ∈ Cn" ; t ∈ [0; T ] (1.4) F(t; !"; !H") and such that with some positive numbers h and C0 H ≤ C0 |"| for all " ∈ Cn" ; |"| ≤ h; t ∈ [0; T ]: |F(t; "; ")| (1.5) H are invariant with respect Moreover; suppose that the continuous functions aj;  (t; "; ")" to gauge transformation; H = aj;  (t; "; ") H for all ! ∈ U (1); " ∈ Cn" ; t ∈ [0; T ] aj;  (t; !"; !H") H are Lipschitz continuous at the origin; and that functions aj;  (t; "; ") H − aj;  (0; 0; 0)| ≤ C0 (t + |"|) when |"| ≤ h; t ∈ [0; T ]: |aj;  (t; "; ") If the solution u(t; x) ≡ 0 is a stable solution to the Cauchy problem (1:2); (0:2) possessing the cone of dependence; then all roots of the characteristic equation
m + aj;  (0; 0; 0)  j = 0 (1.6) j+||=m; j¡m

are real for every  ∈ Rn . In De:nition 1.1 one can replace the uniqueness condition and existence in the cone of dependence K& (x0 ; T ) by local uniqueness and by local existence, respectively. Thus Theorem 1.1 remains true if we replace De:nition 1.1 by the following one

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

163

Denition 1.2. The solution u(t; x) ≡ 0 to the problem (1.2), (0.2) is said to be stable solution of the problem (1.2), (0.2) if there are positive numbers T; M , and (0 and neighbourhoods O; O1 ⊂ Rn of the origin such that for every positive ( and T1 , ( ≤ (0 , T1 ≤ T , there exists a positive number h(() such that for every vector K = ( 0 ; 1 ; : : : ; m−1 ) ∈ C ∞ (O) of initial data satisfying ||K||C M (O) ≤ h(() there exists a unique solution u = uK (t; x), uK ∈ C m ([0; T1 ] × O1 ), and that solution satis:es ||uK ||C m−1 ([0; T1 ]×O1 ) ≤ (. Remark 1.2. One can relax in De:nition 1.1 the condition on the uniqueness of solution in C m if one requires more smoothness from coe4cients aj;  and from the right-hand side F. For example, if aj;  and F are C ∞ functions, then in order to prove Theorem 1.1 the uniqueness of solution in the space C ∞ is enough. For Eq. (1.2) with coe4cients aj;  = aj;  (t) depending on time only the last theorem is proved in [19]. 1.2. Fully nonlinear equations The standard reduction of the fully nonlinear equation to the system of quasilinear equations allows to consider these ones in the similar way. Denition 1.3. The solution u(t; x) ≡ 0 to the problem (1.1), (0.2) is said to be stable solution of the problem (1.1), (0.2) if there are positive numbers T; &; N , and (0 such that for every positive ( ≤ (0 and for every x0 ∈ Rn there exists an h((), h(() ¿ 0, such that for every vector K = ( 0 ; 1 ; : : : ; m−1 ) ∈ C0∞ (Rn ) of initial data satisfying ||K||C N (D& (x0 ;T )) ≤ h(() there exists a unique solution u=uK (t; x), uK ∈ C m+1 (K& (x0 ; T )), and that solution satis:es ||uK ||C m (K& (x0 ;T )) ≤ (. De:ne numbers aj;  , j + || = m, by   @F(0; {@kt @x u}k+||≤m; k¡m ; 0) aj;  := @(@tj @x u) {@k @ u} t

x

: k+||≤m =0

H ∈ C([0; T ] × Cn" × Cn" ); for Theorem 1.2. Let F; @F=@t; @F=@(@tj @x u); @F=@(@tj @x u) every j; ; j + || ≤ m. Assume that Eq. (1:1) is invariant with respect to gauge transformation. If the solution u(t; x) ≡ 0 is a stable solution to the Cauchy problem (1:1); (0:2); then all roots of the characteristic equation
m + aj;    j = 0 (1.7) j+||=m; j¡m

are real for every  ∈ Rn . Now one can easily generalise above results to the fully nonlinear systems. We formulate this result to give a complete picture. Thus consider a system of the :rst-order

164

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

partial di1erential equations @t u1 + F1 (t; {@x u1 }||≤1 ; : : : ; {@x um }||≤1 ; u 1 ; : : : ; u m ) = 0; .. . @t um + Fm (t; {@x u1 }||≤1 ; : : : ; {@x um }||≤1 ; u 1 ; : : : ; u m ) = 0;

(1.8)

where the functions F1 ; : : : ; Fm are continuous with respect to t ∈ J :=[0; T ], and to vectors ({@x u1 }||≤1 ; : : : ; {@x um }||≤1 ) ∈ Cm(1+n) and (uH 1 ; : : : ; uH m ) ∈ Cm . Denote u(t; x):= (u1 (t; x); : : : ; um (t; x)). We assume that (1.8) is invariant with respect to gauge transformation, that is the functions F1 ; : : : ; Fm are gauge invariant. For the system (1.8) consider the Cauchy problem with data ul (0; x) =

l (x);

l = 1; : : : ; m:

(1.9)

Denition 1.4. The solution u(t; x) ≡ 0 to the problem (1.8), (1.9) is said to be stable solution of the problem (1.8), (1.9) if there are positive numbers T; &; M , and (0 such that for every positive ( ≤ (0 and for every x0 ∈ Rn there exists an h((), h(() ¿ 0, such that for every vector K = ( 1 ; 1 ; : : : ; m ) ∈ C0∞ (Rn ) of initial data satisfying ||K||C M (D& (x0 ;T )) ≤ h(() there exists a unique solution u = uK (t; x), uK ∈ C 2 (K& (x0 ; T )), and that solution satis:es ||uK ||C 1 (K& (x0 ;T )) ≤ (. (k) ; j; l = 1; : : : ; m, by De:ne matrices Ak ; k = 1; : : : ; n, with elements ajl   @Fj (0; {@x u1 }||≤1 ; : : : ; {@x um }||≤1 ; 0; : : : ; 0) (k) ajl := : @(@xk ul ) {@ u1 }||≤1 =···={@ um }||≤1 =0 x

x

Theorem 1.3. Let Fj ; @Fj =@t; @Fj =@(ul ); @Fj =@(ul ); @Fj =@(@xk ul ) ∈ C([0; T ] × Cm(2+n) ); while @Fj =@(@xk ul ) are Lipschitz continuous at the origin. Assume that the system (1:8) is invariant with respect to gauge transformation. If the solution u(t; x) ≡ 0 is a stable solution to the Cauchy problem (1:8); (1:9) possessing the cone of dependence; then all roots of the characteristic equation   n
det I + Ak  k = 0 k=1

are real for every  ∈ Rn . 2. Proof of Theorem 1.1 2.1. Choice of initial data for the solutions, violating stability For the simplicity we consider the neighbourhood of the origin, x = 0, t = 0, and ˆ = 1. The proof will be given by contradiction. To we set  = 0ˆ ∈ Rn ; 0 ∈ R+ ; || this end we assume that u(t; x) ≡ 0 is a stable solution of the problem in the sense of

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

165

ˆ of the principal part of De:nition 1.1 and that there exist ˆ ∈ Rn and some zero l () Eq. (0.1), that is a solution to equation
 m + aj;  (0; 0; 0)ˆ  j = 0 j+||=m; j¡m

ˆ = 0. which has non-vanishing imaginary part, Im l () Then we choose the vector K = ( 0 ; 1 ; : : : ; m−1 ) of initial data as follows: k (x)

= eix· 0k+1−m−M −3 1(x)dk ;

k = 0; : : : ; m − 1;

(2.1)

where 1 ∈ C ∞ is cuto1 function, 1(x) = 1 if |x| ≤ R, 1(x) = 0 if |x| ≥ 2R, constants dk ∈ C will be chosen later, M is of De:nition 1.1, while 0 is a large parameter. For every given positive number h(() we have ||K||C M (D& (x0 ;T )) ≤ h(()

(2.2)

if 0 = || is large enough. Then, according to De:nition 1.1, there exist positive numbers T , &, M , and (0 such that for every positive ( ≤ (0 there is a positive h(() such that for every vector of initial data, K = ( 0 ; 1 ; : : : ; m−1 ), of (2.1) with (2.2), there is a unique solution uK ∈ C m (K& (x0 ; T )), and moreover, ||uK ||C m−1 (K& (x0 ;T )) ≤ (: 2.2. Ordinary diAerential equation with the parameter Consider now the vector-valued function   m−1   0 u˜ U1  U2   0m−2 @t u˜      U =t (U1 ; U2 ; : : : ; Um ) =  .  =  : .. .   .   . Um @m−1 u ˜ t One can write for U a following system of ordinary di1erential equations: dU = 0A(t; ; U; UH )U + R(t; ; U; UH ); dt where the matrix A(t; ; U; UH ) is the following:  0 1 .. ..   . .  A(t; ; U; UH ):=  0 0 

 a˜  0−m a˜  0−m+1 0; 

||=m

1; 

||=m−1

 ::: 0 ..  ..  . .  ; :::
1  ::: a˜m−1;   0−1  ||=1

a˜l;  = a˜l;  (t; {(i) @kt u}k+||≤m−1 ; {(−i) @kt u}k+||≤m−1 );

l = 0; : : : ; m − 1:

166

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

Here a˜j;  :=(−1) j+1 im+j aj;  . The vector-valued function R(t; ; U; UH ) is given by R(t; ; U; UH ):=im F(t; {(i) @kt u}; ˜H m ; em :=t (0; : : : ; 0; 1); ˜ {(−i) @kt u})e ˆ  0||+k+1−m Uk+1 })em : ˆ  0||+k+1−m Uk+1 }; {(−i) R(t; ; U; UH ) = im F(t; {(i) Let us denote by 1 (); 2 (); : : : ; m (), the roots of the equation
m + a˜j;  (0; 0; 0)  j = 0: j+||=m; j¡m

One can order these roots as follows: Re 1 () ¿ 0; Re 2 () ¿ 0; : : : ; Re r1 () ¿ 0; Re r1 +1 () ¡ 0; Re r1 +2 () ¡ 0; : : : ; Re r1 +r2 () ¡ 0; Re r1 +r2 +1 () = 0; Re r1 +r2 +2 () = 0; : : : ; Re m () = 0:

(2.3)

Due to the assumption of the theorem, we can suppose that either the set {1 (); : : : ; r1 ()} or set {r1 +1 (); : : : ; r1 +r2 ()} is not empty. Consider the case with r1 ≥ 1. Let S() be a matrix reducing the principal part A(0; ; 0; 0) to Jordan form: S()−1 A(0; ; 0; 0)S() = D() + L(); where D() is a diagonal matrix with elements 1 ()=0; 2 ()=0; : : : ; m ()=0, while L() is a block matrix consisting of matrices Lj (); j = 1; : : : ; r, of the form   0 lj 0 : : : 0  0 0 lj : : : 0    |Re i ()| 1   Lj () =  ... ... ... . . . ...  ; 0 ≤ lj ≤ min : (2.4) 2   4m i≤r1 +r2 ||  0 0 0 : : : lj  0 0 0 ::: 0 Here r ≤ m. In the next lemma for the constants dk of (2.1) we set dk = 2(−i)k

r1


Sk+1; i ();

k = 0; 1; : : : ; m − 1:

(2.5)

i=1

2.3. Key lemma Lemma 2.1. There exists a positive number T such that the solution u(t; ˜ ) of the Cauchy problem for the ordinary diAerential equation
@m a˜j;  (t; {(i) @kt u} ˜ k+||≤m−1 ; {(−i) @kt u} ˜H k+||≤m−1 ) @tj u˜ t u˜ + j+||≤m; j¡m

˜ k+||≤m−1 ; {(−i) @kt u} ˜H k+||≤m−1 ) = im F(t; {(i) @kt u}

(2.6)

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

167

with initial data @kt u(0) ˜ = 20k+1−m−M −3

r1


Sk+1; i ();

k = 0; 1; : : : ; m − 1

(2.7)

i=1

exists for all t ∈ [0; T ] and all suBciently large 0 ∈ R+ . Furthermore; if u˜ ∈ C m ([0; T ]) solves (2:6); (2:7); then the function ˜ ) u(t; x) = eix· u(t;

(2.8)

solves Eq. (0:1) for all t ∈ [0; T ] in a small neighbourhood of the origin of Rnx . Proof. The last statement of the lemma is obvious. Further, if T is su4ciently small (or R su4ciently large), then the solution uK = u(t; x; ) to (1.2) with the above initial conditions (2.1), in the cylinder [0; T ] × {x ∈ R; |x| ≤ &T } has the form (2.8), where u(t; ˜ ) solves an equation (2.6) for all t ∈ [0; T ]. Indeed, this is a consequence of the local existence theorem for the Cauchy problem for the ordinary di1erential equation (2.6) and the uniqueness condition of De:nition 1.3. Last one guarantees that the solution has the form (2.8) for t ∈ [0; T ∗ ); T ∗ ¡ T , while the existence condition of De:nition 1.3 implies that one can continue a solution of (2.6) over any point t = T∗ ¡ T. 2.4. Lower bound for the energy One can write A(t; ; U; UH ) = A(0; ; 0; 0) + A1 (t; ; U; UH ), where ||A1 (t; ; U; UH )|| ≤ O(t + ||U ||) as t + ||U || → 0 uniformly with respect to 0 ∈ R. Hence S()−1 A(t; ; U; UH )S() = D() + L() + S()−1 A1 (t; ; U; UH )S(): Set Y :=S()−1 U , then U = S()Y . The vector-valued function Y solves the system dY = 0(D() + L())Y + 0S()−1 A1 (t; ; S()Y; S()Y )S()Y dt + S()−1 R(t; ; S()Y; S()Y ): Consider the term S()−1 R(t; ; S()Y; S()Y ). According to condition (1.5) of the theorem, there exist positive numbers h and C0 such that ˆ  0||+k+1−m Uk+1 })| ˆ  0||+k+1−m Uk+1 }; (−i) if ||U || ≤ h; then |F(t; {(i) ≤ C0 ||U ||: Hence there is a C such that for all su4ciently large |0| we have ||S()−1 R(t; ; S()Y; S()Y )|| ≤ CC0 ||S()Y || if ||S()Y || ≤ h: To get a lower bound for the energy of Y we consider the Lyapunov function 1 V (Y (t; )):=V (t):= {|Y1 |2 + · · · + |Yr1 |2 − |Yr1 +1 |2 − · · · − |Ym |2 }: 2

168

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

If we chose T and h small while 0 large enough, then we obtain 0 dV (t) ≥ 80 V (t) for all t ∈ [0; T ] such that ||S()Y (t; )|| ≤ h 2 dt with some positive number 80 independent of T and 0. We remind that h does not exceed the one from (1.5). Hence, if ||S()Y (t ; )|| ≤ h for all t ∈ [0; t], then V (t) ≥ V (Y (0)) exp(80 0t=2): For the positive number M of De:nition 1.3 we choose  −M −3 if i = 1; : : : ; r1 ; 20 t Y (0) = (Y1 (0); Y2 (0); : : : ; Ym (0)); Yi (0) = 0 if i = r1 + 1; : : : ; m so that V (Y (0)) = 2r1 0−2M −6 ¿ 0 implies V (Y (t)) ≥ 2r1 0−2M −6 exp(80 0t=2) provided that ||S()Y (t ; )|| ≤ h for all t ∈ [0; t]. Hence, ||Y (t)|| ≥ 0−M −3 exp(80 0t=4) for all t with the above-mentioned property. Then, ||U (0; )|| ≤ C0−M −3 ;

U (0; ) = S()Y (0; );

@kt u(0) = eix· 20k+1−m−M −3

r1


Sk+1; i ();

k = 0; : : : ; m − 1:

i=1

We have ||Y (t)|| ≤ ||S()−1 || ||U (t)||. It follows:   80 1 ||U (t)|| ≥ exp 0t 0−M −3 4 ||S()−1 ||

(2.9)

provided that ||U (t ; )|| ≤ h for all t ∈ [0; t]. On the other hand if 0 is large, then ||U (0)|| ≤ C−M −3 ¡ h: Due to (2.9), for increasing 0 there is a :rst point t such that ||U (t )|| = h and

t → 0 as || → ∞

while ||U (t; )|| ≤ h for all t ≤ t : Thus ||U (t )|| = h ¿ 0 while ||U (0)|| ≤ const 0−M −3 → 0 as 0 → ∞. Therefore u(t ; ·)||C 0 ≥ h ||u(t ; ·)||C m−1 + ||@t u(t ; ·)||C m−2 + · · · + ||@m−1 t

(2.10)

while u(0; x)||C M −m+1 ||u(0; x)||C M + ||@t u(0; x)||C M −1 + · · · + ||@m−1 t ≤ const 0−2 → 0

as 0 → ∞:

(2.11)

Comparing (2.10), (2.11) with De:nition 1.3 for 0 su4ciently large we get a contradiction.

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

169

3. Proof of Theorem 1.2 To prove by contradiction we suppose that u(t; x) ≡ 0 is a stable solution of the problem in the sense of De:nition 1.3 in the neighbourhood of the origin, x = 0; t = 0. ˆ = 1, and some zero l () ˆ of equation Then we assume that there exist ˆ ∈ Rn ; ||
 aj;  ˆ  j = 0 m + j+||=m; j¡m

ˆ = 0. which has non-vanishing imaginary part, Im l () Further we choose the vector K = ( 0 ; 1 ; : : : ; m−1 ) of initial data as in (2.1), where 1 ∈ C ∞ is a cuto1 function, 1(x) = 1 if |x| ≤ R; 1(x) = 0 if |x| ≥ 2R, the constants dk ∈ C will be chosen later, M is of De:nition 1.3,  = 0ˆ ∈ Rn , while 0 ∈ R+ is a large parameter. For every given positive number h(() the inequality (2.2) holds if 0 = || is large enough. Then, according to De:nition 1.3, there exist positive numbers T; &; M , and (0 such that for every positive ( ≤ (0 there is a positive h(() such that for every vector of initial data, K = ( 0 ; 1 ; : : : ; m−1 ), of (2.1) with (2.2), there is a unique solution uK ∈ C ∞ (K& (x0 ; T )), and moreover, ||uK ||C m (K& (x0 ;T )) ≤ (: Lemma 3.1. There exists a positive number T such that the solution u(t; ˜ ) of the Cauchy problem for the ordinary diAerential equation m  j  j @m t u + i F(t; {(i) @t u}j+||≤m; j¡m ; {(−i) @t u}j+||≤m−1 ) = 0

(3.1)

with initial data (r1 ≥ 1) @kt u(0) ˜ = 20k−m−M −2 if k = 0; : : : ; r1 − 1;

@kt u(0) ˜ = 0 if k = r1 ; : : : ; m − 1 (3.2)

exists for all t ∈ [0; T ] and all suBciently large 0 ∈ R+ . Furthermore; if u˜ ∈ C m ([0; T ]) solves (3:1); (3:2); then the function u(t; x) = ei x· u(t; ˜ )

(3.3)

solves Eq. (1:1) for all t ∈ [0; T ] in a small neighbourhood of the origin of Rnx . Proof. We omit a proof since it is quite a repetition of the one of Lemma 2.1. Reduction to a system of quasilinear equations. If u = u(t; x); u ∈ C m+1 (K& (x0 ; T )); solves an Eq. (1.1) then, by di1erentiation, vr = @xr u(t; x); r = 1; : : : ; n, solves the quasilinear equation
Dtm vr + aj;  (t; {@kt @x u}k+||≤m; j¡m ; {@kt @x u}k+||≤m−1 )Dtj Dx vr j+||=m; j¡m

= Fr (t; {@kt @x u}k+||≤m; j¡m ; {@kt @x u}k+||≤m );

r = 1; : : : ; n:

170

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

If we set v0 :=u;

F0 := − (−i)m F;

then we can rewrite Eq. (1.1) and above equations as a system Dtm v0 = F0 (t; {@kt @x vl }k+||≤m−1; l=0; :::; n ; {@kt @x vl }k+||≤m−2; l=0; :::; n );
aj;  (t; {@kt @x vl }k+||≤m−1; l=0; :::; n ; Dtm vr + j+||=m; j¡m

{@kt @x vl }k+||≤m−2; l=0; :::; n )Dtj Dx vr = Fr (t; {@kt @x vl }k+||≤m−1; l=0; :::; n ; {@kt @x vHl }k+||≤m−1; l=0; :::; n );

r = 1; : : : ; n;

˜ ); r = 1; : : : ; n. for (v0 ; v1 ; : : : ; vn ). If u(t; x) has a form (3.3) then vr (t; x) = ir eix· u(t; One can write also, vr (t; x) = eix· v˜r (t; ); r = 1; : : : ; n. Therefore,  kH  k ˜ @m t v˜0 = F 0 (t; {(i) @t v˜l }k+||≤m−1; l=0; :::; n ; {(−i) @t v˜l }k+||≤m−2; l=0; :::; n );
a˜j;  (t; {(i) @kt v˜l }k+||≤m−1; l=0; :::; n ; @m t v˜r + j+||=m; j¡m

{(−i) @kt v˜Hl }k+||≤m−2; l=0; :::; n ) @tj v˜r = F˜ r (t; {(i) @kt v˜l }k+||≤m−1; l=0; :::; n ; {(−i) @kt v˜Hl }k+||≤m−1; l=0; :::; n );

r = 1; : : : ; n:

Let us consider vector-valued functions de:ned by  (r)   m−1  U1 0 v˜r  U (r)   0m−2 @t v˜r    2   U (r) =  .  :=   ; r = 0; : : : ; n: ..   ..   . @m−1 v˜r t

Um(r)

One can write for U (r) the following system of ordinary di1erential equations: dU (0) dU (r) = R(0) (; U; UH ); = 0A(t; ; U; UH )U + R(r) (t; ; U; UH ); r = 1; : : : ; n; dt dt where U :=t (U (0) ; U (1) ; : : : ; U (n) ), while the matrix A(t; ; U; UH ) is the following: A(t; ; U; UH ) 

0 .. .

1 .. .

   :=  
0
0  −m − a˜0;   0 − a˜1;   0−m+1 ||=m

||=m−1

 ::: 0 ..  ..  . .  ; ::: 1 
 −1  ::: − a˜m−1;   0 ||=1

a˜l;  = a˜l;  (t; {(i) @kt v˜l }k+||≤m−1; l=0; :::; n ; {(−i) @kt v˜Hl }k+||≤m−2; l=0; :::; n ); l = 0; : : : ; m − 1:

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

171

The vector-valued function R(r) (t; ; U; UH ) is given by (r) R(r) (t; ; U; UH ) := im F˜ (t; {(i) @kt v˜l }k+||≤m−1; l=0; :::; n ; {(−i) @kt v˜Hl }k+||≤m−1; l=0; :::; n )em ;

where em :=t (0; : : : ; 0; 1). It can be written also in the following way: (r)

ˆ  0||+k+1−m U (l) }k+||≤m−1; l=0; :::; n ; R(r) (t; ; U; UH ) = im F˜ (t; {(i) k+1 ˆ  0||+k+1−m U (l) }k+||≤m−1; l=0; :::; n )em : {(−i) k+1 Let us denote by 1 (); 2 (); : : : ; m () the roots of Eq. (1.7). One can order these roots like in (2.3). Due to the assumption of the theorem, we can suppose that either the set {1 (); : : : ; r1 ()} or the set {r1 +1 (); : : : ; r1 +r2 ()} is not empty. Consider the case with r1 ≥ 1. Let S() be a matrix reducing the principal part A(0; ; 0; 0) to Jordan form: S()−1 A(0; ; 0; 0)S() = D() + L(); where D() is a diagonal matrix with elements 1 ()=0; 2 ()=0; : : : ; m ()=0, while L() is a block matrix consisting of the matrices Lj (); j = 1; : : : ; r, of (2.4) satisfying 0 ≤ lj ≤

|Re i ()| n2 : min 2 i≤r +r 4m || 1 2

These matrices depend on ˆ only. Now we choose (2.5) for the constants dk of (2.1). One can write A(t; ; U; UH ) = A(0; ; 0; 0) + A1 (t; ; U; UH ), where ||A1 (t; ; U; UH )|| ≤ O(t + ||U ||) as t + ||U || → 0 uniformly with respect to 0 ∈ R+ . Hence S()−1 A(t; ; U; UH )S() = D() + L() + S()−1 A1 (t; ; U; UH )S(): Set Y :=t (Y (0) ; Y (1) ; : : : ; Y (n) ):=Sd ()−1 U :=t (U (0) ; S()−1 U (1) ; : : : ; S()−1 U (n) ), then U = Sd ()Y . The vector-valued functions Y (r) ; r = 0; 1; : : : ; n, solve the systems dY (0) = R(0) (t; ; Sd ()Y; Sd ()Y ); dt dY (r) = 0(D() + L())Y (r) + 0S()−1 A1 (t; ; Sd ()Y; Sd ()Y )S()Y (r) dt + S()−1 R(r) (t; ; Sd ()Y; Sd ()Y ); r = 1; : : : ; n: Consider the term S()−1 R(r) (t; ; Sd ()Y; Sd ()Y ). According to condition (1.5), there exist positive numbers h and C0 such that if ||U || ≤ h; then (r) ˆ  0||+k+1−m U (l) }k+||≤m−1; l=0; :::; n ; |F˜ (t; {(i) k+1

ˆ  0||+k+1−m U (l) }k+||≤m−1; l=0; :::; n )| ≤ C0 ||U || {(−i) k+1

172

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

for all t ∈ T . Hence for 0 large enough we have ||S()−1 R(r) (t; ; Sd ()Y; Sd ()Y )|| ≤ ||S()−1 em ||C0 ||Sd ()Y || ≤ CC0 ||Sd ()Y || if ||Sd ()Y || ≤ h

(3.4)

for r = 0; : : : ; n, where C is independent of 0 and of t ∈ [0; T ]. To get a lower bound for the energy of Y , we consider the Lyapunov function n


1 (l) 1 V (t):= − ||Y (0) ||2 + |2 − |Yr(l) |2 − · · · − |Ym(l) |2 ): (|Y |2 + · · · + |Yr(l) 1 1 +1 2 1 2 l=1

It is easily seen that m


(0) dV = −Re YH i (S()−1 R(0) (t; ; Sd ()Y; Sd ()Y ))i dt i=1

+ 0Re + Re

r1 n



l=1 i=1 r1 n



(l) (|Yi(l) |2 Dii () + YH i (L()Y (l) )i ) (l)

(0YH i (S()−1 A1 (t; ; Sd ()Y; Sd ()Y )S()Y (l) )i

l=1 i=1 (l) +YH i (S()−1 R(l) (t; ; Sd ()Y; Sd ()Y ))i ) n m

(l) −0Re (|Yi(l) |2 Dii () + YH i (L()Y (l) )i ) l=1 i=r1 +1 n m

(l) (0YH i (S()−1 A1 (t; ; Sd ()Y; Sd ()Y )S()Y (l) )i −Re l=1 i=r1 +1 (l)

+YH i (S()−1 R(l) (t; ; Sd ()Y; Sd ()Y ))i ):

(3.5)

Furthermore,   r1 
 (l)   −1 (l) YH i (S() A1 (t; ; Sd ()Y; Sd ()Y )S()Y )i  Re   i=1   m  
(l)   + Re YH i (S()−1 A1 (t; ; Sd ()Y; Sd ()Y )S()Y (l) )i    i=r1 +1

≤ O(t + h)||Y (t; )||2

for all t ∈ [0; T ]; 0 ∈ R+ if ||Sd ()Y || ≤ h;

(3.6)

where O(t + h) as t + h → 0. Then due to (3.4) we get n
m


(l)

|YH i (S()−1 R(l) (t; ; Sd ()Y; Sd ()Y ))i | ≤ CC0 ||Y ||2

l=0 i=1

where the constant C is independent of 0 and of t ∈ [0; T ].

if ||Sd ()Y || ≤ h;

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

173

On the other hand, with an appropriately chosen positive number 8 we have Re

r1
i=1



≥

m


(l) (|Yi(l) |2 Dii () + YH i (L()Y (l) )i ) − Re

min

i≤r1 +r2

≥ 28

r
1+r2 i=1

|Re i ()| 0

 r
1 +r2

i=r1 +1

 |Yi(l) |2 − max |lj | ||Y (l) ||2 j

i=1

m


|Yi |2 − 8



(l) (|Yi(l) |2 Dii () + YH i (L()Y (l) )i)

|Yi |2

for all t ∈ [0; T ]; 0 ∈ R+ ; l = 1; : : : ; n:

(3.7)

r1 +r2 +1

If we choose T and h of (1.5) small while 0 large enough, then on account of (3.6) to (3.7) we obtain dV 0 (t) ≥ 80 V (t; ) for all t ∈ [0; T ] such that ||Sd ()Y (t; )|| ≤ h dt 2 with some positive number 80 independent of T and 0. Hence, if ||Sd ()Y (t ; )|| ≤ h for all t ∈ [0; t], then V (t) ≥ V (Y (0)) exp(80 0t=2): For the positive number M of De:nition 1.3 we choose Y (0) (0) =t (Y1(0) (0); Y2(0) (0); : : : ; Ym(0) (0));  −M −3 if i = 1; : : : ; r1 ; 20 (0) Yi (0) = 0 if i = r1 + 1; : : : ; m; Y (l) (0) =t (Y1(l) (0); Y2(l) (0); : : : ; Ym(l) (0));

Yi(l) (0) = ˆl 0Yi(0) (0);

l = 1; : : : ; n

so that V (Y (0)) = 2r1 0−2M −4 − 20−2M −6 ¿ 0 implies V (Y (t)) ≥ (2r1 0−2M −4 − 20−2M −6 ) exp(80 0t=2) provided that ||Sd ()Y (t ; )|| ≤ h for all t ∈ [0; t]. Hence, ||Y (t)|| ≥ 0−M −2 exp(80 0t=4) for all t with the above mentioned property. Then, U (0) (0; ) = Y (0) (0; );

||U (0) (0; )|| ≤ 2r1 (n + 1)0−M −3 ;

@kt u(0) = 20k−m−M −2 if k = 0; : : : ; r1 − 1; We have ||Y (t)|| ≤ ||Sd ()−1 ||||U (t)||. It follows:   80 1 ||U (t)|| ≥ exp 0t 0−M −2 4 ||S()−1 ||

@kt u(0) = 0 if k = r1 ; : : : ; m − 1:

(3.8)

provided that ||U (t ; )|| ≤ h for all t ∈ [0; t]. On the other hand if 0 is large enough then ||U (0)|| ≤ C−M −2 ¡ h:

174

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

Due to (3.8), for increasing 0 there is a :rst point t such that ||U (t )|| = h and

t → 0 as || → ∞;

while the condition ||U (t; )|| ≤ h for all t ≤ t is ful:lled. Thus ||U (t )|| = h ¿ 0 while ||U (0)|| ≤ const 0−M −2 → 0 as 0 → ∞. Therefore ||u(t ; ·)||C m−1 + ||@t u(t ; ·)||C m−2 + · · · + ||@m−1 u(t ; ·)||C 0 ≥ h0−3=2 t

(3.9)

while ||u(0; x)||C M + ||@t u(0; x)||C M −1 + · · · + ||@m−1 u(0; x)||C M −m+1 t ≤ const 0−2 → 0

as 0 → ∞:

(3.10)

Comparing (3.9), (3.10) with De:nition 1.3 for 0 su4ciently large we get a contradiction. The proof of the Theorem 1.3 is quite a repetition of those of Theorem 1.1 and Theorem 1.2, therefore we omit it. Acknowledgements This paper was written during the work of the author as a Foreign Professor at the University of Tsukuba. He is very grateful to all the members of Institute of Mathematics of the University of Tsukuba for the helpful and warm hospitality. Special thanks to Prof. K. Kajitani and Prof. S. Wakabayashi for the invitation to the University of Tsukuba and for many useful discussions. References [1] H. Bahouri, P. GVerard, High frequency approximation of solutions to critical nonlinear wave equation, Amer. J. Math. 121 (1999) 131–175. [2] J. Bourgain, Global wellposedness of defocusing critical nonlinear SchrWodinger equation in the radial case, J. Amer. Math. Sci. 12 (1) (1999) 145–171. [3] M. Dreher, M. Reissig, Local solutions of fully nonlinear weakly hyperbolic di1erential equations in Sobolev spaces, Hokkaido Math. J. 27 (2) (1998) 337–381. [4] L. HWormander, Lectures on Nonlinear Hyperbolic Di1erential Equations, Mathematiques and Applications, Vol. 26, Springer, Berlin, 1997. [5] V. Ivrii, Conditions for the correctness in Gevrey classes of the Cauchy problem for non-strictly hyperbolic operators, Sibirsk. Mat. Zh. 17 (3) (1976) 547–563. [6] K. JWorgens, Das Anfangswertproblem im GroZen fWur eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961) 295–308. [7] K. Kajitani, K. Yagdjian, Quasilinear hyperbolic operators with characteristics of variable multiplicity, Tsukuba J. Math. 22 (1) (1998) 49–85. [8] S. Kichenassamy, Nonlinear Wave Equations, Marcel Dekker, Inc., New York, 1996. [9] T. Kinoshita, On the Cauchy problem with small analytic data for nonlinear weakly hyperbolic systems, Tsukuba J. Math. 21 (2) (1997) 397– 420.

K. Yagdjian / Nonlinear Analysis 49 (2002) 159 – 175

175

[10] H. Komatsu, Irregularity of characteristic elements and hyperbolicity, Publ. Res. Inst. Math. Sci. Kyoto Univ. 12 (Suppl.) (1977) 233–245. [11] P. Lax, Asymptotic solutions of oscillatory initial value problem, Duke Math. J. 24 (4) (1957) 627–646. [12] Li Ta-tsien, Global Classical Solutions for Quasilinear Hyperbolic Systems, Wiley, New York, 1994. [13] S. Mizohata, Some remarks on the Cauchy problem, J. Math. Kyoto Univ. 1 (1961) 109–127. [14] T. Nishitani, On the Lax-Mizohata theorem in the analytic and Gevrey classes, J. Math. Kyoto Univ. 18 (3) (1978) 509–521. [15] Yu.N. Ovchinnikov, I.M. Sigal, Ginzburg-Landau Equation I, Static vortices, Partial Di1erential Equations and their Applications, Toronto, ON, 1995, CRM Proceedings of the Lecture Notes, Vol. 12, American Mathematical Society, Providence, RI, 1997, pp. 199 –220. [16] R. Racke, Lectures on Nonlinear Evolution Equations, Aspects of Mathematics, Vieweg, Braunschweig=Wiesbaden, 1992. [17] S. Spagnolo, Dipendenza continua dai dati in sistemi di evoluzione, Boll. Un. Mat. Ital. A (7) 11-A (1997) 279–295. [18] K. Yagdjian, The Cauchy Problem for Hyperbolic Operators, Multiple Characteristics, Micro-Local Approach, Akademie Verlag, Berlin, 1997. [19] K. Yagdjian, A note on Lax-Mizohata theorem for quasilinear equations, Comm. Partial Di1erential Equations 23 (5&6) (1998) 1111–1122.