Sharp blow-up criteria for the Davey-Stewartson system in R 3

Transcription

1 Dynamics of PDE, Vol.8, No., 9-60, 011 Sharp blow-up criteria for the Davey-Stewartson system in R Jian Zhang Shihui Zhu Communicated by Y. Charles Li, received October 7, 010. Abstract. In this paper, we study the blow-up solutions for the Davey- Stewartson system iu t + u + u p 1 u + E( u )u = 0, t 0, x R. (DS) Using the profile decomposition of the bounded sequences in H 1 (R ), we give some new variational characteristics for the ground states generalized Gagliardo-Nirenberg inequalities. Then, we obtain the precise expressions on the sharp blow-up criteria to (DS) for p < 5. Contents 1. Introduction 9. Preliminary 41. Sharp Gagliardo-Nirenberg Inequalities 4 4. Sharp Blow-up Criteria Properties of Blow-up Solutions 58 References Introduction This paper is concerned with the Cauchy problem of the following Davey- Stewartson system (1.1) iu t + u + u p 1 u + E( u )u = 0, t 0, x R N, (1.) u(0, x) = u 0, 1991 Mathematics Subject Classification. 5B44; 5Q55. Key words phrases. Davey-Stewartson system, profile decomposition, blow-up solution, sharp criteria. Supported by National Natural Science Foundation of P.R.China (No ). 9 c 011 International Press

2 40 JIAN ZHANG AND SHIHUI ZHU where i = 1; = x 1 + x + + x N is the Laplace operator on R N ; u = u(t, x): [0, T) R N C is the complex valued function 0 < T + ; the parameter 1 < p < 1 (where = + for N = ; = N N for N ); E is the singular integral operator with symbol σ 1 (ξ) = ξ 1 ξ, ξ R N, E(ψ) = F 1 ξ 1 ξ Fψ, F F 1 are the Fourier transform Fourier inverse transform on R N, respectively. For the Cauchy problem (1.1)-(1.), Ghidaglia Saut [8], Guo Wang [9] established the local well-posedness in the energy space H 1 (R N ) for N = N = respectively (see [1, ] for a review). Cipolatti [] showed the existence of the sting waves. Cipolatti [], Ohta [15, 16], Gan Zhang [6] showed the stability instability of the sting waves. Ghidaglia Saut [8], Guo Wang [9] showed the existence of the blow-up solutions. Ozawa [17] gave the exact blow-up solutions. Wang Guo [] studied the scattering of solutions. Richards [19], Papanicolaou etal [18], Gan Zhang [5, 6], Shu Zhang [0] studied the sharp conditions of blow-up global existence. Li etal [1], Richards [19] obtained the mass-concentration properties of the blow-up solutions in L - critical case in R. We note that in R, Richards [19] Papanicolaou etal [18] gave the precise expression on the sharp blow-up criteria in L -critical case that is N = p =. But in R Equation (1.1) has not any L -critical case because of influence of the nonlocal term E( u )u. Although in [5,6,0] some sharp thresholds of blow-up global existence are gotten, we also note that the upper bound d of the energy functional I(u) is not determined. This motivates us to investigate the precise expression on the sharp blow-up criteria in R. In the present paper, at first, we study the sharp blow-up criteria for the Cauchy problem (1.1)-(1.) in R for p < 5. Motivated by Holmer Roudenko s studies [11] for the classical L -super nonlinear Schrödinger equation, we consider the following elliptic equations (1.) (1.4) Q 1 Q + Q Q + E( Q )Q = 0, Q H 1 (R ) R 1 R + E( R )R = 0, R H 1 (R ). Applying the profile decomposition of the bounded sequences in H 1 (R ), we obtain the following generalized Gagliardo-Nirenberg inequalities (1.5) f 4 + E( f ) f dx R Q f L f L, f H1 (R ) (1.6) E( f ) f dx R R f L f L, f H1 (R ) L where Q is the solution of Equation (1.) R is the solution of Equation(1.4). Using the above Gagliardo-Nirenberg inequalities, we obtain the sharp blow-up criteria to the Cauchy problem (1.1)-(1.) for p < 5 by overcoming the loss of scaling invariance. We remark that we get a clear bound value of energy functional, which corresponds to d in [6]. Furthermore, we prove that there is no

3 SHARP BLOW-UP CRITERIA 41 L strong limit of the blow-up solutions to the Cauchy problem (1.1)-(1.) provided 1 < p. There are two major difficulties in the analysis of blow-up solutions to the Davey-Stewartson system (1.1)-(1.) in H 1 (R ): One is the nonlinearity containing the singular integral operator E; The other is the loss of scaling invariance to Equation (1.1) for p, which destroys the balance between u p 1 u E( u )u. Due to the singular integral operator E, we have to establish the corresponding generalized Gagliardo-Nirenberg inequalities variational structures. Since there is no scaling invariance for p, improving Holmer Roudenko s method [11] we use the ground state of the classical nonlinear Schrödinger equation to describe the sharp blow-up criteria to the Davey-Stewartson system (1.1)-(1.). Finally, for the time being, as we have mentioned, the results in the present paper are new for the Davey-Stewartson system (1.1)-(1.). In particular, the sharp blow-up criteria are different from [6], the sharp blow-up criteria obtained in this paper are more precisely, which is very useful from the viewpoint of physics. We conclude this section with several notations. We abbreviate L q (R ), Lq (R ), H s (R ) R dx by L q, q, H s dx. The various positive constants will be simply denoted by C.. Preliminary For the Cauchy problem (1.1)-(1.), the energy space is defined by H 1 := {u L ; u L }, which is a Hilbert space. The norm of H 1 is denoted by H 1. Moreover, we define the energy functional H(u) in H 1 by H(u) := 1 u(t, x) dx 1 u(t, x) p+1 dx 1 E( u ) u dx. p The functional H is well-defined according to the Sobolev embedding theorem the properties of the singular operator E. Guo Wang [9] established the local well-posedness of the Cauchy problem (1.1)-(1.) in energy space H 1. Proposition.1. Let u 0 H 1. There exists an unique solution u(t, x) of the Cauchy problem (1.1)-(1.) on the maximal time [0, T) such that u(t, x) C([0, T); H 1 ) either T = + (global existence), or T < + lim t T u(t, x) H 1 = + (blow-up). Furthermore, for all t [0, T), u(t, x) satisfies the following conservation laws (i) Conservation of mass (.1) u(t, x) = u 0, (ii) Conservation of energy (.) H(u(t, x)) = H(u 0 ). For more specific results concerning the Cauchy problem (1.1)-(1.), we refer the reader to [8, ]. In addition, by some basic calculations, we have the following proposition(see also Ohta [16]).

4 4 JIAN ZHANG AND SHIHUI ZHU Proposition.. Assume that u 0 H 1, x u 0 L the corresponding solution u(t, x) of the Cauchy problem (1.1)-(1.) on the interval [0, T). Then, for all t [0, T) we have x u(t, x) L. Moreover, let J(t) := x u(t, x) dx, we have (.) J (t) = 4I xu udx (.4) J (t) = 8 u dx 1 p 1 p + 1 u p+1 dx 6 E( u ) u dx. We give some known facts of the singular integral operator E (see Cipolatti [,]), as follows. by Lemma.. Let E be the singular integral operator defined in Fourier variables F[E(ψ)](ξ) = σ 1 (ξ)f[ψ](ξ), where σ 1 (ξ) = ξ 1 ξ, ξ R F denotes the Fourier transform in R. For 1 < p < +, E satisfies the following properties: (i) E L(L p, L p ), where L(L p, L p ) denotes the space of bounded linear operators from L p to L p. (ii) If ψ H s (R ), then E(ψ) H s (R ), s R. (iii) If ψ W m,p, then E(ψ) W m,p k E(ψ) = E( k ψ), k = 1,. (iv) E preserves the following operations: translation: E(ψ( + y))(x) = E(ψ)(x + y), y R ; dilatation: E(ψ(λ ))(x) = E(ψ)(λx), λ > 0; conjugation: E(ψ) = E(ψ), where ψ is the complex conjugate of ψ. At the end of this section, we shall give the profile decomposition of bounded sequences in H 1 proposed by Gérard [7], Hmidi Keraani [10], which is important to study the variational characteristic of the ground state. Proposition.4. Let {v n } n=1 be a bounded sequence in H1. Then there is a subsequence of {v n } n=1 (still denoted by {v n } n=1 ) a sequence {V j } in H1 a family of {x j n} R such that (i) for every j k, x j n x k n n + ; (ii) for every l 1 every x R (.5) v n (x) = with V j (x x j n ) + vl n (x) (.6) lim sup n vl n l r 0, for every r (, 6).

5 SHARP BLOW-UP CRITERIA 4 Moreover, we have, as n, (.7) v n = V j + vl n + o(1) (.8) v n = V j + vl n + o(1).. Sharp Gagliardo-Nirenberg Inequalities In order to study the sharp blow-up criteria to the Cauchy problem (1.1)-(1.), we have to establish the generalized Gagliardo-Nirenberg inequalities corresponding to the Davey-Stewartson system. For p =, we consider the sting wave solutions of Equation (1.1) in the form u(t, x) = Q( satisfies (.1) x)e i t. It is easy to check that Q(x) Q 1 Q + Q Q + E( Q )Q = 0, Q H 1. We say that Q H 1 is a ground state solution of (.1), if it satisfies S(v) = inf{ S(v) v H 1 \ {0} is a solution of (.1)}, where S(v) is defined for v H 1 by S(v) = 1 v dx 1 4 ( v 4 + E( v ) v )dx + 1 v dx. For any solution Q of Equation (.1), we claim the following Pohozhaev identities: (.) 1 Q dx Q dx + Q 4 + E( Q ) Q dx = 0 (.) Q dx + Q dx Q 4 + E( Q ) Q dx = 0. Indeed, multiplying (.1) by Q integrating by parts, we have that (.) is true. On the other h, multiplying (.1) by x Q integrating by parts, we have Qx Qdx 1 Qx Qdx+ Q Qx Qdx+ E( Q )Qx Qdx = 0. It follows from some basic calculations that Qx Qdx = Q dx + x ( Q )dx = 1 Q dx, Qx Qdx = Q dx, Q Qx Qdx = Q 4 dx 4 E( Q )Qx Qdx = 4 E( Q )Q dx. Collecting the above identities, we have that (.) is true.

6 44 JIAN ZHANG AND SHIHUI ZHU In this section, using the profile decomposition of bounded sequence in H 1, we shall give a new simple proof of the existence of the ground state of (.1). Moreover, we compute the best constant of a generalized Gagliardo-Nirenberg inequality in dimension three, which is corresponding to Equation (1.1). More precisely, we have the following theorem. Theorem.1. Let f H 1, then (.4) f 4 + E( f ) f dx Q f f, where Q is the solution of Equation (.1). We shall give the proof of Theorem.1 by three steps. First, from the definition of E the classical Gagliardo-Nirenberg inequality, we observe that R ( v 4 + E( v ) v )dx C R ( v 4 + v 4 )dx C v v, which motivates us to investigate the best constant C of the above inequality. Thus, we consider the variational problem (.5) J := inf{j(u) u H 1 } where J(u) := ( u dx) 1 ( u dx) ( u 4 + E( u ) u )dx. It is obvious that if W is the minimizer of J(u), then W is also a minimizer. Hence we can assume that W is an real positive function. Indeed, W = W e iθ(x) we have W W in the sense of distribution. On the other h, if W H 1, then W H 1 J( W ) J(W). Second, by some basic calculations, if W is the minimizer of J(u), we have the following lemma. (.6) Lemma.. If W is the minimizer of J(u), then W satisfies W W W 1 W W + J( W + E( W ))W = 0. W Proof. Since W is a minimizing function of J(u) in H 1, we have v C 0 (R ) d (.7) dε J(W + εv) ε=0 = 0. By some basic calculations, we have (.8) (.9) d dε { d dε { (W + εv) (W + εv) } ε=0 = 1 W RWvdx W W W R W vdx (W +εv) 4 +E( W +εv ) W +εv dx} ε=0 = 4 ( W +E( W )RWvdx.

7 SHARP BLOW-UP CRITERIA 45 By (.7)-(.9), we have (.10) = 1 W W RWvdx W W R Wvdx W W R W 4 +E( W ) W dx ( W + E( W ))RWvdx, which implies that (.6) is true. The Euler-Lagrange equation (Lemma.) shows that any minimizer of J(u) is a solution of (.1). Since any smaller mass solution would yield a lower value of J(u), the Pohozhaev identities show that it is in fact a minimal mass solution of (.1). Therefore, to prove the existence of a ground state it suffices to prove the existence of a minimizer for J(u). Thirdly, we use the profile decomposition of bounded sequences in H 1 to prove the following proposition, which give a proof of the existence of a minimizer for J(u). Moreover, Theorem.1 is a direct conclusion of the following proposition. Proposition.. J is attained at a function U(x) with the following properties: (.11) U(x) = aq(λx + b) for some a C, λ > 0 b R where Q is the solution of Equation(.1). Moreover, we have (.1) J = Q. Proof. If we set u λ,µ = µu(λx), where λ = u u, µ = u 1, we have u u λ,µ = 1, u λ,µ = 1 J(u λ,µ ) = J(u). Now, choosing a minimizing sequence {u n } n=1 H1 such that J(u n ) J as n, after scaling, we may assume (.1) u n = 1 u n = 1, we have (.14) J(u n ) = 1 un 4 + E( u n ) u n J, as n. dx Note that {u n } n=1 is bounded in H 1. It follows form the profile decomposition (Proposition.4) that u n (x) = U j (x x j n ) + rl n (x), (.15) Un j 1 Un j 1, where U j n = U j (x x j n). Moreover, using the Hölder s inequality for r l n the properties of E, we have

8 46 JIAN ZHANG AND SHIHUI ZHU ( r l n 4 + E( r l n ) r l n )dx C( r l n E( r l n ) r l n 4) (.16) C( rn l rl n 4 4 ) 0, as l. Applying the orthogonal conditions the properties of E, we have the following claims: (i) (.17) U j (x x j n) 4 dx U j 4 dx, as n. (ii) (.18) E( U j (x x j n ) ) U j (x x j n ) dx E( U j ) U j dx, as n. Indeed, for (.17), it suffices to show that (.19) I n = U j1 n U j n U j n U j4 n dx 0, as n for 1 j k l at least two j k are different. Assuming for example j 1 j, by the Hölder s inequality, we can estimate I n C U j1 n Uj n, where C = Π 4 k= Uj k n 4. Without loss of generality, we can assume that both U j1 U j are continuous compactly supported. Now, we use the pairwise orthogonal conditions, we have the following estimation (.0) I n C U j1 (y)u j (y (x j n x j1 n )) dy 0, as n. This completes the proof of Claim (i). For (.18), we have C ( [ E( U j (x x j n) ) E( Un j ) + which implies that U j (x x j n) dx 1 j,k l,j k E( Un juk n )][ Un j + 1 j,k l,j k U j n Uk n ]dx),

9 SHARP BLOW-UP CRITERIA 47 (.1) E( U j (x x j n ) ) U j (x x j n ) dx l E( U j ) U j dx C l E( U j n ) Un j dx + E( U j n ) Un k dx 1 j,k l,j k +C E( U i n Un ) U j n k dx + E( U i n ) UnU j n dx k 1 i,j,k l,i j 1 i,j,k l,j k +C E( U i n Un ) U j nu k n m dx = I + II + III. 1 i,j,k,m l,i j,k m Without loss of generality, we can assume that U i, U j, U k U m are continuous compactly supported. Using the orthogonal conditions the properties of the singular operator E(u), we have I = C E( U j n ) Un k dx 1 j,k l,j k = E( U j )(x x j n ) Uk (x x k n ) dx 1 j,k l,j k = C E( U j )(x) U k (x (x k n xj n )) dx 1 j,k l,j k II 0, as n, C [ E( UnU i n ) j Un k 4 + E( Uk n ) L UnU i n j ] 1 i,j,k l,i j C 1 i,j,k l,i j U i nu j n U k n 4 0, as n III C E( UnU i n ) j UnU k n m 0, as n. 1 i,j,k,m l,i j,k m The last step estimations of I, II III follows from the proof of Claim (i) this completes the proof of Claim (ii). Therefore, by (.14) (.16)- (.18), we have (.) ( U j 4 + E( U j ) U j )dx 1, as n. J On the other h, by the definition of J, we have (.) J ( U j 4 + E( U j ) U j )dx U j U j. Since the series j Uj is convergent, there exists a j 0 1 such that (.4) U j0 = sup{ U j j 1}.

10 48 JIAN ZHANG AND SHIHUI ZHU It follows from (.)-(.4) that (.5) ( ) 1 J ( U j 4 + E( U j ) U j )dx sup{ U j j 1} ( ) U j ( ) U j0 U j U j0. It follows from (.15) that U j0 = 1, which implies that there exists only one term U j0 0 such that (.6) U j0 = 1, U j0 = 1 ( U j0 4 + E( U j0 ) U j0 )dx = 1 J. Therefore, we show that U j0 is the minimizer of J(u). It follows from Lemma. that (.7) 1 + J( U Uj0 Uj0 j0 + E( U j0 ))U j0 = 0. We may assume that U j0 is an real positive function by the definition of J(u). Now, we take U j0 = aq(λx + b) with Q is the positive solution of (.1). By some computations, we have that U j0 = a λ Q = 1, Uj0 = a λ Q = 1 ( U j0 4 + E( U j0 ) U j0 )dx = a4 λ ( Q 4 + E( Q ) Q )dx = 1 J. Applying Claim (.) (.), we have ( Q 4 + E( Q ) Q )dx = which implies that Q dx = Q dx, (.8) J = λ a 4 1 ( Q 4 + E( Q ) Q )dx = 1 a = Q. This completes the proof Proposition.. Next, we consider the following elliptic equation (.9) R 1 R + E( R )R = 0, R H 1. By the same argument in Theorem.1, we have the following theorem. Theorem.4. Let f H 1, then (.0) E( f ) f dx R f f, where R is the solution of Equation(.9). Remark.5. (i) To our knowledge, the uniqueness of the ground state Q(x) R(x) is still an open problem. Indeed, the known proofs of the uniqueness rely on radiality (see [1]). Since the E operator does not commute with rotation, one cannot deduce the radiality of minimizers to J(u) by symmetric rearrangement. (ii) The best constants of the generalized Gagliardo-Nirenberg inequalities (.4) (.0) are dependent on the space dimension N, but they are independent of the choices of the ground state solution Q(x) R(x). Cipolatti [] showed the

11 SHARP BLOW-UP CRITERIA 49 existence of a ground state solution of (.1) in dimension two three. Papanicolaou etal [18] computed the best constant of the generalized Gagliardo-Nirenberg inequality in dimension two. We point out that the method used in this paper is different from [, 18]. Moreover, our method can also be applied in the dimension N =. In the end, we collect Weinstein s results [4], we consider the following elliptic equation (p 1) (.1) P 5 p 4 4 P + P p 1 P = 0, P H 1. Strauss [1] showed the existence of equation (.1). Weinstein [4] showed the best constant of the Gagliardo-Nirenberg inequality, as follows Proposition.6. Let f H 1 1 < p < 5, then (.) f p+1 p+1 p + 1 P p 1 where P is the solution of Equation (.1). f (p 1) f 5 p 4. Sharp Blow-up Criteria In this section, using the sharp Gagliardo-Nirenberg inequalities obtained in Section, we obtain the sharp blow-up criteria to the Cauchy problem (1.1)-(1.). More precisely, establishing four classes of invariant evolution flows according to the value of p, we obtain the sharp blow-up criteria to the Cauchy problem (1.1)-(1.) for all p < 5. Sharp Criteria for p = Theorem 4.1. Let p =, u 0 H 1 satisfy (4.1) H(u 0 ) u 0 < 7 Q 4. Then, we have that (i) If (4.) u 0 u 0 < Q,, then the solution u(t, x) of the Cauchy problem (1.1)-(1.) exists globally. Moreover, u(t, x) satisfies (4.) u(t, x) u(t, x) < Q. (ii) If (4.4) u 0 u 0 > Q, x u 0 L, then the solution u(t, x) of the Cauchy problem (1.1)-(1.) blows up in finite time T < +, where Q is the solution of Equation (.1).

12 50 JIAN ZHANG AND SHIHUI ZHU Proof. Applying the generalized Gagliardo-Nirenberg inequality (Theorem.1), we have H(u) = 1 u dx 1 4 u 4 + E( u ) u dx (4.5) 1 u u u Q. Now, we define a function f(y) on [0, + ) by f(y) = 1 y u 0 Q y, then we have f(y) is continuous on [0, + ) (4.6) f (y) = y u 0 Q y = y(1 u 0 Q y). It is obvious that there are two roots for equation f (y) = 0: y 1 = 0, y = Q u 0. Hence, we have that y 1 y are two minimizers of f(y), f(y) is increasing on the interval [0, y ) decreasing on the interval [y, + ). Note that f(0) = 0 f max = f(y ) = Q 4 7 u 0. By the conservation of energy assumption (4.1), we have (4.7) f( u ) H(u) = H(u 0 ) < Q 4 7 u 0 = f(y ). Therefore, using the convexity monotony of f(y) the conservation laws, we obtain two invariant evolution flows generated by the Cauchy problem (1.1)-(1.), as follows. K 1 := {u H 1 0 < u u < Q, 0 < H(u) u < Q 4 7 } K := {u H 1 u u > Q, 0 < H(u) u < Q 4 7 }. Indeed, by the conservation of mass energy, we have u = u 0 H(u) = H(u 0 ). If u 0 K 1, we have 0 < H(u) u < Q 4 7 u 0 u 0 < Q, which implies that u 0 < y. Since f(y) is continuous increasing on [0, y ) f(y) < f max = Q 4 7 u 0, we have that for all t I(maximal existence interval) which implies that K 1 is invariant. If u 0 K, we have u(t, x) < y, u 0 u 0 > Q, which implies that u 0 > y. Since f(y) is continuous decreasing on [y, + ) f(y) < f max = Q 4 7 u 0, we have that for all t I(maximal existence interval) (4.8) u(t, x) > y u(t, x) u(t, x) > Q, which implies that K is invariant.

13 SHARP BLOW-UP CRITERIA 51 Now, we return to the proof the Theorem 4.1. By (4.1) (4.), we have u 0 K 1. Applying the invariant of K 1, we have that (4.) is true the solution u(t, x) of the Cauchy problem (1.1)-(1.) exists globally. This completes the part (i) of the proof. By (4.1) (4.4), we have u 0 K. Applying the invariant of K, we have (4.8) is true. If we assume x u 0 L, then we have x u(t, x) L by the local well-posedness. Thus, we recall the virial identity the conservation of energy H(u(t)) = H(u 0 ), we have (4.9) J (t) = d dt x u(t, x) dx = 8 u dx 6 u 4 + E( u ) u dx = 4H(u 0 ) 4 u. Multiplying both side of (4.9) by u 0, applying the conservation laws, (4.1) (4.8), we have (4.10) u 0 d dt x u(t, x) dx = 4H(u 0 ) u 0 4 u u 0 By the classical analysis identity (4.11) J(t) = J(0) + J (0)t + < 16 9 Q Q 4 = 0. t 0 J (s)(t s)ds, we have that the maximal existence interval I of u(t, x) must be finite, which implies that the solution u(t, x) of the Cauchy problem (1.1)-(1.) blows up in finite time T < +. This completes the proof. Sharp Criteria for p = Theorem 4.. Let p = 1 + 4, u 0 H 1 satisfy (4.1) u 0 < P H(u 0 ) < R 4 ( P 4 u 0 4 ) 7 u 0 P. 4 Then, we have (i) If (4.1) u 0 u 0 < R ( P 4 u 0 4 ), P 4 then the solution u(t, x) of the Cauchy problem (1.1)-(1.) exists globally. Moreover, for all time t, u(t, x) satisfies (4.14) u(t, x) u(t, x) < R ( P 4 u 0 4 ). P 4 (ii) If (4.15) u 0 u 0 > R ( P 4 u 0 4 ), P 4

14 5 JIAN ZHANG AND SHIHUI ZHU x u 0 L, then the solution u(t, x) of the Cauchy problem (1.1)-(1.) blows up in finite time T < +, where P is the solution of Equation (.1) R is the solution of Equation (.9). (4.16) Proof. Applying Theorem.4 Proposition.6, we have H(u) = 1 u dx 1 + u + dx E( u ) u dx 1 u u 4 P 4 u u R u. Now, we define a function f(y) on [0, + ) by f(y) = ( 1 u 0 4 )y u 0 P 4 R y, then we have f(y) is continuous on [0, + ) (4.17) f (y) = (1 u 0 4 )y u 0 P 4 R y. It is obvious that there are two roots for equation f (y) = 0: y 1 = 0, y = R 4 4 ( P u 0 ) u 0 P 4 > 0. Hence, we have that y 1 y are two minimizers of f(y), f(y) is increasing on the interval [0, y ) decreasing on the interval [y, + ). Note that f(y 1 ) = 0 f max = f(y ) = R 4 ( P 4 u 0 4 ) 7 u 0 P. 4 By the conservation of energy the assumption (4.1), we have (4.18) f( u ) H(u) = H(u 0 ) < f(y ). Therefore, using the convexity monotony of f(y) the conservation laws, we obtain two invariant evolution flows generated by the Cauchy problem (1.1)-(1.), as follows. K := {u H 1 0 < u u < R ( P 4 u 0 4 ), u P 4 < P, 0 < H(u) < D} K 4 := {u H 1 u u > R ( P 4 u 4 ), u P 4 < P, 0 < H(u) < D}, where D = R 4 ( P 4 u 4 ) 7 u 0 P. Indeed, by the conservation of mass energy, 4 we have u = u 0 H(u) = H(u 0 ). If u 0 K, we have u < P, 0 < H(u) < D u 0 < R ( P 4 u 0 4 ) u 0 P 4, which implies that u 0 <

15 SHARP BLOW-UP CRITERIA 5 y. Since f(y) is increasing on [0, y ) 0 < f(y) < D, we have that for all t I(maximal existence interval) u(t, x) u(t, x) < R ( P 4 u 0 4 ), P 4 which implies that K is invariant. If u 0 K 4, we have u < P, 0 < H(u) < D u 0 > R ( P 4 u 0 4 ), u 0 P 4 which implies u 0 > y. Since f(y) is decreasing on [y, ) 0 < f(y) < D, we have that for all t I(maximal existence interval) (4.19) u(t, x) > y u(t, x) u(t, x) > R ( P 4 u 0 4 ), P 4 which implies that K 4 is invariant. Now, we return to the proof the Theorem 4.. By (4.1) (4.1), we have u 0 K. Applying the invariant of K, we have that (4.14) is true the solution u(t, x) of the Cauchy problem (1.1)-(1.) exists globally. This completes the part (i) of the proof. By (4.1) (4.15), we have u 0 K 4. Applying the invariant of K 4, we have (4.19) is true. If we assume x u 0 L, then we have x u(t, x) L by the local well-posedness. Thus, we recall the virial identity the conservation of energy H(u(t)) = H(u 0 ), we have (4.0) J (t) = d dt x u(t, x) dx = 8 u dx 16 + u E( u ) u dx = 4H(u 0 ) 4 u dx u + dx 4 4 4H(u 0 ) 4[1 u0 4 P 4 ] u. By the assumption (4.1), it follows from (4.19) (4.0) that (4.1) u 0 d dt x u(t, x) dx 4H(u 0 ) u 0 4[1 u0 4 P 4 ] u u 0 < 16 R 4 ( P 4 u 0 4 ) 9 P 4 16 R 4 ( P 4 u 0 4 ) 9 P 4 = 0, which implies that the solution u(t, x) of the Cauchy problem (1.1)-(1.) blows up in finite time T < +. This completes the proof.

16 54 JIAN ZHANG AND SHIHUI ZHU In order to study the sharp thresholds of blow-up global existence for the Cauchy problem (1.1)-(1.) for < p < < p < 5, we need the following preparations. Let us define a function g(y) on [0, + ) (4.) g(y) = 1 (p 1) u 0 5 p 4 P p 1 y (p 1) u 0 P y, where P is the solution of Equation (.1). We claim that there exists an unique positive solution y 0 for the equation g(y) = 0. Indeed, by some computations, we have for y > 0 (4.) g (y) = (p 1)(p 7) u 0 5 p 8 P p 1 y (p 1) u 0 P < 0, which implies that g(y) is decreasing on [0, + ). Notice that g(0) = 1 > 0, g( P (p 1) ) = ( u 0 6 p u 0 4 )p 7 P 6 p < 0. Since g(y) is continuous on [0, + ), there exists an unique positive y 0 [0, P u 0 ] such that g(y 0 ) = 0. Sharp Criteria for < p < Theorem 4.. Let < p <, u 0 H 1 satisfy (4.4) 0 < H(u 0 ) < p 7 6(p 1) y 0. Then, we have (i) If (4.5) u 0 < y 0, then the solution u(t, x) of the Cauchy problem (1.1)-(1.) exists globally. Moreover, for all time t, u(t, x) satisfies (4.6) u(t, x) < y 0. (ii) If (4.7) u 0 > y 0, x u 0 L, then the solution u(t, x) of the Cauchy problem (1.1)-(1.) blows up in finite time T < +, where y 0 is the unique positive solution of the equation g(y) = 0 g(y) is defined in (4.).

17 SHARP BLOW-UP CRITERIA 55 Proof. Applying the Gagliardo-Nirenberg inequality (Proposition.6), we have H(u) = 1 u dx 1 1+p u p+1 dx 1 4 E( u ) u dx (4.8) 1 u dx 1 1+p u p+1 dx 1 4 u 4 dx 5 p 1 u u P p 1 Now, we define a function f(y) on [0, + ) by f(y) = 1 y u 0 5 p P p 1 then we have f(y) is continuous on [0, + ) (4.9) f (y) = y[1 (p 1) u 0 5 p 4 P p 1 By the properties of g(y), we have u (p 1) u u P. y (p 1) u 0 P y, u 0 P y] = yg(y). y (p 1) (4.0) f (0) = f (y 0 ) = y 0 g(y 0 ) = 0 f (y 0 ) = g(y 0 ) + y 0 g (y 0 ) < 0, which implies that 0 y 0 are two minimizers of f(y), f(y) is increasing on the interval [0, y 0 ) decreasing on the interval [y 0, + ). Note that f max = f(y 0 ) f(0) = 0. Since g(y 0 ) = 0, we have (4.1) 5 p f max = f(y 0 ) = 1 y 0 u0 P p 1 y (p 1) 0 u0 y P 0 y P 0 = [ 1 (p 1) ]y [ p 1 1 ] u0 p 7 6(p 1) y 0. By the conservation of energy the assumption (4.4), we have (4.) H(u) = H(u 0 ) < p 7 6(p 1) y 0 < f max. Therefore, using the convexity monotony of f(y) the conservation laws, we obtain two invariant evolution flows generated by the Cauchy problem (1.1)-(1.), as follows. We set K 5 := {u H 1 0 < u < y 0, 0 < H(u) < p 7 6(p 1) y 0 } K 6 := {u H 1 u > y 0, 0 < H(u) < p 7 6(p 1) y 0 }. Indeed, by the conservation of mass energy, we have u = u 0 H(u) = H(u 0 ). If u 0 K 5, we have 0 < H(u) < p 7 6(p 1) y 0 u 0 < y 0. Since f(y) is continuous increasing on [0, y 0 ) f(y) < p 7 6(p 1) y 0 < f max, we have that for all t I(maximal existence interval) which implies that K 5 is invariant. u(t, x) < y 0,

18 56 JIAN ZHANG AND SHIHUI ZHU If u 0 K 6, we have 0 < H(u) < p 7 6(p 1) y 0 u 0 > y 0. Since f(y) is continuous decreasing on [y 0, + ) f(y) < p 7 6(p 1) y 0 < f max, we have that for all t I(maximal existence interval) (4.) u(t, x) > y 0, which implies that K 6 is invariant. Now, we return to the proof the Theorem 4.. By (4.4) (4.5), we have u 0 K 5. Applying the invariant of K 5, we have that (4.6) is true the solution u(t, x) of the Cauchy problem (1.1)-(1.) exists globally. This completes the part (i) of the proof. By (4.4) (4.7), we have u 0 K 6. Applying the invariant of K 6, we have (4.) is true. If we assume x u 0 L, then we have x u(t, x) L by the local well-posedness. Thus, we recall the virial identity the conservation of energy H(u(t)) = H(u 0 ), we have (4.4) J (t) = d dt x u(t, x) dx = 8 u dx 1(p 1) p+1 u p+1 dx 1 4 E( u ) u dx = 1(p 1)H(u 0 ) [6(p 1) 8] u + [(p 1) 6] E( u ) u dx [(p 1) 4]y 0 [6(p 1) 8]y 0 = 0, for < p <, which implies that the solution u(t, x) of the Cauchy problem (1.1)-(1.) blows up in finite time T < +. This completes the proof. Sharp Criteria for < p < 5 Theorem 4.4. Let < p < 5, u 0 H 1 satisfy (4.5) 0 < H(u 0 ) < 1 6 y 0. Then, we have that (i) If (4.6) u 0 < y 0, then the solution u(t, x) of the Cauchy problem (1.1)-(1.) exists globally. Moreover, for all time t, u(t, x) satisfies (4.7) u(t, x) < y 0. (ii) If (4.8) u 0 > y 0, x u 0 L, then the solution u(t, x) of the Cauchy problem (1.1)-(1.) blows up in finite time T < +, where y 0 is the unique positive solution of the equation g(y) = 0 g(y) is defined in (4.).

19 SHARP BLOW-UP CRITERIA 57 Proof. Applying the Gagliardo-Nirenberg inequality (Proposition.6), we have H(u) = 1 u dx 1 1+p u p E( u ) u dx (4.9) 5 p 1 u u P p 1 Now, we define a function f(y) on [0, + ) f(y) = 1 y u 0 5 p P p 1 then we have f(y) is continuous on [0, + ) (4.40) f (y) = y[1 (p 1) u 0 5 p 4 P p 1 u (p 1) u u P. y (p 1) u 0 P y, u 0 P y] = yg(y), y (p 1) where g(y) is defined in (4.). By the properties of g(y), we have (4.41) f (0) = f (y 0 ) = y 0 g(y 0 ) = 0 f (y 0 ) = g(y 0 ) + y 0 g (y 0 ) < 0, which implies that 0 y 0 are two minimizers of f(y), f(y) is increasing on the interval [0, y 0 ) decreasing on the interval [y 0, + ). Note that f(0) = 0 f max = f(y 0 ). Since g(y 0 ) = 0, we have (4.4) f max = f(y 0 ) 5 p = 1 y 0 u0 P p 1 y (p 1) 0 u0 P y 0 = [ 1 1 ]y 0 + [ p ] u0 1 6 y 0. 5 p P p 1 y (p 1) 0 By the conservation of energy the assumption (4.5), we have (4.4) 0 < H(u) = H(u 0 ) < 1 6 y 0 < f max. Therefore, using the convexity monotony of f(y) the conservation laws, we obtain two invariant evolution flows generated by the Cauchy problem (1.1)-(1.), as follows. We set K 7 := {u H 1 0 < u < y 0, 0 < H(u) < 1 6 y 0} K 8 := {u H 1 u > y 0, 0 < H(u) < 1 6 y 0}. Indeed, by the conservation of mass energy, we have u = u 0 H(u) = H(u 0 ). If u 0 K 7, we have u 0 < y 0. Since f(y) is continuous increasing on [0, y 0 ) y [0, + ), f(y) < 1 6 y 0 < f max, we have that for all t I(maximal existence interval) which implies that K 7 is invariant. u(t, x) < y 0,

20 58 JIAN ZHANG AND SHIHUI ZHU If u 0 K 8, we have u 0 > y 0. Since f(y) is continuous decreasing on [y 0, + ) y [0, + ), f(y) < 1 6 y 0 < f max, we have that for all t I(maximal existence interval) (4.44) u(t, x) > y 0, which implies that K 8 is invariant. Now, we return to the proof the Theorem 4.4. By (4.5) (4.6), we have u 0 K 7. Applying the invariant of K 7, we have that (4.7) is true the solution u(t, x) of the Cauchy problem (1.1)-(1.) exists globally. This completes the part (i) of the proof. By (4.5) (4.8), we have u 0 K 8. Applying the invariant of K 8, we have (4.44) is true. If we assume x u 0 L, then we have x u(t, x) L by the local well-posedness. Thus, we recall the virial identity the conservation of energy H(u(t)) = H(u 0 ), we have (4.45) J (t) = d dt x u(t, x) dx = 8 u dx 1(p 1) p+1 = 4H(u 0 ) 4 u + 4 1(p 1) p+1 < 4y0 4y 0 = 0, u p+1 dx 1 4 E( u ) u dx u p+1 dx for < p < 5, which implies that the solution u(t, x) of the Cauchy problem (1.1)-(1.) blows up in finite time T < +. This completes the proof. 5. Properties of Blow-up Solutions In this section, we shall investigate the blow-up properties of the solutions to the Cauchy problem (1.1)-(1.). We prove the nonexistence of the L strong limit to the blow-up solutions of the Cauchy problem (1.1)-(1.) for 1 < p, as follows. Theorem 5.1. Let 1 < p the initial data u 0 H 1. If the solution of the Cauchy problem (1.1)-(1.) u(t, x) blows up in finite time T < +, Then for any sequence {t n } n=1 such that t n T as n, {u(t n, x)} n=1 does not have any strong limit in L as n. Proof. We prove this result by contradiction. Suppose that {u(t n, x)} n=1 has a strong limit in L along a sequence {t n } n=1 such that t n T as n. Since the solution u(t, x) of the Cauchy problem (1.1)-(1.) blows up at finite time T in H 1, we have u(t n ) + as n +. By the conservation of energy H(u) := 1 u(t, x) dx 1 p + 1 for 1 < p, we claim n m, u(t, x) p+1 dx 1 4 (5.1) u(t n ) C u(t n ) u(t m ) C u(t m ) C. E( u ) u dx = H(u 0 ), Indeed, if p =, by the conservation of energy, we have u(t n ) L H(u 0 ) + 1 u(t n) E( u(tn ) ) u(t n ) dx H(u 0 ) + C u(t n ) C C u(t n ) u(t m ) C u(t m ) C.

21 SHARP BLOW-UP CRITERIA 59 If 1 < p <, using the Gagliardo-Nirenberg inequality Hölder inequality, we have ε > 0 u(t n ) p+1 p+1 C u(t n) (p 1) u 0 5 p ε u(t n ) + C(ε). By the conservation of energy, we have u(t n ) H(u 0 ) + ε u(t n ) + C u(t n ) C(ε) ε u(t n ) + C u(t n) u(t m ) C u(t m) C, for ε < 1, which implies that Claim (5.1) is true. Since < 4 < 6, applying the Hölder s inequality for 1 4 = θ + 1 θ 6, θ (0, 1), we have (5.) u(t n ) u(t m ) 4 4 C u(t n ) u(t m ) 4θ u(t n) u(t m ) 4(1 θ) 6 C u(t n ) u(t m ) (u(t n ) u(t m )). It follows from (5.1) (5.) that for m n large enough (5.) u(t n ) C u(t n ) u(t m ) (u(t n ) + C m, where C m depends on m. On the other h, since the sequence {u(t n )} n=1 converges strongly in L, there is a positive integer k such that for all n k, m k C u(t n ) u(t m ) 1. Therefore, choosing m = k in the inequality (5.), we obtain that for all n n k (5.4) u(t n ) 1 u(t n) + C k, which implies that the sequence { u(t n )} n=1 is bounded in L. This is contradictory to that u(t, x) blows up in finite time T < +. This completes the proof. Acknowledgement We thank the referee for pointing out some misprints helpful suggestions. References [1] Cazenave, T. Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, NYU, CIMS, AMS 00. [] Cipolatti, R. On the existence of sting waves for a Davey-Stewartson system, Commun. in PDE, 17(199), [] Cipolatti, R. On the instability of ground states for a Davey-Stewartson system, Ann. Inst. Henri Poincaré. Phys. Theor., 58(199), [4] Davey, A. Stewartson, K. On three-dimensional packets of surfaces waves, Proc. R. Soc. A., 8(1974), [5] Gan, Z. H. Zhang, J. Sharp conditions of global existence for the generalized Davey- Stewartson system in three dimensional space, Acta Math. Scientia, Ser. A, 6(006), [6] Gan, Z. H. Zhang, J. Sharp threshold of global existence instability of sting wave for a Davey-Stewartson system, Commun. Math. Phys., 8(008), [7] Gérard, P. Description du defaut de compacite de l injection de Sobolev, ESAIM Control Optim. Calc. Var., (1998), 1-. [8] Ghidaglia, J.-M. Saut, J. C. On the initial value problem for the Davey-Stewartson systems, Nonlinearity, (1990), [9] Guo, B. L. Wang, B. X. The Cauchy problem for Davey-Stewartson systems, Commun. Pure Appl. Math., 5(1999),

22 60 JIAN ZHANG AND SHIHUI ZHU [10] Hmidi, T. Keraani, S. Blowup theory for the critical nonlinear Schrödinger equations revisited, Internat. Math. Res. Notices, 46(005), [11] Holmer, J. Roudenko, S. On blow-up solutions to the D cubic nonlinear Schrödinger equation, Appl. Math. Research express, Vol. 007, Article ID 004, 9 pages. [1] Kwong, M. K. Uniqueness of positive solutions of u u + u p = 0 in R n, Arch. Rational. Mech. Anal., 105(1989), [1] Li, X. G., Zhang, J., Lai, S. Y. Wu, Y. H. The sharp threshold limiting profile of blow-up solutions for a Davey-Stewartson system, J. Differential Equations, 50(011), [14] Merle, F. Raphaël, P. Blow-up dynamic upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Annals of Math., 16(005), [15] Ohta, M. Stability of sting waves for the generalized Davey-Stewartson system, J. Dynam. Differ. Eqs., 6(1994), 5-4. [16] Ohta, M. Instability of sting waves the generalized Davey-Stewartson systems, Ann. Inst. Henri Poincare, Phys. Theor., 6(1995), [17] Ozawa, T. Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems, Proc. Roy. Soc. London Ser. A, 46(199), [18] Papanicolaou, G.C., Sulem, C., Sulem, P-L. Wang, X.P. The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves, Physica D, 7(1994), [19] Richards, G. Mass concentration for the Davey-Stewartson system, / v1 [0] Shu, J. Zhang, J. Sharp conditions of global existence for the generalized Davey- Stewartson system, IMA J. Appl. Math., 7(007), 6-4. [1] Strauss, W. A. Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55(1977), [] Sulem, C. Sulem, P. L. The nonlinear Schrödinger equation. Self-focusing wave collapse, Appl. Math. Sci., vol. 19, Springer-Verlag, New York, [] Wang, B. X. Guo, B. L. On the initial value problem scattering of solutions for the generalized Davey-Stewartson systems, Science in China, 44(001), [4] Weinstein, M. I. Nonlinear Schrödinger equations sharp interpolation estimates, Commun. Math. Phys., 87(198), College of Mathematics Software Science, Sichuan Normal University, Chengdu, , China College of Mathematics Software Science, Sichuan Normal University, Chengdu, , China; College of Mathematics, Sichuan University, Chengdu, , China address: zhu shihui008@16.com