Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential

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1 Appl. Math. Mech. -Engl. Ed. 31(4), (2010) DOI /s c Shanghai University and Springer-Verlag Berlin Heidelberg 2010 Applied Mathematics and Mechanics (English Edition) Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential Run-zhang XU ( ) 1, Chuang XU ( ) 2 (1. College of Science, Harbin Engineering University, Harbin , P. R. China; 2. Department of Mathematics, Harbin Institute of Technology, Harbin , P. R. China) (Communicated by Li-qun CHEN) Abstract This paper discusses a class of nonlinear Schrödinger equations with combined power-type nonlinearities and harmonic potential. By constructing a variational problem the potential well method is applied. The structure of the potential well and the properties of the depth function are given. The invariance of some sets for the problem is shown. It is proven that, if the initial data are in the potential well or out of it, the solutions will lie in the potential well or lie out of it, respectively. By the convexity method, the sharp condition of the global well-posedness is given. Key words sharp criterion, invariant manifold, harmonic potential, combined powertype nonlinearities Chinese Library Classification O Mathematics Subject Classification 35Q55 1 Introduction In this paper, we deal with the following initial value problem for the nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential: iϕt +Δϕ x 2 ϕ λ 1 ϕ p1 ϕ λ 2 ϕ p2 ϕ =0, t > 0, x R N, (1) ϕ(0,x)=ϕ 0 (x). When λ 1 = 1, λ 2 =0, and p 1 = 3, (1) describes attractive Bose-Einstein condensation. The slowing of atoms by use of cooling apparatus produces a singular quantum state known as a Bose condensate or Bose-Einstein condensate. It is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near to absolute zero. Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, and all wave functions overlap each other, at which Received Jul. 8, 2009 / Revised Mar. 10, 2010 Project supported by the National Natural Science Foundation of China (Nos and ), the Natural Science Foundation of Heilongjiang Province (Nos. A and A200810), the Science and Technology Foundation of Education Office of Heilongjiang Province (No ), and the Foundational Science Foundation of Harbin Engineering University Corresponding author Run-zhang XU, Ph. D., xurunzh@yahoo.com.cn

2 522 Run-zhang XU and Chuang XU point quantum effects become apparent on a macroscopic scale. It is also well-known that iϕt +Δϕ + ϕ p 1 ϕ =0, t > 0, x R N, ϕ(0,x)=ϕ 0 (x) (2) is one of the basic evolution models for nonlinear waves in various branches of physics, such as the quantum mechanics and the field theory. Many papers have studied (2). Zhang [1] studied the global existence of (2) and the relationship between the Schrödinger equation and its ground state. Ginibre and Velo [2] established the local existence of the Cauchy problems in the energy class H 1 (R N ). Glassey [3], Ogawa and Tsutsumi [4 5] proved that for some initial data, especially for a class of sufficiently large data, the solutions of the Cauchy problem for (2) blow up in finite time. It is found that for sufficiently small initial data, the solutions of the Cauchy problem for (2) globally exist on t (0, ) (see. [6 10]). Strauss [11] and Cazenave [12] also mentioned this topic in their monographs. For the problem with harmonic potential, that is, iϕt +Δϕ x 2 ϕ + ϕ p ϕ =0, t > 0, x R N, (3) ϕ(0,x)=ϕ 0 (x), or iϕt +Δϕ V (x)ϕ + ϕ p ϕ =0, t > 0, x R N, ϕ(0,x)=ϕ 0 (x). (4) There are many mathematicians who addressed these problems. Chen and Zhang [13] derived a global existence condition for the supercritical case for (3), which coincides with the critical case. For (4), Fujiwara [14] proved that the smoothness of the Schrödinger kernel for potentials of quadratic growth. Yajima [15] showed that for super-quadratic potentials, the Schrödinger kernel is nowhere C 1.FromOh [16], it is known that quadratic potentials are the highest order potentials for local well-posedness of the equation. When 1 <p<1+4/n, Zhang [17] proved that global solutions of the Cauchy problem of (3) exist for any initial data in the energy space. When p =1+4/N, Zhang [18] showed that there exists a sharp condition of the global existence. When p>1+4/n, Cazenave [12],Carles [19 20], and Tsurumi and Wadati [21] showed that the solutions of the Cauchy problem of (3) blow up in finite time for some initial data, especially for a class of sufficiently large initial data; but the solutions of the Cauchy problem of (3) globally exist for other initial data, especially for a class of sufficiently small initial data, see [19 21]. Moreover, Shu and Zhang [22],XuandLiu [23] also studied (3) for its global existence and blowup. For nonlinear Schrödinger equations with combined power-type nonlinearities, iϕt +Δϕ = λ 1 ϕ p1 ϕ + λ 2 ϕ p2 ϕ, t > 0, x R N, (5) ϕ(0,x)=ϕ 0 (x), Tao, Visan and Zhang described the global existence and blowup and most importantly obtained scattering results in [24]. In [25], Shu and Zhang described the existence of standing wave associated with the ground states of (5) with λ 1 = 1 andλ 2 = 1. Furthermore, they obtained a sharp condition for blowup and global existence to the solutions of the Cauchy problem, showed that for small initial data, the solutions globally exist, and proved the instability of standing wave. Xu et al. [26] attained the vacuum isolating of solutions and the threshold result of global existence and nonexistence of solutions of (6), i.e., the reaction-diffusion equation u t Δu = a u p 1 u b u q 1 u, t > 0, x Ω, u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x Ω, (6) u(x, t) =0, x Ω, t 0.

3 Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential 523 From the work mentioned above, we are apt to consider the Schrödinger equation with combined power-type nonlinearities and harmonic potential. In this paper, we construct a variational problem and derive sharp conditions for global existence and blowup by introducing the potential well originated from [27] and by the concavity method to prove the blowup originated from [3]. Most importantly, we generalize the results in [22]. This paper is organized as follows. In the second section, we give some concerned preliminaries. In the third section, we give the variational problem and invariant manifolds. In the last section, we give the sharp conditions for the global existence. 2 Preliminaries In this section, we like to introduce some functionals that are useful later and a space. We will use them to construct a variational problem. For (1), we first equip the following space: } H = ψ H 1 (R N ) x 2 ψ 2 dx < (7) with the inner product as follows: ψ, φ := ψ φ + ψφ + x 2 ψφdx, (8) whose associated norm is H. Further, we define the energy functional, 1 E(ϕ) = 2 ϕ x 2 ϕ 2 + λ 1 p 1 +2 ϕ p1+2 + λ 2 p 2 +2 ϕ p2+2) dx, (9) and the following functionals: 1 P (ϕ) = 2 ϕ ϕ x 2 ϕ 2 + λ 1 p 1 +2 ϕ p1+2 + λ 2 p 2 +2 ϕ p2+2) dx, (10) I(ϕ) = ϕ 2 + ϕ 2 + x 2 ϕ 2 + λ 1Np 1 2(p 2 +2) ϕ p2+2) dx. (11) We introduce the following lemma, with the equality which will be used to prove the blowup phenomenon in the third section. Lemma 1 Let ϕ 0 H and ϕ be a solution of the Cauchy problem (1) in C([0,T]; H). Set V (t) = x 2 ϕ 2 dx. Then, one has V (t) =8 ϕ 2 x 2 ϕ 2 + λ 1Np 1 2(p 2 +2) ϕ p2+2) dx. (12) 3 The variational problem and invariant manifolds Now, we define the manifold We consider the following variational problem: M := ψ H\O} : I(ψ) =0}. d = inf P (ψ). (13) ψ H\0}:I(ψ)=0} One can prove the following lemma easily by the similar argument to that in [22].

4 524 Run-zhang XU and Chuang XU Lemma 2 d>0. Theorem 1 Define W := ψ H, P(ψ) <d,i(ψ) > 0}. (14) Then, W is an invariant manifold of (1), that is, if ϕ 0 W, then the solution ϕ(x, t) of the Cauchy problem (1) also satisfies ϕ(x, t) W for any t [0,T). Proof Let ϕ 0 W. Then, we have P (ϕ 0 ) <d. By the local well-posedness [28],there exists a unique ϕ(x, t) C([0,T); H) witht such that ϕ(x, t) is a solution of the Cauchy problem (1). If there exists a t 1 such that P (ϕ(x, t 1 )) d, then by the conservation laws for both energy and mass (see [3, 29, 30]), we have which contradicts P (ϕ 0 ) <d. Therefore, P (ϕ 0 )=P (ϕ(x, t 1 )) d, (15) P (ϕ) <d, t [0,T). (16) Now, we show I(ϕ) > 0fort [0,T). First, note that I(ϕ 0 ) > 0 due to ϕ 0 W. Arguing by contradiction, from the continuity of I(ϕ), there is a t 2 [0,T) such that I(ϕ(x, t 2 )) = 0. If ϕ(x, t 2 ) = 0, then by the mass-conservation law, we have 0 = ϕ(x, t 2 ) 2 dx = ϕ 0 2 dx, which indicates ϕ 0 =0. So,wehaveI(ϕ 0 ) = 0, which contradicts I(ϕ 0 ) > 0. If ϕ(x, t 2 ) 0, by (13), we have P (ϕ(x, t 2 )) d, which contradicts (16). Therefore, I(ϕ) > 0 for all t [0,T). Fromtheabove,weprovethatϕ(x, t) W for any t [0,T). This completes the proof of the theorem. By the same argument as Theorem 1, we can get the following result. Theorem 2 Define V := ψ H, P(ψ) <d,i(ψ) < 0}. Then, V is an invariant manifold of (1). 4 Sharp conditions for global existence Theorem 3 If ϕ 0 W, then the solution ϕ(x, t) of the Cauchy problem (1) globally exists on t [0, ). Proof First, we let ϕ 0 W. Thus, Theorem 2 implies that the solution ϕ(x, t) ofthe Cauchy problem (1) satisfies that ϕ(x, t) W for t [0, T). For fixed t [0, T), denote ϕ(x, t) =ϕ. Thus,wehaveP (ϕ) <dand I(ϕ) > 0. From I(ϕ) > 0, we have It follows from (10) and (11) that λ1 Np 1 2(p 1 +2) ϕ p1+2 + λ 2Np 1 2(p 2 +2) ϕ p2+2) dx λ1 Np 1 > 2(p 2 +2) ϕ p2+2) dx > ϕ 2 + ϕ 2 + x 2 ϕ 2) dx. (17) d>p(ϕ) 1 = 2 ϕ ϕ x 2 ϕ 2 + λ 1 p 1 +2 ϕ p1+2 + λ 2 p 2 +2 ϕ p2+2) dx 1 > 2 ϕ ϕ 2 + x 2 ϕ 2 2 ϕ 2 + ϕ 2 + x 2 ϕ 2) dx Np 1 ( 1 = 2 2 ) ϕ 2 + ϕ 2 + x 2 ϕ 2) dx, (18) Np 1

5 Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential 525 which implies ϕ 2 + ϕ 2 + x 2 ϕ 2) dx < 2Np 1d Np 1 4. Therefore, the local well-posedness implies that ϕ globally exists on t [0, ). This completes the proof of this theorem. Theorem 4 If ϕ 0 V, then the solution ϕ(x, t) of the Cauchy problem (1) blows up in finite time. Proof From ϕ 0 V, Theorem 2 implies that the solution ϕ(x, t) of the Cauchy problem (1) satisfies that ϕ(x, t) V for t [0,T). For V (t) = x 2 ϕ 2 dx, (12) and (11) imply that V (t) =8 ϕ 2 x 2 ϕ 2 + λ 1Np 1 2(p 2 +2) ϕ p2+2) dx < 8 ϕ 2 + ϕ 2 + x 2 ϕ 2 + λ 1Np 1 2(p 1 +2) ϕ p1+2) dx 16 x 2 ϕ 2 dx < 16 x 2 ϕ 2 dx = 16V (t). (19) Now, we show that there exists a T 1 (0, ) such that V (t) > 0fort [0,T 1 ]andv (T 1 )=0. Arguing by contradiction, suppose t [0, ), V (t) > 0. Set It is easy to show that g(t) = V (t) V (t). g (t) = V (t) ( V V (t) (t) ) 2 < 16 g 2 (t). (20) V (t) Next, we show g(t) 0foranyt [0, ). Arguing by contradiction again, suppose that there is a t 0 such that g(t 0 ) = 0, i.e., V (t 0 )=0. ByV (t) < 0, we have V (t) < 0fort (t 0, ). Hence, we have g(t) < 0fort (t 0, ). For any fixed t 1 >t 0, dividing (20) by g 2 (t), we have g (t) g 2 (t) < 16 g 2 (t) 1 < 1. Further, we derive namely, t t 1 g (τ) t g 2 (τ) dτ < 1dτ, t 1 1 g(t) > 1 g(t 1 ) +(t t 1), (21) which indicates that there exist t 2 >t 1 such that g(t) > 0, t (t 2, ). (22)

6 526 Run-zhang XU and Chuang XU This contradicts g(t) < 0fort (t 0, ). Hence, we have g(t) 0foranyt [0, ). By (21) and (22), for t (0, ), we have 1 g(t) > 1 g(0) + t. Hence, V 1 (t) > 0fort [ g(0), ). Therefore, V (t) increases on [ 1 g(0), ), and V (t) < 16V (t) < 0. Further, we have t 0 V (τ)dτ < 16V (0)t, i.e., that is, Again, we have t 0 V (t) V (0) < 16V (0)t, V (t) <V (0) 16V (0)t. V (τ)dτ < V (0)t 8V (0)t 2. Namely, V (t) V (0) <V (0)t 8V (0)t 2. Therefore, we have V (t) <V(0) + V (0)t 8V (0)t 2, which contradicts V (t) > 0fort [0, ). So far, we have shown that there exists a T 1 (0, ) such that V (t) > 0fort [0,T 1 ]andv(t 1 ) = 0. Again, by the inequality [30] ϕ 2 2 N ϕ xϕ, we get lim ϕ =, t T 1 which indicates lim ϕ H =, t T 1 i.e., the solution of the Cauchy problem (1) blows up in finite time. This completes the proof of the theorem. Remark 1 It is clear that ψ H, P(ψ) <d} = W V O}. Thus, Theorem 3 and Theorem 4 are both sharp. By Theorem 3, we get another condition for the global existence of the solution of (1). Corollary 1 If ϕ 0 satisfies ϕ 0 2 H < 2d, then the solution ϕ of the Cauchy problem (1) globally exists on t [0, ). Proof If ϕ 0 = 0, the mass conservation implies that ϕ = 0. Hence, it is obvious that the solution ϕ of the Cauchy problem (1) globally exists on t [0, ). If ϕ 0 0,from ϕ 0 2 H < 2d we have P (ϕ 0 ) <d. Moreover, we claim that I(ϕ 0 ) > 0. Otherwise, there is a 0 <μ 1such that I(μϕ 0 )=0. Thus,P (μϕ 0 ) d. On the other hand, μϕ 0 2 H = μ 2 ϕ 0 2 H < 2μ 2 d<2d. It follows that P (μϕ 0 ) <d. Thisisacontradiction. Therefore, we have ϕ 0 W.ThusTheorem 3 implies this corollary. Remark 2 It is easy to show that by setting λ 1 =0,λ 2 = 1 (orλ 2 =0,λ 1 = 1), the results in [22] can be derived. Acknowledgements We thank the anonymous referee for his or her valuable suggestions.

7 Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential 527 References [1] Zhang, J. Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations. Nonlinear Analysis 48(2), (2002) [2] Ginibre, J. and Velo, G. On a class of nonlinear Schrödinger equations. Journal of Functional Analysis 32(1), 1 71 (1979) [3] Glassey, R. T. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. Journal of Mathematical Physics 18(9), (1977) [4] Ogawa, T. and Tsutsumi, Y. Blow-up of H 1 solution for the nonlinear Schrödinger equation. Journal of Differential Equations 92(2), (1991) [5] Ogawa,T.andTsutsumi,Y.Blow-upofH 1 solution for the nonlinear Schrödinger equation with critical power nonlinearity. Proceedings of the American Mathematical Society 111(2), (1991) [6] Kenig, C., Ponce, G., and Vega, L. Small solution to nonlinear Schrödinger equations. Annales de I Institut Henri Poincaré Nonlinear Analysis 10(3), (1993) [7] Hayashi, N., Nakamitsu, K., and Tsutsumi, M. On solutions of the initial value problem for the nonlinear Schrödinger equations. Journal of Functional Analysis 71, (1987) [8] Hayashi, N. and Tsutsumi Y. Scattering theory for Hartree type equations. Annales de I Institut Henri Poincaré, Physique Théorique 46, (1987) [9] Ginibre, J. and Velo, G. The global Cauchy problem for the nonlinear Schrödinger equation. Annales de I Institut Henri Poincaré Nonlinear Analysis 2(4), (1985) [10] Ginibre, J. and Ozawa, T. Long range scattering for nonlinear Schrödinger and Hartree equations in space dimensions n 2. Communications in Mathematical Physics 151(3), (1993) [11] Strauss, W. A. Nonlinear Wave Equations, Conference Board of the Mathematical Sciences, American Mathematical Society, Providence, Rhode Island (1989) [12] Cazenave, T. An Introduction to Nonlinear Schrödinger Equations, Textos de Metodos Matematicos, Rio de Janeiro (1989) [13] Chen, G. G. and Zhang, J. Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential. Journal of Mathematical Analysis and Applications 320(2), (2006) [14] Fujiwara, D. Remarks on convergence of the Feynman path integrals. Duke Mathematical Journal 47(3), (1980) [15] Yajima, K. On fundamental solution of time dependent Schrödinger equations. Contemporary Mathematics 217, (1998) [16] Oh, Y. G. Cauchy problem and Ehrenfest s law of nonlinear Schrödinger equations with potentials. Journal of Differential Equations 81(2), (1989) [17] Zhang, J. Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials. Zeitschrift für Angewandte Mathematik und Physik 51(3), (2000) [18] Zhang, J. Stability of attractive Bose-Einstein condensates. Journal of Statistical Physics 101(3-4), (2000) [19] Carles, R. Critical nonlinear Schrödinger equation with and without harmonic potential. Mathematical Models and Methods in Applied Sciences 12(10), (2002) [20] Carles, R. Remarks on the nonlinear Schrödinger equation with harmonic potential. Annales Henri Poincaré 3(3-4), (2002) [21] Tsurumi, T. and Wadati, M. Collapses of wave functions in multidimensional nonlinear Schrödinger equations under harmonic potential. Journal of the Physical Society of Japan 66(10), (1997) [22] Shu, J. and Zhang, J. Nonlinear Schrödinger equation with harmonic potential. Journal of Mathematical Physics 47(6), (2006) [23] Xu, R. Z. and Liu, Y. C. Remarks on nonlinear Schrödinger equation with harmonic potential. Journal of Mathematical Physics 49(4), (2008)

8 528 Run-zhang XU and Chuang XU [24] Tao, T., Visan, M., and Zhang, X. Y. The nonlinear Schrödinger equation with combined powertype nonlinearities. Commumications in Partial Differential Equations 32(7-9), (2007) [25] Shu, J. and Zhang, J. Instability of standing waves for a class of nonlinear Schrödinger equations. Journal of Mathematical Analysis and Applications 327(2), (2007) [26] Xu, R. Z., Zhang, W. Y., and Wu, W. N. Nonlinear Analysis Research Trends, Nova Science Publishers, Incorporation, Hauppauge, New York (2008) [27] Payne, L. E. and Sattinger, D. H. Saddle points and instability of nonlinear hyperbolic equations. Israel Journal of Mathematics 22(3-4), (1975) [28] Kato, T. On nonlinear Shrödinger equations. Annales de I Institut Henri Poincaré, Physique Théorique 49(1), (1987) [29] Cazenave, T. Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, Providence, Rhode Island (2003) [30] Tsutsumi, Y. and Zhang, J. Instability of optical solitons for two-wave interaction model in cubic nonlinear media. Advances in Mathematical Sciences and Applications 8(2), (1998)