Volume 2013, Issue 1 238410

Research Article

Open Access

Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

Xiaowei An,

Xiaowei An

School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China tju.edu.cn

Department of Basic Curriculum, The Chinese People’s Armed Police Force Academy, Lang Fang, He Bei 065000, China

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Desheng Li,

Desheng Li

Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China tju.edu.cn

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Xianfa Song,

Corresponding Author

Xianfa Song

[email protected]

Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China tju.edu.cn

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Xiaowei An,

Xiaowei An

School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China tju.edu.cn

Department of Basic Curriculum, The Chinese People’s Armed Police Force Academy, Lang Fang, He Bei 065000, China

Search for more papers by this author

Desheng Li,

Desheng Li

Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China tju.edu.cn

Search for more papers by this author

Xianfa Song,

Corresponding Author

Xianfa Song

[email protected]

Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China tju.edu.cn

Search for more papers by this author

First published: 30 November 2013

https://doi.org/10.1155/2013/238410

Academic Editor: Sining Zheng

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Abstract

We consider the following Cauchy problem: −iu_t = Δu − V(x)u + f(x, |u|²)u + (W(x)⋆|u|²)u, x ∈ ℝ^N, t > 0, u(x, 0) = u₀(x), x ∈ ℝ^N, where V(x) and W(x) are real-valued potentials and V(x) ≥ 0 and W(x) is even, f(x, |u|²) is measurable in x and continuous in |u|², and u₀(x) is a complex-valued function of x. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem.

1. Introduction

In this paper, we consider the following Cauchy problem:

()

where V(x) and W(x) are real-valued potentials, V(x) ≥ 0 and W(x) is even, f(x, |u|²) is measurable in x and continuous in |u|²,

()

u₀(x) is a complex-valued function of x, and Σ is the Hilbert space:

()

with the inner product

()

and the norm

()

Model (1) appears in the theory of Bose-Einstein condensation, nonlinear optics and theory of water waves (see [1, 2]).

For convenience, denote 1/(N − 2) ⁺ = +∞ when N = 1,2 and (N − 2) ⁺ = N − 2 when N ≥ 3. We also give some assumptions on V(x), f(x, s), and W(x) as follows.

(V1)
V(x) ≥ 0 and V(x) ∈ L^r(ℝ^N) + L^∞(ℝ^N) for r ≥ 1, r > N/2.
(V2)
V(x) ≥ 0, , and |D^αV| is bounded for all |α| ≥ 2. Here is the complementary set of S₁ = {V(x) satisfies (V1)}.
(f1)
f(x, s) : ℝ^N × ℝ → ℝ is measurable in x and continuous in |u|² with f(x, 0) = 0.

Assume that, for every k > 0, there exists L(k) < +∞ such that |f(x, s₁) − f(x, s₂)| ≤ L(k)|s₁ − s₂| for all 0 ≤ s₁ < s₂ < k. Here

()

(W1)
W(x) is even and W(x) ∈ L^q(ℝ^N) + L^∞(ℝ^N) for some q ≥ 1, q > N/4.

First, we consider the local well-posedness of (1). We have a proposition as follows.

Proposition 1 (local existence result). Assume that (f1) and (W1) are true, V(x) satisfies (V1) or (V2), and u₀ ∈ Σ. Then there exists a unique solution u of (1) on a maximal time interval [0, T_max) such that u ∈ C(Σ; [0, T_max)) and either T_max = +∞ or else

()

Definition 2. If u ∈ C(Σ; [0, T)) with T = ∞, we say that the solution u of (1) exists globally. If u ∈ C(Σ; [0, T)) with T < +∞ and lim _t→T∥u(·,t)∥_Σ → +∞, we say that the solution u of (1) blows up in finite time.

This paper is directly motivated by [1, 3–5]. Since Cazevave established some results on blowup and global existence of the solutions to (1) with (V1), (f1), and (W1) in [1], we are interested in the problems such as “What are the results about the blowup and global existence of the solutions to (1) with (V2), (f1), and (W1)?” On the other hand, since Gan et al. had established some sharp thresholds for global existence and blowup of the solution to the related problems to (1) (see [3–5] and the references therein), it is a natural way to consider the sharp threshold for global existence and blowup of the solution to (1).

About the topic of global existence and blowup in finite time, there are many results on the special cases of (1). We will recall some results on the following Cauchy problem:

()

In [6], Glassey established some blowup results for (8). In [7], Berestyki and Cazenave established the sharp threshold for blowup of (8) with supercritical nonlinearity by considering a constrained variational problem. In [8], Weinstein presented a relationship between the sharp criterion for the global solution of (8) and the best constant in the Gagliardo-Nirenberg inequality. In [9], Cazenave and Weisseler established the local existence and uniqueness of the solution to (8) with f(|u|²)u = |u|^4/Nu. Very recently, Tao et al. in [10] studied the Cauchy problem (8) with

, where μ and ν are real numbers, 0 < p₁ < p₂ < 4/(N − 2) with N ≥ 3, and established the results on local and global well-posedness, asymptotic behavior (scattering), and finite time blowup under some assumptions. Other sharp thresholds were established by Chen et al. in [11, 12]. The following Cauchy problem

()

is also a special case of (1), where 0 < p < q < 4/(N − 2) with N ≥ 3. In [2], Oh obtained the local well-posedness and global existence results of (9) under some conditions. In [3, 5], Gan et al. and Zhang, respectively, established the sharp thresholds for the global existence and blowup of the solutions to (9) under some conditions. In [4], Gan et al. dealt with

()

with E₁(ξ) a singular integral operator, where 0 < p < 4/(N − 2) with N ≥ 3. They got the sharp threshold for global existence and blowup of the solution to (10) and the instability of the wave solutions. Very recently, Miao et al. also obtained some results on the blowup and global existence of the solution to a Hartree equation (see [13–15]). Naturally, we want to establish some new sharp thresholds for global existence and blowup of the solution to (1) in this paper and generalize these results above. Although the methods of our paper are inspired by the references above, our results, which will be stated in Section 2, are new and cover theirs.

This paper is organized as follows. In Section 2, we will recall some results of [1] and state our main results; then we will prove Proposition 1 and give some other properties. In Section 3, we will prove Theorems 3 and 4. In Section 4, we establish the sharp threshold for (1) with V(x) ≡ 0. In Section 5, we will prove Theorem 7.

2. Our Main Results

Now we will introduce some notations. Denote

()

mass (L² norm)

()

energy

()

In [1], Cazenave obtained some sufficient conditions on blowup and global existence of the solution to (1) with (V1), (f1), and (W1). The following two theorems can be looked at as the parallel results to Corollary 6.1.2 and Theorem 6.5.4 of [1], respectively.

Theorem 3 (global existence). Assume that u₀ ∈ Σ, (V2) and (f1) are true, and

()

for some q ≥ 1, q ≥ N/2 (and q > 1 if N = 2). Here W⁺ = max (W(x), 0). Suppose further that there exist constants c₁ and c₂ such that F(x, |u|²) ≤ c₁|u|² + c₂|u|^2p+2 with 0 < p < 2/N. Then the solution of (1) exists globally. That is,

()

Theorem 4 (blowup in finite time). Assume that u₀ ∈ Σ and |x|u₀ ∈ L²(ℝ^N), (V2), (f1), and (W1) are true. Suppose further that

()

If (1) E(u₀) < 0 or (2) E(u₀) = 0 and

, then the solution of (1) will blow up in finite time. That is, there exists T_max < ∞ such that

()

Denote

()

We will establish the first type of sharp threshold as follows.

Theorem 5 (sharp threshold I). Assume that V(x) ≡ 0 and W(x) ∈ L^q(ℝ^N) with N/4 < q < N/2. Suppose further that f(x, 0) = 0 and there exist constants c₁, c₂, c₃ > 0 and 2/N < p₁, p₂, l < 2/(N − 2) ⁺ such that

()

Let ω be a positive constant satisfying

()

where Q(u) is defined by (22). Suppose that u₀ ∈ H¹(ℝ^N) satisfies

()

Then

(1)
if Q(u₀) > 0, the solution of (1) exists globally;
(2)
if Q(u₀) < 0, |x | u₀ ∈ L²(ℝ^N), and , the solution of (1) blows up in finite time.

Remark 6. Theorem 5 is only suitable for (1) with V(x) ≡ 0. To establish the sharp threshold for (1) with V(x) ≠ 0, we will construct a type of cross-constrained variational problem and establish some cross-invariant manifolds. First, we introduce some functionals as follows:

()

Denote the Nehari manifold

()

and cross-manifold

()

Define

()

In Section 5, we will prove that d_II is always positive. Therefore, it is reasonable to define the following cross-manifold:

()

We give the second type of sharp threshold as follows.

Theorem 7 (sharp threshold II). Assume that (f1), (W1), and (23). Suppose that

()

and there exists a positive constant c such that

()

with the same l in (23). Assume further that the function f(x, |u|²) satisfies f(x, 0) = 0 and

()

for k > 1. Here

is the value of the partial derivative of f(x, s) with respect to s at the point (x, z). If u₀ ∈ Σ and |x | u₀ ∈ L²(ℝ^N) with

, then the solution of (1) blows up in finite time if and only if u₀ ∈ 𝒦.

Remark 8. (1) f(x, |u|²) ≤ f(x, k² | u|²) implies that k²F(x, |u|²) ≤ F(x, k² | u|²) for k > 1.

(2) The blowup of solution to (1) will benefit from the role of the potential V if V(x) ≥ 0. In some cases, the blowup of the solution to (1) can be delayed or prevented by the role of potential (see [16] and the references therein).

In the sequel, we use C and c to denote various finite constants; their exact values may vary from line to line.

First, we will give the proof of Proposition 1.

Proof of Proposition 1. If (V1) is true, then there exist V₁(x) ∈ L^r(ℝ^N) with r ≥ 1, r > N/2, and V₂(x) ∈ L^∞(ℝ^N) such that

()

Noticing that 0 < 2r/(r − 1) < 2N/(N − 2), using Hölder’s and Sobolev’s inequalities, we have

()

for any u ∈ H¹(ℝ^N). Consequently, we have

()

By the results of Theorem 3.3.1 in [1], we have the local well-posedness result of (1) in Σ.

If (V2), (f1), and (W1) are true, similar to the proof of Theorem 3.5 in [2], we can establish the local well-posedness result of (1) in Σ. Roughly, we only need to replace |u|^p+1u by f(x, |u|²)u + (W(x)⋆|u|²)u in the proof, and we can obtain similar results under the assumptions of (V2), (f1), and (W1).

Noticing that and h(u) = H^′(u), following the method of [6] and the discussion in Chapter 3 of [1], one can obtain the conservation of mass and energy. We give the following proposition without proof.

Proposition 9. Assume that u(x, t) is a solution of (1). Then

()

for any 0 ≤ t < T_max.

We will recall some results on blowup and global existence of the solution to (1) with (V1), (f1), and (W1).

Theorem A (Corollary 6.12 of [1]). Assume that (V1), (f1), and (16). Suppose that there exist A ≥ 0 and 0 ≤ p < 2/N such that

()

Then the maximal strong H¹-solution of (1) is global and

for every u₀ ∈ H¹(ℝ^N).

Theorem B (Theorem 6.54 of [1]). Assume that (V1), (f1), (W1), and (18)–(20). If u₀ ∈ H¹(ℝ^N), |x | u₀ ∈ L²(ℝ^N), and E(u₀) < 0, then the H¹-solution of (1) will blow up in finite time.

Let

. After some elementary computations, we obtain

()

We have the following proposition.

Proposition 10. Assume that u(x, t) is a solution of (1) with u₀ ∈ Σ and |x | u₀ ∈ L²(ℝ^N). Then the solution to (1) will blow up in finite time if either

(1)
there exists a constant c < 0 such that J^′′(t) = 4Q(u) ≤ c < 0 or
(2)
J^′′(t) = 4Q(u) ≤ 0 and .

Proof. Since u₀ ∈ Σ and |x | u₀ ∈ L²(ℝ^N), we have

()

(1) If J^′′(t) ≤ c < 0, integrating it from 0 to t, we get J^′(t) < ct + J^′(0). Since c < 0, we know that there exists a t₀ ≥ max (0, J^′(0)/−c) such that J^′(t) < J^′(t₀) < 0 for t > t₀. On the other hand, we have

()

which implies that there exists a T_max < +∞ satisfying

()

Using the inequality

()

and noticing that

, we have

()

Consequently,

()

(2) Similar to (46), we can get

()

which implies that the solution will blow up in a finite time T_max ≤ J(0)/−J^′(0).

3. The Sufficient Conditions on Global Existence and Blowup in Finite Time

In this section, we will prove Theorems 3 and 4, which give some sufficient conditions on global existence and blowup of the solution to (1).

We would like to give some examples of V(x), f(x, |u|²), and W(x). It is easy to verify that they satisfy the conditions of Theorem 3.

Example 11. Consider that V(x) = |x|², , and f(x, |u|²) = b | u|^2p with b a real constant and 0 < p < 2/N.

Example 12. Consider that V(x) = |x|², W(x) = |x|²/(1+|x|²), and f(x, |u|²) = b | u|^2pln (1+|u|²) with b a real constant and 0 < p < 2/N.

Proof of Theorem 3. Letting W⁺(x) = W₁(x) + W₂(x), where W₁ ∈ L^∞(ℝ^N) and W₂ ∈ L^q(ℝ^N) with q > N/2, using Hölder’s and Young’s inequalities, we obtain

()

with r = 4q/(2q − 1). Specifically, we have

()

Using (53) and Gagliardo-Nirenberg’s inequality, we get

()

Using Young’s inequality, from (54), we have

()

for some ε > 0. Noticing that F(x, |u|²) ≤ c₁ | u|² + c₂ | u|^2p+2, using Gagliardo-Nirenberg’s inequality and (55) with ε = 1/4, we get

()

Since

, from (56), we can obtain

()

Since p < 2/N means that (pN/2) − 1 < 0, (57) implies that

is always controlled by

. That is, the solution of (1) exists globally.

We would like to give some examples of V(x), W(x), and f(x, |u|²). It is easy to verify that they satisfy the conditions of Theorem 4.

Example 13. Consider that V(x) = |x|², W(x) = |x|⁻², and f(x, |u|²) = b | u|^2p with b > 0 and p > 2/N with N ≥ 3.

Example 14. Consider that V(x) = |x|², W(x) = |x|⁻², and f(x, |u|²) = b | u|^2pln (1+|u|²) with b > 0 and p ≥ 2/N with N ≥ 3.

Proof of Theorem 4. Set

()

Using (18)–(20), we have

()

From (58) and (59), we obtain

()

Since

, whether (1) or (2), (60) will be absurd for t > 0 large enough. Therefore, the solution of (1) will blow up in finite time.

4. The Sharp Threshold for Global Existence and Blowup of the Solution to (1) with V(x) ≡ 0 and W ∈ L^q(ℝ^N) with N/4 < q < N/2

In this section, we will establish the sharp threshold for global existence and blowup of the solution to (1) with V(x) ≡ 0 and W ∈ L^q(ℝ^N) with N/4 < q < N/2.

Before giving the proof of Theorem 5, we would like to give some examples of f(x, |u|²) and W(x). It is easy to verify that they satisfy the conditions of Theorem 5.

Example 15. Consider that W(x) ≡ 0, with c < 0, d > 0 and q₂ > 2/N, q₂ > q₁ > 0.

Example 16. Consider that W(x) ≡ 0, f(x, |u|²) = b | u|^2pln (1+|u|²) with b > 0 and p > 2/N.

Example 17. Let f(x, |u|²) be one of those in Examples 15 and 16. Let

()

where 2 < Nl < N/q < K and φ(x) satisfies

()

when 1≤|x | ≤ 2 and makes W(x) smooth. Obviously, W ∈ L^q(ℝ^N).

Proof of Theorem 5. We will proceed in four steps.

Step 1. We will prove d_I > 0. u ∈ H¹(ℝ^N)∖{0} and Q(u) = 0 mean that

()

Using Gagliardo-Nirenberg’s and Hölder’s inequalities, we can get

()

That is,

()

if Q(u) = 0 and u ∈ H¹(ℝ^N)∖{0}.

On the other hand, if Q(u) = 0, we have

()

that is,

()

Using (67), we can obtain

()

from (65). Hence

()

Step 2. Denote

()

We will prove that K₊ and K₋ are invariant sets of (1) with V(x) ≡ 0 and W ∈ L^q(ℝ^N) with N/4 < q < N/2. That is, we need to show that u(·, t) ∈ 𝒦 for all t ∈ (0, T_max) if u₀ ∈ K₊. Since ∥u∥₂ and E(u) are conservation quantities for (1), we have

()

for all t ∈ (0, T_max) if u₀ ∈ K₊. We need to prove that Q(u(·, t)) > 0. Otherwise, assume that there exists a t₁ ∈ (0, T_max) satisfying Q(u(·, t₁)) = 0 by the continuity. Note that (71) implies

()

However, the inequality above and Q(u(·, t₁)) = 0 are contradictions to the definition of d_I. Therefore, Q(u(·, t)) > 0. Consequently, (71) and Q(u(·, t)) > 0 imply that u(·, t) ∈ K₊. That is, K₊ is an invariant set of (1) with V(x) ≡ 0 and W ∈ L^q(ℝ^N) with N/4 < q < N/2. Similarly, we can prove that K₋ is also an invariant set of (1) with V(x) ≡ 0 and W ∈ L^q(ℝ^N) with N/4 < q < N/2.

Step 3. Assume that Q(u₀) > 0 and . By the results of Step 2, we have Q(u(·, t)) > 0 and . That is,

()

The two inequalities imply that

()

which means that

()

that is, the solution exists globally.

Step 4. Assume that Q(u₀) < 0 and . By the results of Step 2, we obtain Q(u(·, t)) < 0 and . Hence we get

()

By the results of Proposition 10, the solution will blow up in finite time.

As a corollary of Theorem 5, we obtain the sharp threshold for global existence and blowup of the solution of (8) as follows.

Corollary 18. Assume that f(x, 0) = 0 and (23). Let ω be a positive constant satisfying

()

Here

()

Suppose that u₀ ∈ H¹(ℝ^N) satisfies

()

Then

(1)
if Q₁(u₀) > 0, the solution of (8) exists globally;
(2)
if Q₁(u₀) < 0, |x | u₀ ∈ L²(ℝ^N), and , the solution of (8) blows up in finite time.

Remark 19. In Theorem 1.5 of [10], Tao et al. proved the following.

Assume that u(x, t) is a solution of (8) with , where μ > 0, ν > 0, 4/N ≤ p₁ < p₂ ≤ 4/(N − 2) with N ≥ 3, , |x | u₀ ∈ L²(ℝ^N), and E(u₀) < 0. Then blowup occurs.

Corollary 18 covers the result above under some conditions. In fact, if , then

()

hence E(u₀) < 0 implies that Q₁(u₀) < 0. That is, our blowup condition is weaker than theirs. On the other hand, our conclusion is still true if

with Q₁(u₀) < 0,

, and |x | u₀ ∈ L²(ℝ^N). In other words, our result is stronger than theirs if

with Q₁(u₀) < 0,

, and |x | u₀ ∈ L²(ℝ^N).

5. Sharp Threshold for the Blowup and Global Existence of the Solution to (1)

Theorem 7 is inspired by [5], but it extends the results to more general case. We need subtle estimates and more sophisticated analysis in the proof.

First, we would like to give some examples of V(x), f(x, |u|²), and W(x). It is easy to verify that they satisfy the conditions of Theorem 7.

Example 20. Consider that V(x) = |x|², W(x) = a | x|^−K with 2 < Nl < K < N/q < 4 for x ∈ ℝ^N and with a ≥ 0, b > 0, c > 0, and p₂ > p₁ > 2/N.

Example 21. Consider that V(x) = |x|², W(x) = a | x|^−K with 2 < Nl < K < N/q < 4 for x ∈ ℝ^N and with a ≥ 0, c is a real number, d > 0, and q₂ > 2/N, q₂ > q₁ > 0.

Example 22. Consider that V(x) = |x|²/(1 + |x|²), W(x) = a | x|^−K with 2 < Nl < K < N/q < 4 for x ∈ ℝ^N and f(x, |u|²) = b | u|^2pln (1+|u|²) with a ≥ 0, b > 0, and p > 2/N.

5.1. Some Invariant Manifolds

In this subsection, we will prove that d_𝒩, d_ℳ, d_II > 0 and construct some invariant manifolds.

Proposition 23. Assume that the conditions of Theorem 7 hold. Then d_𝒩 > 0.

Proof. Assume that u ∈ Σ∖{0} satisfying S_ω(u) = 0. Using Gagliardo-Nirenberg’s and Young’s inequalities, we have

()

Using Hölder’s inequality, from (81), we can obtain

()

Equation (82) implies that

()

for some positive constant C.

On the other hand, if S_ω(u) = 0, we get

()

From (84), we obtain

()

Consequently,

()

Now, we will give some properties of I_ω(u), S_ω(u), and Q(u). We have a proposition as follows.

Proposition 24. Assume that Q(u) and S_ω(u) are defined by (22) and (28). Then one has the following.

(i)
There at least exists a w^⋆ ∈ Σ∖{0} such that
()
(ii)
There at least exists a u^* ∈ Σ∖{0} such that
()

Proof. (i) Noticing the assumptions on V(x), W(x), and f(x, |u|²), similar to the proof of Theorem 1.7 in [17], it is easy to prove that there exists a w^⋆ ∈ Σ∖{0} satisfying

()

Multiplying (89) by w^⋆ and integrating over ℝ^N by part, we can get S_ω(w^⋆) = 0.

Multiplying (89) by (x · ∇w^⋆) and integrating over ℝ^N by part, we obtain Pohozaev’s identity:

()

From S_ω(w^⋆) = 0 and (90), we can get Q(w^⋆) = 0.

(ii) Letting v_k,λ(x) = kw^⋆(λx) for k > 0 and λ > 0, we can obtain

()

Looking at S_ω(v_k,λ) and Q(v_k,λ) as the functions of (k, λ), setting g(k, λ) = S_ω(v_k,λ) and η(k, λ) = Q(v_k,λ), we get that g(1,1) = 0 and η(1,1) = 0. We want to prove that there exists a pair of (k, λ) such that g(k, λ) = S_ω(v_k,λ) < 0 and η(k, λ) = Q(v_k,λ) = 0. Since η(1,1) = 0, we know that the image of η(k, λ) and the plane η = 0 intersect in the space of (k, λ, η) and form a curve η(k, λ) = 0. Hence there exist many positive real number pairs (k, λ) relying on w^⋆ such that Q(v_k,λ) = 0 near (1,1) with k > 1. On the other hand, under the assumptions of V(x) and W(x), it is easy to see that g(k, 1) < 0 for any k > 1. By the continuity, we can choose that a pair of (k, λ) near (1,1) with k > 1 satisfies both Q(v_k,λ) = 0 and S_ω(v_k,λ) < 0. Letting u^* = v_k,λ for this (k, λ), we get that S_ω(u^*) < 0 and Q(u^*) = 0.

Proposition 24 means that 𝒞ℳ is not empty and d_ℳ is well defined. Moreover, we have the following.

Proposition 25. Assume that the conditions of Theorem 7 hold. Then d_ℳ > 0.

Proof. u ∈ Σ∖{0} and S_ω(u) < 0 imply that

()

Similar to (81) and (82), from (93), we have

()

On the other hand, if Q(u) = 0, we have

()

that is,

()

Using (23), (35), (36), (94), and (96), we can get

()

Consequently,

()

By the conclusions of Proposition 23 and Proposition 25, we have

()

Now we define the following manifolds:

()

The following proposition will show some properties of 𝒦, 𝒦₊, and ℛ₊.

Proposition 26. Assume that the conditions of Theorem 7 hold. Then

(i)
𝒦, 𝒦₊, and ℛ₊ are not empty;
(ii)
𝒦, 𝒦₊, and ℛ₊ are invariant manifolds of (1).

Proof. (i) In order to prove 𝒦 is not empty, we only need to find that there at least exists a w ∈ 𝒦. For w^⋆ ∈ Σ∖{0} satisfying S_ω(w^⋆) = 0 and Q(w^⋆) = 0, letting w_ρ = ρw^⋆ for ρ > 0, we have

()

Since and for ρ > 1 and from (38), we can obtain

()

for any ρ > 1. Noticing d_II > 0, we also can choose ρ > 1 closing to 1 enough such that

()

Equations (104) and (105) mean that w_ρ ∈ 𝒦. That is, 𝒦 is not empty.

Similar to (104), we can obtain

()

for any 0 < ρ < 1. Noticing d_II > 0, we also can choose 0 < ρ < 1 closing to 1 enough such that I_ω(w_ρ) < d_II by continuity, which implies that w_ρ ∈ ℛ₊. That is, ℛ₊ is not empty.

For w^* ∈ Σ satisfying S_ω(w^*) < 0 and Q(w^*) = 0, letting w_σ = σw^* for σ > 0, we have

()

Since ϕ(σ) = Q(w_σ) is a smooth function of σ and Q(w^*) = 0, we have ϕ(1) = 0. If ϕ^′(1) ≠ 0, then there exists a σ₀ > 0 such that Q(u_σ) = ϕ(σ) > 0 for σ ∈ (1, σ₀) if σ₀ > 1 (or σ ∈ (σ₀, 1) if σ₀ < 1). By continuity, we can choose such σ₀ closing to 1 enough such that S_ω(w_σ) < 0 and I_ω(w_σ) < d_II for σ ∈ (1, σ₀) if σ₀ > 1 (or σ ∈ (σ₀, 1) if σ₀ < 1). That is, w_σ ∈ 𝒦₊ and 𝒦₊ is not empty.

If ϕ^′(1) = 0, from ϕ(1) = 0 and ϕ^′(1) = 0, we can, respectively, obtain

()

Letting w_σ = σw^*, we have

()

for 0 < σ < 1. By continuity, we can choose such σ closing to 1 enough such that S_ω(w_σ) < 0 and I_ω(w_σ) < d_II. That is to say, w_σ ∈ 𝒦₊ and 𝒦₊ is not empty.

(ii) In order to prove that 𝒦 is the invariant manifold of (1), we need to show that, if u₀ ∈ 𝒦, then solution u(x, t) of (1) satisfies u(x, t) ∈ 𝒦 for any t ∈ [0, T).

Assume that u(x, t) is a solution of (1) with u₀ ∈ 𝒦. Then we can obtain

()

for t ∈ [0, T). Next we prove that S_ω(u(·, t)) < 0 for t ∈ [0, T). Otherwise, by continuity, there exists a t₀ ∈ (0, T) such that S_ω(u(·, t₀)) = 0 because of S_ω(u₀) < 0. Since

and u₀ ∈ Σ∖{0}, it is easy to see that u(·, t₀) ∈ Σ∖{0}. By the definitions of d_𝒩 and d_II, we know that I_ω(u(·, t₀)) ≥ d_𝒩 ≥ d_II, which is a contradiction to I_ω(u(·, t)) < d_II for t ∈ [0, T). Hence S_ω(u(·, t)) < 0 for all t ∈ [0, T).

Now we only need to prove that Q(u(·, t)) < 0 for t ∈ [0, T). Otherwise, since Q(u₀) < 0, there exists a t₁ ∈ (0, T) such that Q(u(·, t₁)) = 0 by continuity. S_ω(u(·, t₁)) < 0 means that u(·, t₁) ∈ 𝒞ℳ. By the definitions of d_ℳ and d_II, we obtain I_ω(u(·, t₁)) ≥ d_ℳ ≥ d_II, which is a contradiction to I_ω(u(·, t)) < d_II for t ∈ [0, T). Hence Q(u(·, t)) < 0 for all t ∈ [0, T).

By the discussions above, we know that u(x, t) ∈ 𝒦 for any t ∈ [0, T) if u₀ ∈ 𝒦, which means that 𝒦 is the invariant manifold of (1).

Similarly, we can prove that 𝒦₊, and ℛ₊ are also invariant manifolds of (1).

Remark 27. By the definitions of d_II, d_𝒩, d_ℳ, 𝒦, 𝒦₊, and ℛ₊, it is easy to see that

()

5.2. Proof of Theorem 7

Proof of Theorem 7 depends on the following two lemmas.

Lemma 28. Assume that the conditions of Theorem 7 hold. Then the solutions of (1) with u₀ ∈ 𝒦 will blow up in finite time.

Proof. Since u₀ ∈ 𝒦 and 𝒦 is the invariant manifold of (1), we have Q(u(x, t)) < 0, S_ω(u(x, t)) < 0, and I_ω(u(x, t)) < d_II.

Under the conditions of Theorem 7, we have J^′′(t) = 4Q(u) < 0 and J^′(0) < 0. By the results of Proposition 10, the solution u(x, t) will blow up in finite time. The conclusion of this lemma is true.

On the other hand, we have a parallel result on global existence.

Lemma 29. Assume that the conditions of Theorem 7 hold. If u₀ ∈ 𝒦₊ or u₀ ∈ ℛ₊, then the solutions of (1) exist globally.

Proof. Case 1. Assume that u(x, t) is a solution of (1) with u₀ ∈ 𝒦₊. Since 𝒦₊ is an invariant manifold of (1), we know that u(·, t) ∈ 𝒦₊, which means that I_ω(u(·, t)) < d_II and Q(u(·, t)) > 0. Q(u(·, t)) > 0 and (23) imply that

()

By the definition of I_ω(u) and using (113), we have

()

Equation (114) means that u(x, t) exists globally.

Case 2. Assume that u(x, t) is a solution of (1) with u₀ ∈ ℛ₊. Since ℛ₊ is also an invariant manifold of (1), we know that u(x, t), ∈ℛ₊, which means that I_ω(u(·, t)) < d_II and S_ω(u(·, t)) > 0. Since S_ω(u) > 0, we can get

()

From (115), we can obtain

()

Equation (116) implies that the solution u(x, t) exists globally.

Proof of Theorem 7. By the results of Lemmas 28 and 29, we know that Theorem 7 is right.

As a corollary of Theorem 7, we obtain a sharp threshold for the blowup in finite time and global existence of the solution of (9) as follows.

Corollary 30. Assume that f(x, |u|²) ≡ 0, V(x) ≡ 0, W(x) > 0 for all x ∈ ℝ^N, W(x) is even, and W(x) ∈ L^∞(ℝ^N) + L^q(ℝ^N) with some q > N/4. Suppose further that there exists l satisfying 2 < Nl and

()

If u₀ ∈ H¹(ℝ^N), |x | u₀ ∈ L²(ℝ^N), and

, then the solution of (9) blows up in finite time if and only if u₀ ∈ 𝒦.

Remark 31. A typical example is

()

which is also a special case of (1) with V(x) ≡ 0, f(x, |u|²) ≡ 0, and W(x) = |x|^−K with 2 < Nl < K < N/q < 4. Letting W(x) = W₁(x) + W₂(x) with

()

we can see that W₁(x) ∈ L^∞(ℝ^N) and W₂(x) ∈ L^q(ℝ^N) with some N/4 < q < N/2. Corollary 30 gives the sharp threshold for blowup and global existence of the solution to (118).

Acknowledgments

The authors are grateful to the referees for their helpful comments. In addition, the second author is supported by the National Natural Science Foundation of China, Grant 11071185 and Natural Science Foundation of Tianjin (09JCYBJC01800). The third author is supported by the National Natural Science Foundation of China, Grant 11071237.

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Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

Abstract

1. Introduction

2. Our Main Results

3. The Sufficient Conditions on Global Existence and Blowup in Finite Time

4. The Sharp Threshold for Global Existence and Blowup of the Solution to (1) with V(x) ≡ 0 and W ∈ L^q(ℝ^N) with N/4 < q < N/2

5. Sharp Threshold for the Blowup and Global Existence of the Solution to (1)

5.1. Some Invariant Manifolds

5.2. Proof of Theorem 7

Acknowledgments

References

References

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Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

Abstract

1. Introduction

2. Our Main Results

3. The Sufficient Conditions on Global Existence and Blowup in Finite Time

4. The Sharp Threshold for Global Existence and Blowup of the Solution to (1) with V(x) ≡ 0 and W ∈ Lq(ℝN) with N/4 < q < N/2

5. Sharp Threshold for the Blowup and Global Existence of the Solution to (1)

5.1. Some Invariant Manifolds

5.2. Proof of Theorem 7

Acknowledgments

References

References

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4. The Sharp Threshold for Global Existence and Blowup of the Solution to (1) with V(x) ≡ 0 and W ∈ L^q(ℝ^N) with N/4 < q < N/2