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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential

Mingyou Zhang / Md Salik Ahmed
Published Online: 2019-08-06 | DOI: https://doi.org/10.1515/anona-2020-0031

Abstract

The Cauchy problem of nonlinear Schrödinger equation with a harmonic potential for describing the attractive Bose-Einstein condensate under the magnetic trap is considered. We give some sufficient conditions of global existence and finite time blow up of solutions by introducing a family of potential wells. Some different sharp conditions for global existence, and some invariant sets of solutions are also obtained here.

Keywords: nonlinear Schrödinger equation; harmonic potential; potential wells; global existence; invariant sets

1 Introduction

In the present paper we study the Cauchy problem of nonlinear Schrödinger equation with a harmonic potential

iφt=Δφ+|x|2φ|φ|p1φ,xRn,t>0,φ(x,0)=φ0(x),xRn,(1.1)

where p satisfies

(A)1+4n<p<n+2n2forn3;1+4n<p<forn=1,2.

The equation in (1.1) may model the Bose-Einstein condensate with attractive inter-particle interactions under the magnetic trap [1, 6, 10, 11].

Fujiwara [7] considered the above nonlinear Schrödinger equation with a general real-valued potential function V (x) which later studied in [5]. When |Dα V (x)| is bounded for all α ≥ 2, the author of [7] gave the smoothness of Schrödinger kernel for potentials of quadratic growth. Further Yajima [20] showed that for super-quadratic potentials, the Schrödinger kernel is nowhere C1. In addition, Oh [16] pointed out that the quadratic potentials are the highest order potentials for local well-posedness of the equation. Therefore V (x) = |x|2 is the critical potential for the local existence of the Cauchy problem.

On the global existence and finite time blow up of solutions for problem (1.1), there have been some results. Firstly, Oh [16] and Cazenave [4] established the local existence of solutions of problem (1.1) in the energy space. Then for 1 < p < 1 + 4n, Zhang [22] proved the global existence of solutions for any initial data in energy space. For the case p = 1 + 4n, Zhang [21] gave a sharp condition of global existence of solution for problem (1.1). For the case p > 1 + 4n, Cazenave [4], Carles [2, 3] and Tsurumi and Wadati [18] showed that the solutions of problem (1.1) blows up in a finite time for some initial data, especially for a class of sufficiently large initial data; but the solutions of problem (1.1) globally exist for other sufficiently small initial data [2, 3, 18].

Chen and Zhang [5] studied problem (1.1) and gave a sufficient condition of global existence of solutions in energy space. They proved that if p satisfies (A), φ0H(ℝn) and satisfies φ0L2hφ0L22+φ0L22, then the solution φ of problem (1.1) exists globally and satisfies

φL22+|x|φL222n(p1)n(p1)4E(φ0),

where

H(Rn)={φH1(Rn)|x|φL2<},(1.2)

h(λ)=n(p1)44(p1)(n2)n(p1)42(n+2)(n2)pQ22(p1)(n+2)(n2)pλn(p1)42(n+2)(n2)p,(1.3)

E(φ)=12φL22+12|x|φL221p+1φLp+1p+1,(1.4)

Q is the ground state solution of equation

Δu+u=|u|p1uinRn.

In addition, in [17] Shu and Zhang gave a sharp condition for global existence of solution for problem (1.1). Moreover, in [23] Zhang studied Cauchy problem of following nonlinear Schrödinger equation with harmonic potential

iφt=12Δφ+2|x|2φ|φ|p1φ,xRn,t>0,

obtained a sharp condition for global existence and finite time blow up of solutions and discussed the instability of standing wave. These works motivated us to study on this and related problems [24, 25, 26, 27, 28, 29, 30, 31, 32].

In this paper we study problem (1.1), where p satisfies (A). By introducing two families of sets Wδ and Vδ like [12, 13, 14], we not only give some different sufficient conditions for global existence and finite time blow up of solutions which are completely different from that given in [2, 3, 4, 5], [17], [18], [23] but also obtain a lot of different sharp conditions for global existence of solutions and some invariant sets of solutions for problem (1.1).

Throughout the present paper, the following notations are used for precise statements: Lp(ℝn) (1 ≤ p ≤ ∞) denotes the usual space of all complex Lp-functions on ℝn with norm ∥φLp(ℝn) = ∥φp and ∥φL2(ℝn) = ∥φ∥. H1 and H denote H1(ℝn) and H(ℝn) respectively. H(ℝn) and E(φ) are defined by (1.2) and (1.4) respectively.

Proposition 1.1

[4, 8, 9, 16] Assume that 1 < p < n+2n2 for n ≥ 3 and 1 < p < ∞ for n = 1, 2, and φ0H1 (ℝn). Then the problem (1.1) admits a unique solution φ(t) ∈ C([0, T); H(ℝn)) for some T ∈ [0, ∞) (Maximal existence time), and φ(t) satisfies the following two conservation laws:

φ(t)=φ0(1.5)

and

Eφ(t)=E(φ0)(1.6)

for all t ∈ [0, T). Furthermore, we have the following alternative: T = ∞ or T < ∞ and

limtTφH(Rn)=.

Proposition 1.2

[4] Let φ(t) be a solution of problem (1.1) with φ0H, and

F(t)=|x|2|φ|2dx.

Then φ(t) ∈ H for 0 ≤ t < T and

F(t)=8|φ|2|x|2|φ|2n(p1)2(p+1)|φ|p+1dx,0t<T

and

φ22nφF12(t),0t<T.

Proposition 1.3

[19] Let 1 < p < n+2n2 for n ≥ 3 and 1 < p < ∞ for n = 1, 2. Then the best constant C > 0 of the Gagliardo-Nirenberg’s inequality,

fLp+1p+1CfL2p+1n(p1)2fL2n(p1)2,(1.7)

is given by

C=2(p+1)n(p1)4(p1)(n2)n(p1)n(p1)44QL2(p1).(1.8)

2 Preliminaries

In this section we shall give some necessaries lemmas and by using them we introduce two families Wδ and Vδ. For problem (1.1) with ∥φ0∥ ≠ 0 we define

φH2=φH12+|x|φL22=φ2+φ2+|x|φ2,H~={φHφ=φ0},J(φ)=12φ21p+1φp+1p+1,Iδ(φ)=δφ2φp+1p+1,δ>0.

From (1.7) we can obtain the following lemma.

Lemma 2.1

Let p satisfy (A), φ. Thenφp+1 ≠ 0 and ∥∇φ∥ ≠ 0.

Next we discuss the relations between ∥∇φ∥ and the sign of Iδ(φ), which are crucial for obtaining the main results in this paper.

Lemma 2.2

Let p satisfy (A). Assume that φH͠ and

r(δ)=δCM01q,M0=φ0p+1n(p1)2,q=n(p1)22.

  1. If ∥∇ φ∥ < r(δ), then Iδ (φ) > 0.

  2. If Iδ(φ) < 0, then ∥∇φ∥ > r(δ).

  3. If Iδ(φ) = 0, then ∥∇φ∥ ≥ r(δ).

Proof

  1. Since φ implies ∥∇ φ∥ ≠ 0, from ∥∇ φ∥ < r(δ) we get

    φp+1p+1Cφp+1n(p1)2φn(p1)2=CM0φqφ2<δφ2

    which gives Iδ(φ) > 0.

  2. From Iδ(φ) < 0 we get

    δφ2<φp+1p+1CM0φqφ2,

    which gives ∥∇ φ∥ > r(δ).

  3. From Iδ(φ) = 0 we get

    δφ2=φp+1p+1CM0φqφ2,

    which together with ∥∇ φ∥ ≠ 0 gives ∥∇φ∥ ≥ r(δ).

As it is well known that in space H1(ℝn), Poincaré inequality does not hold, so that one can not use the important fact that ∥∇ u∥ is equivalent to ∥uH1. In order to overcome this difficulty, we introduce the space (ℝn), so that by (1.5) and (1.7) the norms ∥∇φ∥ and ∥φH1 are equivalent in some sense again.

Definition 2.3

For problem (1.1) with ∥φ0∥ ≠ 0 we define

d(δ)=infφNδJ(φ),Nδ={φH~|Iδ(φ)=0},δ>0.

In the following Lemma 2.4 we estimate the value of d(δ) and give its expression by d(1), which palys an important role in the proof of the main results of this paper.

Lemma 2.4

Let p satisfy (A). Then

  1. d(δ) ≥ a(δ)r2(δ) for a(δ)=12δp+1,0<δ<p+12;

  2. d(δ)=δ4n(p1)4p+12δp1d(1),0<δ<p+12.

Proof

  1. For any φ ∈ 𝓝δ, 0 < δ < p+12 we have ∥∇φ∥ ≥ r(δ) and

    J(φ)=12φ21p+1φp+1p+1=12δp+1φ2+1p+1Iδ(φ)=a(δ)φ2a(δ)r2(δ),

    which gives d(δ) ≥ a(δ)r2(δ) for 0 < δ < p+12.

    1. From the definition of d(1) it follows that for any ε > 0 there exists a φ ∈ 𝓝1 such that

      d(1)J(φ)<d(1)+ε.

      For δ > 0, define λ = λ(δ) by

      δφλ2=φλp+1p+1,φλ(x)=λn2φ(λx).

      Then

      δφ2=λn(p1)42φp+1p+1.

      Hence for each δ > 0 there exists a unique

      λ(δ)=δa(φ)b(φ)2n(p1)4,

      where

      a(φ)=φ2,b(φ)=φp+1p+1.

      Since φ ∈ 𝓝1 implies a(φ) = b(φ) we get

      λ(δ)=δ2n(p1)4.

      Note that ∥φλ∥ = ∥φ∥ = ∥φ0∥, ∀ λ > 0, we have φλ(δ) ∈ 𝓝δ. From the definition of d(δ) we get

      d(δ)J(φλ)=12λ2a(φ)1p+1λn(p1)2b(φ)=12δ4n(p1)4a(φ)1p+1δn(p1)n(p1)4b(φ)=δ4n(p1)412δp+1a(φ).(2.1)

      From (2.1) and

      J(φ)=12a(φ)1p+1b(φ)=p12(p+1)a(φ)

      it follows that

      d(δ)δ4n(p1)412δp+12(p+1)p1J(φ)<δ4n(p1)412δp+12(p+1)p1d(1)+ε,0<δ<p+12.

      From the arbitrariness of ε we obtain

      d(δ)δ4n(p1)4p+12δp1d(1),0<δ<p+12.(2.2)

    2. Let δ > 0. From the definition of d(δ) it follows that for any ε > 0 there exists a φ ∈ 𝓝δ such that

      d(δ)J(φ)<d(δ)+ε.

      Define λ = λ(δ) by

      φλ2=φλp+1p+1.

      Then

      φ2=λn(p1)42φp+1p+1.λ=a(φ)b(φ)2n(p1)4.

      Since φ ∈ 𝓝δ implies that δ a(φ) = b(φ) we get

      λ(δ)=1δ2n(p1)4.

      From φλ ∈ 𝓝1 and the definition of d(1) we have

      d(1)J(φλ)=12λ2a(φ)1p+1λn(p1)2b(φ)=121δ4n(p1)4a(φ)1p+11δn(p1)n(p1)4b(φ)=1δ4n(p1)412a(φ)1p+11δb(φ)=1δ4n(p1)4p12(p+1)a(φ).(2.3)

      From (2.3) and

      J(φ)=12a(φ)1p+1b(φ)=12δp+1a(φ)

      it follows that

      d(δ)1δ4n(p1)4p12(p+1)12δp+11J(φ)<1δ4n(p1)4p12(p+1)12δp+11d(δ)+ε,0<δ<p+12

      and

      d(δ)+ε>δ4n(p1)412δp+12(p+1)p1d(1),0<δ<p+12.(2.4)

      From (2.4) and the arbitrariness of ε we get

      d(δ)δ4n(p1)4p+12δp1d(1),0<δ<p+12.(2.5)

      From (2.2) and (2.5) we obtain (ii) of this lemma. □

Corollary 2.5

Let p satisfy (A). Then

  1. limδ→0d(δ) = 0, limδp+12d(δ)=0;

  2. d(δ) is continuous on 0 < δ < p+12;

  3. d(δ) is increasing on 0 < δa, decreasing on aδ < p+12 and takes the maximum d(a) at δ = a = 2(p+1)n(p1).

Proof

Conclusions (i) and (ii) follow from (ii) in Lemma 2.4 immediately.

Conclusion (iii) follows from (ii) in Lemma 2.4 and

d(δ)=A(aδ)δα,A=2nn(p1)4d(1),α=δn(p1)n(p1)4.

Definition 2.6

For problem (1.1) with ∥φ0∥ ≠ 0 we define

I(φ)=Ia(φ),d=d(a),a=2(p+1)n(p1),W={φH~|I(φ)>0,J(φ)<d},V={φH~|I(φ)<0,J(φ)<d},Wδ={φH~|Iδ(φ)>0,J(φ)<d(δ)},0<δ<p+12,Vδ={φH~|Iδ(φ)<0,J(φ)<d(δ)},0<δ<p+12.

Remark 2.7

Lemma 2.4 shows that the depth d of the potential well W defined by Definition 2.6 depends on ∥φ0∥ and d → +∞ as ∥φ0∥ → 0. This property of d is completely different from the depth of other known potential wells defined for other nonlinear evolution equations. In addition, W or any Wδ do not include φ = 0.

3 Invariant sets and vacuum isolating of solutions

In this section we discuss the invariant sets and vacuum isolating of solutions for problem (1.1). First we consider the case 0 < E(φ0) < d, and the case E(φ0) = d will be considered later.

Theorem 3.1

Let p satisfy (A), φ0H. Assume that 0 < e < d, δ1 < δ2 are two roots of equation d(δ) = e. Then

  1. All solutions of problem (1.1) with E(φ0) = e belong to Wδ for δ ∈ [δ1, δ2], provided I(φ0) > 0.

  2. All solutions of problem (1.1) with E(φ0) = e belong to Vδ for δ ∈ [δ1, δ2], provided I(φ0) < 0.

Proof

  1. Let φ(t) ∈ C([0, T); ) be any solution of problem (1.1) with E(φ0) = e and I(φ0) > 0, T be the existence time of φ(t). Firstly we prove φ0Wδ for δ ∈ [δ1, δ2]. From

    12|x|φ02+J(φ0)=E(φ0)=ed(δ),δ[δ1,δ2](3.1)

    we get J(φ0) < d(δ) for δ ∈ [δ1, δ2]. On the other hand, I(φ0) > 0 implies ∥φ0∥ ≠ 0. Hence from (3.1) we can get Iδ(φ0) > 0 for δ ∈ [δ1, δ2]. Otherwise there exists a δ̄ ∈ [δ1, δ2] such that Iδ̄(φ0) = 0 which together with ∥φ0∥ ≠ 0 gives J(φ0) ≥ d(δ̄). This contradicts (3.1). Next we prove that φ(t) ∈ Wδ for δ ∈ [δ1, δ2], t ∈ (0, T). Arguing by contradiction, we suppose that there exists a t0 ∈ (0, T) such that φ(t0) ∈ ∂ Wδ for some δ ∈ [δ1, δ2], i.e. Iδ(φ(t0)) = 0 or J(φ(t0)) = d(δ). From (1.6) we get

    12|x|φ2+J(φ)=E(φ0)d(δ),δ[δ1,δ2],t(0,T).(3.2)

    Hence J(φ(t0)) = d(δ) is impossible. If Iδ(φ(t0)) = 0, then by ∥φ(t0)∥ = ∥φ0∥ ≠ 0 we get J(φ(t0)) ≥ d(δ) which contradicts (3.2).

  2. Let φ(t) ∈ C([0, T); ) be any solution of problem (1.1) with E(φ0) = e, I(φ0) < 0, T be the existence time of φ(t). From I(φ0) < 0 and (3.1) we can get φ0Vδ for δ ∈ [δ1, δ2]. The remainder of this proof is similar to that in part (i). □

From (3.2) it follows that if 0 < E(φ0) = e < d, δ1 < δ2 are two roots of equation d(δ) = e, then for any δ ∈ [δ1, δ2], φ ∈ 𝓝δ is impossible. Therefore for the set of all solutions of problem (1.1) with 0 < E(φ0) = e < d there exists a vacuum region

Ue=Nδ1δ2=δ1δδ2Nδ={φH~|Iδ(φ)=0,δ1δδ2}

such that φ(t) ∉ Ue for any solution φ(t) of problem (1.1) with 0 < E(φ0) = e < d.

Now we consider the invariant sets of solutions of problem (1.1) with E(φ0) = d.

Theorem 3.2

Let p satisfy (A), φ0H. Then

  1. All solutions of problem (1.1) with E(φ0) = d belong to W, provided I(φ0) > 0.

  2. All solutions of problem (1.1) with E(φ0) = d belong to V, provided I(φ0) < 0.

Proof

  1. Let φ(t) ∈ C([0, T); ) be any solution of problem (1.1) with E(φ0) = d and I(φ0) > 0, T be the existence time of φ(t). Firstly I(φ0) > 0 gives ∥φ0∥ ≠ 0 and ∥|x|φ0∥ ≠ 0. Hence from

    J(φ0)=E(φ0)12|x|φ02=d12|x|φ02<d

    we get φ0W. Next we prove that φ(t) ∈ W for t ∈ (0, T). Arguing by contradiction, we assume that there exists a t0 ∈ (0, T) such that φ(t0) ∈ ∂ W, i.e. I(φ(t0)) = 0 or J(φ(t0)) = d. From ∥φ(t0)∥ = ∥φ0∥ ≠ 0 we get ∥|x|φ(t0)∥ ≠ 0. Hence from

    J(φ(t0))=E(φ0)12|x|φ(t0)2=d12|x|φ(t0)2<d

    we see that J(φ(t0)) = d is impossible. If I(φ(t0)) = 0, then again we have J(φ(t0)) ≥ d which contradicts J(φ(t0)) < d.

  2. The proof is similar to that of part (i) of this lemma. □

Next we discuss the invariant sets of solutions for problem (1.1) with E(φ0) ≤ 0.

Theorem 3.3

Let p satisfy (A), φ0H. Assume that E(φ0) < 0 or E(φ0) = 0, ∥φ0∥ ≠ 0. Then all solutions of problem (1.1) belong to Vδ for δ ∈ (0, p+12).

Proof

Let φ(t) be any solution of problem (1.1) with E(φ0) < 0 or E(φ0) = 0, ∥φ0∥ ≠ 0. Since E(φ0) < 0 implies ∥φ0∥ ≠ 0. Hence for two cases we always have ∥φ(t)∥ = ∥φ0∥ ≠ 0 and ∥∇φ(t)∥ ≠ 0 for 0 ≤ t < T. Thus from

12δp+1φ2+1p+1Iδ(φ)=J(φ)=E(φ0)12|x|φ2,0<δ<p+12,0t<T,

we can get Iδ(φ) < 0 and J(φ) < 0 < d(δ) for δ ∈ (0, p+12), t ∈ (0, T), which gives φ(t) ∈ Vδ for δ ∈ (0, p+12), t ∈ [0, T). □

4 global existence and finite time blow up of solutions

In this section we shall prove the global existence and finite time blow up of solutions and give some sharp conditions for global existence and finite time blow up of solutions for problem (1.1) which are completely different from those given in [2, 3, 4, 5], [17, 18, 23].

Theorem 4.1

Let p satisfy (A), φ0H. Assume thatφ0∥ = 0 or E(φ0) ≤ d, I(φ0) > 0. Then problem (1.1) admits a unique global solution φ(t) ∈ C([0, ∞); ) such that

  1. φ(t)∥ = ∥φ0∥ = 0 for 0 ≤ t < ∞ ifφ0∥ = 0.

    or

  2. φ(t) ∈ W for 0 ≤ t < ∞ if E(φ0) ≤ d, I(φ0) > 0.

Proof

Firstly from Proposition 1.1 it follows that problem (1.1) admits a unique local solution φ(t) ∈ C([0, T); H) satisfying (1.5), (1.6). Next we prove T = +∞.

  1. If ∥φ0∥ = 0, then by (1.5) we have ∥φ(t)∥ = ∥φ0∥ = 0, 0 ≤ t < T, which gives ∥|x|φ(t)∥ = 0 and ∥∇φ(t)∥ = 0, i.e. ∥φ(t)∥H = 0 for 0 ≤ t < T. Hence by Proposition 1.1 we get T = +∞.

  2. If E(φ0) ≤ d, I(φ0) > 0, then

    E(φ0)=12|x|φ02+n(p1)42n(p1)φ02+1p+1I(φ0)>0.

    Consequently, from Theorem 3.1 and Theorem 3.2 we have φ(t) ∈ W for 0 ≤ t < T. Thus, from

    12|x|φ2+n(p1)42n(p1)φ2+1p+1I(φ)=12|x|φ2+J(φ)=E(φ0),0t<T,

    we get

    |x|φ2+φ22n(p1)n(p1)4E(φ0),0t<T

    and

    |x|φ2+φ2+φ22n(p1)n(p1)4E(φ0)+φ02,0t<T,

    which gives T = +∞.

Since I(φ0) > 0 gives E(φ0) > 0, the following corollary is the improvement of Theorem 4.1.

Corollary 4.2

If in Theorem 4.1 the assumption “E(φ0) ≤ d, I(φ0) > 0” is replaced by “0 < E(φ0) < d, Iδ2(φ0) > 0”, where δ1 < δ2 are two roots of equation d(δ) = E(φ0), then problem (1.1) admits a unique global solution φ(t) ∈ C([0, ∞);) and φ(t) ∈ Wδ for δ ∈ [δ1, δ2], 0 ≤ t < ∞.

Proof

From Theorem 4.1 and Theorem 3.1 it follows that it is enough to prove I(φ0) > 0. In fact, if it is false, then there exists a δ̄ ∈ [a, δ2] such that Iδ̄ (φ0) = 0 and J(φ0) ≥ d(δ̄), which contradicts (3.1). □

Corollary 4.3

Let p satisfy (A), φ0H, a < δ0 < p+12. Assume that E(φ0) ≤ d(δ0) and Iδ0(φ0) > 0. Then problem (1.1) admits a unique global solution φ(t) ∈ C([0, ∞); ) and φ(t) ∈ Wδ0 for 0 ≤ t < ∞.

Proof

First we have E(φ0) ≤ d(δ0) < d(a) = d and

E(φ0)=12|x|φ02+12δ0p+1φ02+1p+1Iδ0(φ0)>0.

If δ1 < δ2 are two roots of equation d(δ) = E(φ0), then we have δ2δ0. Hence Iδ0(φ0) > 0 gives Iδ2(φ0) > 0. So by Corollary 4.2, problem (1.1) admits a unique global solution φ(t) ∈ C([0, ∞); ) and φ(t) ∈ Wδ for δ ∈ [δ1, δ2]. Since δ0 ∈ (a, δ2] ⊂ [δ1, δ2] we obtain φ(t) ∈ Wδ0 for 0 ≤ t < ∞. □

In the following theorem we give two results on global existence of solutions for problem (1.1) regarding ∥∇φ0∥.

Theorem 4.4

If in Corollary 4.2 the assumptionIδ2(φ0) > 0” is replaced by “∥∇φ0∥ < r(δ2)”, then problem (1.1) admits a unique global solution φ(t) ∈ C([0, ∞); ) satisfying

φ2E(φ0)a(δ1),|x|φ22E(φ0),0t<,(4.1)

where r(δ) and a(δ) are defined in Lemma 2.2 and Lemma 2.4 respectively.

Proof

Firstly E(φ0) > 0 gives ∥φ0∥ ≠ 0 and (by Lemma 2.1) ∥∇φ0∥ ≠ 0. Furthermore from 0 < ∥∇φ0∥ < r(δ2) we get Iδ2(φ0) > 0. Hence from Corollary 4.2 it follows that problem (1.1) admits a unique global solution φ(t) ∈ C([0, ∞); ) and φ(t) ∈ Wδ for δ ∈ (δ1, δ2), 0 ≤ t < ∞. Finally in

12|x|φ2+a(δ)φ2+1p+1Iδ(φ)=E(φ0)

letting δ = δ1 we obtain (4.1). □

Theorem 4.5

Let p satisfy (A), φ0H. Assume that E(φ0) = d, ∥∇φ0∥ < r(a). Then problem (1.1) admits a unique global solution φ(t) ∈ C([0, ∞); ) satisfying

φ22n(p1)n(p1)4d,|x|φ22d,0t<.(4.2)

Proof

Firstly by E(φ0) = d > 0 we have ∥φ0∥ ≠ 0. Hence by Lemma 2.2 we get I(φ0) > 0. From Theorem 4.1 it follows that problem (1.1) admits a unique global solution φ(t) ∈ C([0, ∞); ) and φ(t) ∈ W for 0 ≤ t < ∞. Inequality (4.2) follows from

12|x|φ2+n(p1)42n(p1)φ2+1p+1I(φ)=E(φ0)=d,0t<.

Next we discuss the finite time blow of solution for problem (1.1).

Theorem 4.6

Let p satisfy (A), φ0H. Assume that E(φ0) < d and I(φ0) < 0. Then the solution of problem (1.1) blows up in finite time.

Proof

First Proposition 1.1 gives the existence of unique local solution φC([0, T); ), where T is the existence time of φ. Let us prove T < ∞. Arguing by contradiction, suppose T = ∞. Let

F(t)=|x|2|φ|2dx.

Then by Proposition 1.2 we have

F(t)=8|φ|2|x|2|φ|2n(p1)2(p+1)|φ|p+1dx8|φ|2n(p1)2(p+1)|φ|p+1dx=8aIa(φ).(4.3)

In order to finish this proof we consider the following two cases:

  1. 0 < E(φ0) < d.

    In this case from Theorem 3.1 we have φVδ for δ1 < δ < δ2 and 0 ≤ t < ∞, where δ1 < δ2 are two roots of equation d(δ) = E(φ0). Clearly we have δ2 > a > 1. Hence we have Iδ (φ) < 0 and ∥∇φ∥ > r(δ) for a < δ < δ2, 0 ≤ t < ∞. And Iδ2(φ) ≤ 0, ∥∇φ∥ ≥ r(δ2) for 0 ≤ t < ∞. Thus from (4.3) we get

    F(t)8aIa(φ)=8a(aδ2)φ2+Iδ2(φ)8a(aδ2)φ28a(aδ2)r2(δ2)=C(δ2)<0,

    F(t)C(δ2)t+F(0),0t<,

    Hence there exists a t0 ≥ 0 such that F′(t) < F′(t0) < 0 for t0 < t < ∞ and

    F(t)F(t0)(tt0)+F(t0),t0t<.(4.4)

    Since I(φ0) < 0 implies F(0) > 0 from (4.4) it follows that there exists a T1 > 0 such that F(t) > 0 for 0 ≤ t < T1 and

    limtT1F(t)=0,

    which together with

    φ02=φ22nφF12(t)

    gives

    limtT1supφ=+.

    This contradicts T = +∞.

  2. E(φ0) ≤ 0.

    Since I(φ0) < 0 implies ∥φ0∥ ≠ 0, from Theorem 3.3 we have φVδ for 0 < δ < p+12, 0 ≤ t < ∞. If in the proof of part (i) δ2 is replaced by p+12, then we also obtain T < ∞.

Finally from Proposition 1.1 we get

limtTφH=+.

Theorem 4.6 is proved. □

Remark 4.7

The proof of Theorem 4.6 strongly depends on the fact that φ(t) ∈ Vδ for some δ > a, where φ(t) is the solution of problem (1.1) with E(φ0) < d, I(φ0) < 0. Therefore the introducing of Vδ is crucial for the proof of Theorem 4.6.

Corollary 4.8

Let p satisfy (A), φ0H and a < δ < p+12. Assume that E(φ0) ≤ d(δ) and Iδ (φ0) < 0. Then solution of problem (1.1) blows up in finite time.

Proof

On the one hand, from E(φ0) ≤ d(δ) and Corollary 2.5 we get E(φ0) < d. On the other hand, from Iδ (φ0) < 0 we get I (φ0) < 0. So by Theorem 4.6 solution of problem (1.1) blows up in finite time. □

Corollary 4.9

Let p satisfy (A), φ0H. Assume that E(φ0) < 0 or E(φ0) = 0, φ0 ≠ 0. Then solution of problem (1.1) blows up in finite time.

Proof

This corollary follows from Theorem 3.3 and Theorem 4.6. □

Next we discuss the blow up of solutions for problem (1.1) with E(φ0) = d.

Theorem 4.10

Let p satisfy (A), φ0H. Assume that E(φ0) = d, I(φ0) < 0 and F′(0) ≤ 0. Then solution of problem (1.1) blows up in finite time.

Proof

Let φ(t) be any solution of problem (1.1) with E(φ0) = d, I(φ0) < 0 and F′(0) ≤ 0. Let us prove T < ∞. Arguing by contradiction, again suppose T = +∞. From Theorem 3.2 we have φ(t) ∈ V for 0 ≤ t < ∞. Hence we get

F(t)8aI(φ)<0,0t<.

From this and F′(0) ≤ 0 it follows that for any t0 > 0 we have F′(t0) < 0 and

F(t)<F(t0)(tt0)+F(t0),t0<t<.

Hence there exists a T1 > 0 such that F(t) > 0 for 0 ≤ t < T1 and

limtT1F(t)=0.

The remainder of this proof is same as the proof of Theorem 4.6. □

From Theorem 4.1 and Theorem 4.6 we can obtain a sharp condition for global existence of solutions for problem (1.1) with E(φ0) < d as follows:

Theorem 4.11

Let p satisfy (A), φ0H. Assume that E(φ0) < d. Then when I(φ0) > 0, solution of problem (1.1) exists globally; and when I(φ0) < 0, solution of problem (1.1) blows up in finite time.

From Corollary 4.3 and Corollary 4.8 we can obtain a serious of sharp conditions for global existence and finite time blow up of solution for problem (1.1) in the following theorem.

Theorem 4.12

Let p satisfy (A), φ0H, a < δ < p+12. Assume that E(φ0) ≤ d(δ). Then when Iδ (φ0) > 0, the solution of problem (1.1) exists globally; and when Iδ (φ0) < 0, solution of problem (1.1) blows up in finite time.

From Theorem 4.1 and Theorem 4.10 we can obtain the following sharp condition for global existence of solution for problem (1.1) with E(φ0) = d.

Theorem 4.13

Let p satisfy (A), φ0H. Assume that E(φ0) = d, F′(0) ≤ 0. Then when I(φ0) > 0, the solution of problem (1.1) exists globally; and when I(φ0) < 0, solution of problem (1.1) blows up in finite time.

Note that from Lemma 2.4 we have

d=d(a)n(p1)42n(p1)r2(a)=d0.

From this we get the following sharp condition that only depends on ∥∇φ0∥.

Corollary 4.14

Let p satisfy (A), φ0H. Assume that E(φ0) < d0. Then when ∥∇φ0∥ < r(a), solution of problem (1.1) exists globally. When ∥∇φ0∥ ≥ r(a), solution of problem (1.1) blows up in finite time.

Proof

If ∥∇φ0∥ < r(a), then 0 < ∥∇φ0∥ < r(a) or ∥∇φ0∥ = 0, i.e. I(φ0) > 0 or ∥∇φ0∥ = 0. Then Theorem 4.1 gives the existence of unique global solution φ(t) ∈ C([0, ∞);). If ∥∇φ0∥ ≥ r(a), then by

12|x|φ02+n(p1)42n(p1)φ02+1p+1I(φ0)=E(φ0)<d0=n(p1)42n(p1)r2(a),

we get I(φ0) < 0. Hence from Theorem 4.6 it follows that solution of problem (1.1) blows up in finite time. □

Corollary 4.15

Let p satisfy (A), φ0H. Assume that E(φ0) > 0 and

n(p1)2(p+1)φ0p+1n(p1)22n(p1)n(p1)4E(φ0)n(p1)44<C1.(4.5)

Then when

n(p1)2(p+1)φ0p+1n(p1)2φ0n(p1)42<C1,(4.6)

the solution of problem (1.1) exists globally. And when

n(p1)2(p+1)φ0p+1n(p1)2φ0n(p1)42C1,(4.7)

solution of problem (1.1) blows up in finite time.

Proof

This corollary follows from Corollary 4.14 and the fact that (4.5), (4.6) and (4.7) are equivalent to E(φ0) < d0, ∥∇φ0∥ < r(a) and ∥∇φ0∥ ≥ r(a) respectively. □

From E(φ0) < 12(∥|x|φ02 + ∥∇φ02) for φ0 ≠ 0 and Corollary 4.15 we can obtain the following corollary.

Corollary 4.16

Let p satisfy (A), φ0H. Assume that E(φ0) > 0 and

n(p1)2(p+1)φ0p+1n(p1)2n(p1)n(p1)4|x|φ02+φ02n(p1)44C1,(4.8)

then solution of problem (1.1) exists globally.

Proof

One can see that (4.8) gives both (4.5) and (4.6), which proves this corollary. □

Finally we give another series of sharp conditions for globally existence and finite time blow up of solution for problem (1.1) as follows:

Corollary 4.17

Let p satisfy (A), φ0H, a < δ < p+12. Assume that E(φ0) ≤ a(δ) r2(δ), where r(δ) and a(δ) are defined in Lemma 2.2 and Lemma 2.4. Then when ∥∇φ0∥ < r(δ) the solution of problem (1.1) exists globally; and when ∥∇φ0∥ ≥ r(δ) the solution of problem (1.1) blows up in finite time.

Proof

  1. If ∥∇φ0∥ < r(δ), then we have 0 < ∥∇φ0∥ < r(δ) or ∥∇φ0∥ = 0, which gives Iδ(φ0) > 0 or ∥φ0∥ = 0. Hence by Theorem 4.12 and Theorem 4.1 the solution of problem (1.1) exists globally.

  2. If ∥∇φ0∥ ≥ r(δ), then from ∥|x|φ0∥ > 0 and

    12|x|φ02+a(δ)φ02+1p+1Iδ(φ0)=E(φ0)a(δ)r2(δ),

    we get Iδ(φ0) < 0. Again by Theorem 4.12 solution of problem (1.1) blows up in finite time. □

Acknowledgement

We appreciate the reviewers for their precious comments and suggestions, which help us tremendously to improve our manuscript.

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About the article

Received: 2019-01-18

Accepted: 2019-02-25

Published Online: 2019-08-06

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 882–894, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0031.

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© 2020 Mingyou Zhang and Md Salik Ahmed, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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