Blow-Up of Solutions to a Novel Two-Component Rod System

Abstract

We consider a novel two-component rod system which is closely connected to the shallow water theory. The present work is mainly concerned with the blow-up mechanism of strong solutions; we establish new conditions in view of some special classes of initial value to guarantee finite time blow-up of solutions.

1. Introduction

We consider the following variation of the two-component rod system: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥−𝜎2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥+𝜋𝜌𝜌_𝑥𝜋𝜌=0,𝑡>0,𝑥∈𝕊,_𝑡+𝜋𝜌𝑢_𝑥=0,𝑡>0,𝑥∈𝕊. (1) If we introduce a momentum 𝑦=𝑢−𝑢_𝑥𝑥, the previous system possesses the following form: 𝑦_𝑡+𝜎𝑦_𝑥𝑢+2𝜎𝑦𝑢_𝑥−3(𝜎−1)𝑢𝑢_𝑥+𝜋𝜌𝜌_𝑥𝜋𝜌=0,𝑡>0,𝑥∈𝕊,_𝑡+𝜋𝜌𝑢_𝑥=0,𝑡>0,𝑥∈𝕊, (2) where 𝜋𝜌=𝜌−𝜇𝜌=𝜌−¹₀𝜌𝑑𝑥, (3) with 𝑢(𝑥,𝑡) and 𝜌(𝑥,𝑡) depending on a space variable 𝑥∈𝕊=ℝ/ℤ and a time variable 𝑡. It is obvious that system (1) for 𝜋(𝜌)=0 reduces to the rod equation studied in [1–12]. Particularly, for 𝜋(𝜌)=0 and 𝜎=1, (1) becomes the celebrated Camassa-Holm equation which was investigated by many authors [13–21]. It reduces to the two-component rod system for 𝜇(𝜌)=0 studied in [22]. Moreover, system (1) reduces to the two-component Camassa-Holm (CH2) equation for 𝜇(𝜌)=0 and 𝜎=1 which was investigated in [23–30]. The two-component rod system which includes both velocity and density variables in the dynamics possesses the following form: 𝑢_𝑡−𝑢_𝑥𝑥𝑡−𝐴𝑢_𝑥+3𝑢𝑢_𝑥−𝜎2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥+𝜌𝜌_𝑥𝜌=0,𝑡>0,𝑥∈ℝ,_𝑡+𝜌𝑢_𝑥=0,𝑡>0,𝑥∈ℝ. (4) We note that the geometric structure of (1) is different from (4) with the incorporation of the term 𝜇(𝜌); this restricts our discussion only on the unit circle. Furthermore, the discussion of this work shows that system (1) possesses wave breaking phenomenon which is described with different blow-up criteria, while system (4) admits not only breaking wave solutions but also global in time solutions. However, we do not know whether the global solutions of (1) exist or not for the time being. It is the structure of (1) that breaks some properties which previously holds for (4). It is also known that for (4) 𝑑𝑑𝑡_𝕊𝜌𝑑𝑥=−_𝕊𝜌𝑢_𝑥𝑑𝑥=0, (5) which follows that ∫_𝕊𝜌𝑑𝑥 is an invariant with respect to time. Particularly, if 𝜌 has zero mean initially, then the solution 𝜌 to (4) will preserve zero mean for all time. This motivates us to introduce the term 𝜋(𝜌) since it always has zero mean. On the other hand, we consider the system (1) in the spaces 𝐻^𝑠×𝐻^𝑠−1/ℝ for 𝑠>5/2 on the circle, where 𝐻^𝑠=𝐻^𝑠(𝕊) denotes the 𝐿²-Sobolev space of regularity, and denote by 𝐻^𝑠−1/ℝ the space 𝐻^𝑠−1 with two functions being identified if they differ by a constant. The basic idea of this variation is the decomposition of 𝜌∈𝐻^𝑠−1 into 𝜋(𝜌) and 𝜇(𝜌), where 𝜇(𝜌)∈ℝ is independent on variable 𝑥, 𝜋(𝜌) belongs to 𝐻^𝑠−1, a subspace of 𝐻^𝑠−1 containing all zero mean functions. The main purpose of our work is to investigate formation of singularities of solutions to (1) where the conservation laws play crucial roles. The idea is motivated by Guo's recent works [28, 31].

The rest of this paper is organized as follows. In Section 2, we recall the local well-posedness theorem and show some auxiliary results which will be used in the sequel. In Section 3, the detailed blow-up criteria are established via various initial conditions and as a byproduct, the blow-up rate is presented.

2. Preliminaries

In this section, we would like to list some useful results for later use. We now provide the framework in which we shall reformulate (1). Let 𝐺(𝑥)=cosh(𝑥−[𝑥]−(1/2))2sinh(1/2),𝑥∈ℝ. (6) Then (1−𝜕²_𝑥)⁻¹𝑓=𝐺∗𝑓 for all 𝑓∈𝐿²(𝕊) and 𝐺∗𝑦=𝑢, where ∗ is the spatial convolution. With this in hand, we rewrite (1) as follows: 𝑢_𝑡+𝜎𝑢𝑢_𝑥+𝜕_𝑥𝐺∗3−𝜎2𝑢²+𝜎2𝑢²_𝑥+12𝜋𝜌²𝜋𝜌=0,𝑡>0,𝑥∈𝕊,_𝑡+𝜋𝜌𝑢_𝑥=0,𝑡>0,𝑥∈𝕊. (7) Firstly, we recall the elementary result on the local well-posedness theorem for system (7) which was shown in [32] from the geometric formulation.

Theorem 1. Let 𝑠>5/2. There is an open neighborhood 𝑈 containing (0,𝜋(0))∈𝐻^𝑠×𝐻^𝑠−1/ℝ such that for any (𝑢₀,𝜋(𝜌₀))∈𝑈 there exists a maximal 𝑇>0 and a unique solution (𝑢,𝜋(𝜌)) to the initial value problem for system (7) with 𝑢,𝜋𝜌∈𝐶[0,𝑇);𝐻^𝑠×𝐻^𝑠−1ℝ∩𝐶¹[0,𝑇);𝐻^𝑠−1×𝐻^𝑠−2ℝ, (8)(𝑢,𝜋(𝜌))(0)=(𝑢₀,𝜋(𝜌₀)). Moreover, the solution depends continuously on the initial data, that is, the mapping 𝑢₀,𝜋𝜌₀↦𝑢,𝜋𝜌∶𝑈⟶𝐶[0,𝑇);𝐻^𝑠×𝐻^𝑠−1ℝ∩𝐶¹[0,𝑇);𝐻^𝑠−1×𝐻^𝑠−2ℝ (9) is continuous.

We now introduce two characteristics for (7) 𝜕𝑞₁𝑞𝜕𝑡=𝑢₁𝑞,𝑡,0<𝑡<𝑇,𝑥∈ℝ,₁(𝑥,0)=𝑥,𝑥∈ℝ,𝜕𝑞₂𝑞𝜕𝑡=𝜎𝑢₂𝑞,𝑡,0<𝑡<𝑇,𝑥∈ℝ,₂(𝑥,0)=𝑥,𝑥∈ℝ, (10)(11) where 𝑢 denotes the first component of the solution to (7) with certain initial data and 𝑇 is the lifespan of the solution, then 𝑞_𝑖(𝑖=1,2) is a diffeomorphism of the line. These two characteristics are both increasing diffeomorphisms of ℝ. A direct calculation shows 𝑞_1,𝑡𝑥(𝑥,𝑡)=𝑢_𝑥𝑞₁(𝑥,𝑡),𝑡𝑞_1,𝑥(),𝑞𝑥,𝑡_2,𝑡𝑥(𝑥,𝑡)=𝜎𝑢_𝑥𝑞₂(𝑥,𝑡),𝑡𝑞_2,𝑥(𝑥,𝑡). (12) Thus, we have 𝑞_1,𝑥(𝑥,𝑡)=exp^𝑡₀𝑢_𝑥𝑞₁()[),𝑞,𝑠𝑑𝑠>0,𝑥,𝑡∈ℝ×0,𝑇_2,𝑥(𝑥,𝑡)=exp^𝑡₀𝜎𝑢_𝑥𝑞₂()[).,𝑠𝑑𝑠>0,𝑥,𝑡∈ℝ×0,𝑇 (13)(14) This is usually called the particle trajectory method, and it is important in the discussion of blow-up phenomena. We are now in a position to state the following.

Lemma 2. Let 𝑋₀=(𝑢₀,𝜋(𝜌₀))∈𝐻^𝑠×𝐻^𝑠−1/ℝ,𝑠>5/2, and let 𝑇 be the maximal existence time of the solution 𝑋=(𝑢,𝜋(𝜌)) to (7) with the initial data 𝑋₀. Then for all (𝑥,𝑡)∈𝕊×[0,𝑇), we have 𝜋𝜌𝑞₁,𝑡𝑞_1,𝑥(𝑥,𝑡)=𝜋𝜌₀(𝑥). (15)

Proof. Differentiating the left-hand side of (15) with respect to 𝑡, we obtain 𝑑𝜋𝜌𝑞𝑑𝑡¹,𝑡𝑞_1,𝑥(=𝜋𝑥,𝑡)_𝑡𝜌𝑞₁,𝑡+𝜋𝜌_𝑥𝑞_1,𝑡𝑞_1,𝑥()𝜌𝑞𝑥,𝑡+𝜋₁,𝑡𝑞_1,𝑥𝑡()=𝜋𝑥,𝑡_𝑡𝜌𝑞₁,𝑡+𝜋𝜌_𝑥𝑢𝑞₁,𝑡𝑞_1,𝑥()𝜌𝑞𝑥,𝑡+𝜋₁,𝑡𝑢_𝑥𝑞₁,𝑡𝑞_1,𝑥()𝜋𝜌𝑢𝑥,𝑡=−_𝑥𝑞_1,𝑥(𝑥,𝑡)+𝜋𝜌_𝑥𝑢𝑞₁,𝑡𝑞_1,𝑥()𝜌𝑞𝑥,𝑡+𝜋₁,𝑡𝑢_𝑥𝑞₁,𝑡𝑞_1,𝑥(𝑥,𝑡)=0, (16) where we have used 𝑞₁ defined in (10) and the second equation of system (7). Then 𝜋(𝜌(𝑞₁,𝑡))𝑞_1,𝑥(𝑥,𝑡) is independent on time 𝑡. Now we choose 𝑡=0, due to (13) we know 𝑞_1,𝑥(𝑥,0)=1. Therefore, this lemma is easily proved.

The application of (15) leads to the following result.

Theorem 3. Let 𝑋₀=(𝑢₀,𝜋(𝜌₀))∈𝐻^𝑠×𝐻^𝑠−1/ℝ,𝑠>5/2, and let 𝑇 be the maximal existence time of the solution 𝑋=(𝑢,𝜋(𝜌)) to (7) with the initial data 𝑋₀. If there exists 𝑀>0 such that 𝑢_𝑥(⋅,𝑡)_𝐿∞+𝜋𝜌_𝑥(⋅,𝑡)_𝐿∞≤𝑀, (17) then 𝐻^𝑠×𝐻^𝑠−1-norm of 𝑋 does not blow up on [0,𝑇).

Proof. Applying the operator Λ^𝑠 where Λ∶=(1−𝜕²_𝑥)^1/2 to the first equation in (7), and multiplying by Λ^𝑠𝑢, then integrating over 𝕊, we obtain 𝑑‖𝑢‖𝑑𝑡^2𝐻𝑠=−2𝜎𝑢𝑢_𝑥,𝑢_𝑠−2𝑢,𝑓₁(𝑢)_𝑠−2𝑢,𝑓₂(𝑢)_𝑠, (18) where 𝑓₁(𝑢)=𝜕_𝑥𝐺∗3−𝜎2𝑢²,𝑓₂(𝑢)=12𝜕_𝑥𝐺∗𝜎𝑢²_𝑥+𝜋𝜌². (19) From the proof in [25], we get 𝜎𝑢𝑢_𝑥,𝑢_𝑠≤𝑐𝑢_𝑥𝐿∞‖𝑢‖²_𝐻^𝑠,𝑢,𝑓₁(𝑢)_𝑠≤𝑐𝑢_𝑥𝐿∞‖𝑢‖²_𝐻^𝑠. (20) For the third term on the right-hand side of (13), we estimate it as follows: 𝑢,𝑓₂(𝑢)_𝑠≤𝑓₂(𝑢)_𝐻𝑠‖𝑢‖_𝐻^𝑠≤𝑐𝑢²_{𝑥𝐻𝑠−1}+𝜋𝜌²_𝐻𝑠−1‖𝑢‖_𝐻^𝑠≤𝑐𝑢_𝑥𝐿∞𝑢_{𝑥𝐻𝑠−1}+𝜋𝜌_𝐿∞𝜋𝜌_𝐻𝑠−1‖𝑢‖_𝐻𝑠≤𝑐𝑢_𝑥𝐿∞+𝜋𝜌_𝐿∞×‖𝑢‖²_𝐻^𝑠+𝜋𝜌²_𝐻𝑠−1. (21) Combining the previous inequalities, we can get 𝑑‖𝑢‖𝑑𝑡^2𝐻𝑠≤𝑐𝑢_𝑥𝐿∞+𝜋𝜌_𝐿∞×‖𝑢‖²_𝐻^𝑠+𝜋𝜌²_𝐻𝑠−1, (22) where the constant 𝑐 may be different from instance to instance. We do similar estimates for the second component by applying the operator Λ^𝑠−1 to the second equation in (7), and multiplying by Λ^𝑠−1𝜋(𝜌), then integrating over 𝕊 to obtain 𝑑‖𝑢‖𝑑𝑡^2𝐻𝑠=−2𝜋𝜌_𝑥𝑢,𝜋𝜌_𝑠−1−2𝜋𝜌𝑢_𝑥,𝜋𝜌_𝑠−1. (23) Similarly, we have 𝜋𝜌_𝑥𝑢,𝜋𝜌_𝑠−1≤𝑐𝑢_𝑥𝐿∞+𝜋𝜌_𝑥𝐿∞‖𝑢‖²_𝐻^𝑠+𝜋𝜌²_𝐻𝑠−1,𝜋𝜌𝑢_𝑥,𝜋𝜌_𝑠−1≤𝑐𝑢_𝑥𝐿∞+𝜋𝜌_𝐿∞‖𝑢‖²_𝐻^𝑠+𝜋𝜌²_𝐻𝑠−1. (24) Therefore, it follows that 𝑑𝜋𝜌𝑑𝑡^{2𝐻𝑠−1}≤𝑐𝑢_𝑥𝐿∞+𝜋𝜌_𝑥𝐿∞+𝜋𝜌_𝐿∞×‖𝑢‖²_𝐻^𝑠+𝜋𝜌²_𝐻𝑠−1. (25) By (22) and (25) we have 𝑑‖𝑢‖𝑑𝑡^2𝐻𝑠+𝜋𝜌²_𝐻𝑠−1≤𝑐𝑢_𝑥𝐿∞+𝜋𝜌_𝑥𝐿∞×‖𝑢‖²_𝐻^𝑠+𝜋𝜌²_𝐻𝑠−1. (26) It is easy to obtain by Gronwall's inequality, Lemma 2, and (13) that ‖𝑢‖²_𝐻^𝑠+𝜋𝜌²_𝐻𝑠−1≤exp(𝑐𝑀𝑡)𝑢₀²_𝐻𝑠+𝜋𝜌₀²_𝐻𝑠−1. (27) Then the 𝐻^𝑠×𝐻^𝑠−1-norm of 𝑋 does not blow up on [0,𝑇).

Next we present the precise blow-up scenario for sufficiently regular solutions to (7).

Theorem 4. Let 𝑋₀=(𝑢₀,𝜋(𝜌₀))∈𝐻^𝑠×𝐻^𝑠−1/ℝ,𝑠>5/2, and let 𝑇 be the maximal existence time of the solution 𝑋=(𝑢,𝜋(𝜌)) to (7) with the initial data 𝑋₀. Then the solution X blows up in finite time if and only if lim_𝑡→𝑇inf_𝑥∈𝕊𝑢_𝑥(𝑥,𝑡)=−∞𝑜𝑟lim_𝑡→𝑇sup𝜌_𝑥(⋅,𝑡)_𝐿∞=+∞. (28)

Proof. Multiplying the first equation in (2) by 𝑦 and integrating by parts, we get _𝕊𝑦𝑦_𝑡𝑑𝑥+𝜎_𝕊𝑦𝑦_𝑥𝑢𝑑𝑥+2𝜎_𝕊𝑦²𝑢_𝑥−3()𝑑𝑥𝜎−12_𝕊𝑦𝑢²_𝑥𝑑𝑥+_𝕊𝑦𝜋𝜌𝜋𝜌_𝑥𝑑𝑥=0. (29) It then follows that 12𝑑𝑑𝑡_𝕊𝑦²𝑑𝑥=−3𝜎2_𝕊𝑦²𝑢_𝑥𝑑𝑥−3(𝜎−1)_𝕊𝑢𝑢_𝑥𝑢_𝑥𝑥+1𝑑𝑥2_𝕊𝜋𝜌²𝑦_𝑥𝑑𝑥=−3𝜎2_𝕊𝑦²𝑢_𝑥𝑑𝑥−3(𝜎−1)_𝕊𝑢𝑢_𝑥𝑢_𝑥𝑥+1𝑑𝑥2_𝕊𝜋𝜌²𝑢_𝑥1𝑑𝑥−2_𝕊𝜋𝜌²𝑢_𝑥𝑥𝑥𝑑𝑥. (30) Differentiating the first equation in (2) with respect to 𝑥, multiplying by 𝑦_𝑥, and integrating by parts yield 12𝑑𝑑𝑡_𝕊𝑦²_𝑥𝑑𝑥=−5𝜎2_𝕊𝑢_𝑥𝑦²_𝑥𝑑𝑥+𝜎_𝕊𝑦²𝑢_𝑥+3()𝑑𝑥𝜎−12_𝕊𝑢²_𝑥𝑥𝑦_𝑥+𝑑𝑥_𝕊𝑢_𝑥𝑥𝑥𝜋𝜌²_𝑥+𝜋𝜌𝜋𝜌_𝑥𝑥−12𝜋𝜌²𝑑𝑥. (31) Combining (30) and (31) together, we obtain 𝑑𝑑𝑡_𝕊𝑦²+𝑦²_𝑥𝑑𝑥=_𝕊𝜋𝜌²−5𝜎𝑦²_𝑥−𝜎𝑦²𝑢_𝑥()𝑑𝑥+3𝜎−1_𝕊𝑢_𝑥2𝑢²_𝑥+5𝑢²_𝑥𝑥𝑑𝑥+2_𝕊𝑢_𝑥𝑥𝑥𝜋𝜌²_𝑥+𝜋𝜌𝜋𝜌_𝑥𝑥−𝜋𝜌²𝑑𝑥. (32) Similar arguments made on the second equation in (2) yield 𝑑𝑑𝑡_𝕊𝜋𝜌²𝑑𝑥=−_𝕊𝜋𝜌²𝑢_𝑥𝑑𝑑𝑥,𝑑𝑡_𝕊𝜋𝜌²_𝑥𝑑𝑥=−3_𝕊𝜋𝜌²_𝑥𝑢_𝑥+𝑑𝑥_𝕊𝜋𝜌²𝑢_𝑥𝑥𝑥𝑑𝑑𝑥,𝑑𝑡_𝕊𝜋𝜌²_𝑥𝑥𝑑𝑥=−5_𝕊𝜋𝜌²_𝑥𝑥𝑢_𝑥𝑑𝑥−_𝕊2𝜋𝜌𝜋𝜌_𝑥𝑥−3𝜋𝜌²_𝑥𝑢_𝑥𝑥𝑥𝑑𝑥. (33) It follows by combining (32) and (33) that 𝑑𝑑𝑡_𝕊𝑦²+𝑦²_𝑥+𝜋𝜌²+𝜋𝜌²_𝑥+𝜋𝜌²_𝑥𝑥𝑑𝑥=−_𝕊𝑢_𝑥𝜎𝑦²+5𝜎𝑦²_𝑥+3𝜋𝜌²_𝑥+5𝜋𝜌²_𝑥𝑥+3(1−𝜎)2𝑢²_𝑥+5𝑢²_𝑥𝑥+𝑑𝑥_𝕊𝑢_𝑥𝑥𝑥5𝜋𝜌²_𝑥−𝜋𝜌²𝑑𝑥. (34) Assume that there exist 𝑀₁>0 and 𝑀₂>0, such that 𝑢_𝑥(𝑥,𝑡)≥−𝑀₁,()[),𝜋𝜌𝑥,𝑡∈𝕊×0,𝑇_𝑥𝐿∞=𝜌_𝑥𝐿∞≤𝑀₂,∀𝑡∈[0,𝑇). (35) By Lemma 2, we have 𝜋𝜌_𝐿∞≤𝑒^𝑀1𝑇𝜋𝜌_0𝐿∞≤𝑐𝑒^𝑀1𝑇𝜌_0𝐿∞. (36) It then follows that 𝑑𝑑𝑡_𝕊𝑦²+𝑦²_𝑥+𝜋𝜌²+𝜋𝜌²_𝑥+𝜋𝜌²_𝑥𝑥𝑑𝑥≤𝑐𝑀_1𝕊𝑦²+𝑦²_𝑥+𝜋𝜌²_𝑥+𝜋𝜌²_𝑥𝑥𝑑𝑥+𝑐𝑀_2𝕊𝑢²_𝑥𝑥𝑥+𝜋𝜌²+𝜋𝜌²_𝑥𝑑𝑥≤𝑐_𝕊𝑦²+𝑦²_𝑥+𝜋𝜌²+𝜋𝜌²_𝑥+𝜋𝜌²_𝑥𝑥𝑑𝑥. (37) It is easy to obtain by Gronwall's inequality that ‖𝑢‖²_𝐻3+𝜋𝜌²_𝐻2≤𝑦²_𝐻1+𝜋𝜌²_𝐻2≤𝑒^𝑐𝑡𝑦₀²_𝐻1+𝜋𝜌₀²_𝐻2. (38) Sobolev imbedding, (38), and Theorem 3 ensure that the solution 𝑋 does not blow up in finite time.
On the other hand, due to the Sobolev imbedding theorem, we observe that 𝑢_𝑥(𝑥,𝑡)→−∞ or ‖𝜌_𝑥(⋅,𝑡)‖_𝐿∞→+∞ will lead to blow-up of solutions.

After the local well-posedness of strong solutions (see Theorem 1) is established, a natural question is whether this local solution can exist globally. If the solution only exists in finite time, what induces the blow-up phenomenon? On the other hand, finding sufficient conditions to guarantee the finite time singularities or global existence is very interesting, especially for sufficient conditions added on certain initial data. The following results will give some positive answers.

3. Blow-Up

In this section, we pay more attention to the formation of singularities for strong solutions to our system. It will show that wave breaking is one way that singularities arise in smooth solutions. We start this section with the following lemmas.

Lemma 5. Let 𝑋₀=(𝑢₀,𝜋(𝜌₀))∈𝐻^𝑠×𝐻^𝑠−1/ℝ, 𝑠≥5/2, and let 𝑇 be the maximal existence time of the solution 𝑋=(𝑢,𝜋(𝜌)) to (7) with the initial data 𝑋₀. Then we obtain the following conservation laws: 𝐸₁=_𝕊𝑢²+𝑢²_𝑥+𝜋𝜌²𝐸𝑑𝑥,₂=_𝕊𝑢³+𝜎𝑢𝑢²_𝑥+𝑢𝜋𝜌²𝑑𝑥. (39)

Proof. The proofs are direct consequences of the energy method; similar ones are given in [28] for the two-component Camassa-Holm equations; we refer the readers to [28] for the details.

Remark 6. The conservation of 𝐸₁ guarantees the uniform bound of 𝑢(𝑥,𝑡), then Theorem 4 is also interpreted as wave breaking.

Lemma 7 (see [10]). For all 𝑓∈𝐻¹(𝕊), the following inequality holds: 𝐺∗𝑓²+𝑓²_𝑥2(𝑥)≥𝐶₀𝑓²(𝑥), (40) with 𝐶₀=12+arctan(sinh(1/2))2sinh(1/2)+2arctan(sinh(1/2))sinh²(1/2)≈0.869. (41) Moreover, 𝐶₀ is the optimal constant obtained by the function 𝑓₀=1+arctan(sinh(𝑥−[𝑥]−(1/2)))sinh(𝑥−[𝑥]−(1/2))((().1+arctansinh1/2))sinh1/2 (42)

Lemma 8 (see [33]). For all 𝑓∈𝐻¹(𝕊), the following inequality holds: max_[]𝑥∈0,1𝑓²(𝑥)≤𝐶₁𝑓²_𝐻1(𝕊), (43) where 𝐶₁=𝑒^1/2+𝑒^−1/22𝑒^1/2−𝑒^−1/2. (44) Moreover, 𝐶₁ is the minimum value, so in this sense, 𝐶₁ is the optimal constant which is obtained by the associated Green's function 𝐺(𝑥)=cosh(𝑥−[𝑥]−(1/2))().2sinh1/2 (45)

Lemma 9 (see [33]). For any function 𝑓∈𝐻²(𝕊), the following inequality holds: 𝑓(𝑥)²_{𝐿∞(𝕊)}≤_𝕊𝑓(𝑥)𝑑𝑥²+𝑓(𝑥)²_𝐻1(𝕊). (46)

Lemma 10 (see [34]). Assume 𝑓(𝑥)∈𝐻^𝑠(𝕊), 𝑠>2. If ∫_𝕊𝑓(𝑥)𝑑𝑥=0, then 𝑓(𝑥)²_{𝐿∞(𝕊)}≤112_𝕊𝑓²_𝑥(𝑥)𝑑𝑥,_𝕊𝑓²(𝑥)1𝑑𝑥≤12_𝕊𝑓²_𝑥(𝑥)𝑑𝑥,_𝕊𝑓²(𝑥)𝑓²_𝑥(𝑥)1𝑑𝑥≤12_𝕊𝑓²_𝑥(𝑥)𝑑𝑥². (47)

It is now to state our result.

Theorem 11. Let 𝑋₀=(𝑢₀,𝜋(𝜌₀))∈𝐻^𝑠×𝐻^𝑠−1/ℝ, 𝑠≥5/2, and let 𝑇 be the maximal existence time of solution 𝑋=(𝑢,𝜋(𝜌)) to (7) with the initial data 𝑋₀. If the following inequality holds: (1)_𝕊𝑢³_0𝑥𝑑𝑥<−2𝐸₁(0)𝜅𝜎(2),𝑖𝑓0<𝜎<1,_𝕊𝑢³_0𝑥𝑑𝑥<−3(3−𝜎)1−𝐶₀𝜎𝐶₁𝐸³₁(0)+3𝜎𝑀^∗2𝐸²₁(0),𝑖𝑓1≤𝜎<3, (48) where 𝐸₁(0) is the initial value of 𝐸₁ in Lemma 5, and constant 𝜅 is determined later. Then the corresponding solution 𝑋 blows up in finite time.

Proof. Differentiating the first equation in system (7) with respect to 𝑥, we obtain 𝑢_𝑥𝑡+𝜎𝑢²_𝑥+𝑢𝑢_𝑥𝑥+𝜕²_𝑥𝐺∗3−𝜎2𝑢²+𝜎2𝑢²_𝑥+12𝜋𝜌²=0. (49) Applying the relation (1−𝜕²_𝑥)⁻¹𝑓=𝐺∗𝑓 to (49) gives 𝑢_𝑥𝑡+𝜎2𝑢²_𝑥+𝜎𝑢𝑢_𝑥𝑥+𝐺∗3−𝜎2𝑢²+𝜎2𝑢²_𝑥+12𝜋𝜌²−3−𝜎2𝑢²−12𝜋𝜌²=0. (50) If 0<𝜎<1,multiplying (50) with 𝑢²_𝑥 and integrating by parts subsequently, we obtain 13𝑑𝑑𝑡_𝕊𝑢³_𝑥𝜎𝑑𝑥=−6_𝕊𝑢⁴_𝑥𝑑𝑥−_𝕊𝑢²_𝑥𝐺∗3−𝜎2𝑢²+𝜎2𝑢²_𝑥+12𝜋𝜌²+𝑑𝑥_𝕊𝑢²_𝑥3−𝜎2𝑢²+12𝜋𝜌²𝜎𝑑𝑥=−6_𝕊𝑢⁴_𝑥𝑑𝑥−𝜎_𝕊𝑢²_𝑥𝐺∗𝑢²+𝑢²_𝑥2−𝑑𝑥3−3𝜎2_𝕊𝑢²_𝑥𝐺∗𝑢²−1𝑑𝑥2_𝕊𝑢²_𝑥𝐺∗𝜋𝜌²𝑑𝑥+_𝕊𝑢²_𝑥3−𝜎2𝑢²+12𝜋𝜌²𝜎𝑑𝑥≤−6_𝕊𝑢⁴_𝑥𝑑𝑥+3−𝜎−2𝜎𝐶₀2_𝕊𝑢²_𝑥𝑢²1𝑑𝑥+2_𝕊𝑢²_𝑥𝜋𝜌²𝑑𝑥, (51) where we have used the following identity: _𝕊𝑢²_𝑥𝑢𝑢_𝑥𝑥1𝑑𝑥=−3_𝕊𝑢⁴_𝑥𝑑𝑥. (52) Since 3−3𝜎>0, and _𝕊𝑢²_𝑥𝐺∗𝜋𝜌²𝑑𝑥≥0,_𝕊𝑢²_𝑥𝐺∗𝑢²𝑢𝑑𝑥≥0,𝐺∗²+𝑢²_𝑥2(𝑥)≥𝐶₀𝑢²(𝑥), (53) where 𝐶₀≈0.869<1.
In the following, we estimate the three terms on the right-hand side of (51) one by one. The Cauchy-Schwarz inequality implies that _𝕊𝑢³_𝑥𝑑𝑥≤_𝕊𝑢⁴_𝑥𝑑𝑥^1/2_𝕊𝑢²_𝑥𝑑𝑥^1/2, (54) hence _𝕊𝑢⁴_𝑥≥1𝑑𝑥∫_𝕊𝑢²_𝑥𝑑𝑥^𝕊𝑢³_𝑥𝑑𝑥²≥1𝐸₁(0)_𝕊𝑢³_𝑥𝑑𝑥². (55) Using Lemma 8 and the invariant property of 𝐸₁, we have _𝕊𝑢²_𝑥𝑢²𝑑𝑥≤‖𝑢‖²_𝐿^∞_𝕊𝑢²_𝑥𝑑𝑥≤𝐶₁𝐸²₁(0). (56) Note the fact that _𝕊𝜋𝜌𝑑𝑥=_𝕊𝜌(𝑥,𝑡)𝑑𝑥−_𝕊𝕊𝜌(𝑥,𝑡)𝑑𝑥𝑑𝑥=0. (57) This implies that for any 𝑡∈[0,𝑇), 𝜋(𝜌) has at least a zero point 𝜂_𝑡, that is, 𝜋(𝜌(𝜂_𝑡,𝑡))=0. Therefore, we have 𝜋𝜌=^𝑥_𝜂𝑡𝜋𝜌(𝑠,𝑡)_𝑠𝑑𝑠,𝑥∈𝜂_𝑡,𝜂_𝑡+1. (58) Suppose that the solution does not blow-up in finite time, it follows that there exists a constant 𝑀^∗>0 such that 𝑢_𝑥(𝑥,𝑡)>−𝑀^∗ and ‖𝜌_𝑥(⋅,𝑡)‖_{𝐿∞(𝕊)} is bounded by 𝑀^∗; thus 𝜋𝜌=^𝑥_𝜂𝑡𝜋𝜌(𝑠,𝑡)_𝑠𝑑𝑠=^𝑥_𝜂𝑡𝜌_𝑠(𝑠,𝑡)𝑑𝑠≤𝑀^∗. (59) Then _𝕊𝑢²_𝑥𝜋𝜌²𝑑𝑥≤𝑀^∗2_𝕊𝑢²_𝑥𝑑𝑥≤𝑀^∗2𝐸₁(0). (60) By (55)–(60), we have 𝑑𝑑𝑡_𝕊𝑢³_𝑥𝜎𝑑𝑥≤−2_𝕊𝑢⁴_𝑥3𝑑𝑥+3−𝜎−2𝜎𝐶₀2_𝕊𝑢²_𝑥𝑢²+3𝑑𝑥2_𝕊𝑢²_𝑥𝜋𝜌²𝜎𝑑𝑥≤−2𝐸₁(0)_𝕊𝑢³_𝑥𝑑𝑥²+33−𝜎−2𝜎𝐶₀2𝐶₁𝐸²₁(0)+32𝑀^∗2𝐸₁(0). (61) For convenience, we denote 𝜑(𝑡)=_𝕊𝑢³_𝑥3𝑑𝑥,𝜅=3−𝜎−2𝜎𝐶₀2𝐶₁𝐸²₁(0)+32𝑀^∗2𝐸₁(0), (62) where we noticed the fact that 3−𝜎−2𝜎𝐶₀>0. (63) That is, 𝑑𝜑(𝑡)𝜎𝑑𝑡≤−2𝐸₁(0)𝜑²(𝑡)+𝜅. (64) Note that if the initial quantity satisfies 𝜑(0)<−2𝐸₁(0)𝜅𝜎, (65) the standard argument on the Riccati type inequality and the initial hypothesis ensure that there exists a finite time 𝑇 such that lim_{𝑡→𝑇𝕊}𝑢³_𝑥(𝑥,𝑡)𝑑𝑥=−∞. (66) Since _𝕊𝑢³_𝑥𝑑𝑥≥inf𝑢_𝑥(𝑥,𝑡)_𝕊𝑢²_𝑥𝑑𝑥>inf𝑢_𝑥(𝑥,𝑡)𝐸₁(0). (67) This implies that lim_𝑡→𝑇inf𝑢_𝑥(𝑥,𝑡)=−∞. (68) Then it contradicts the assumption 𝑢_𝑥(𝑥,𝑡)>−𝑀^∗. By Theorem 4, we know that the solution must blow up in finite time.
If 1≤𝜎<3, we get 𝑑𝑑𝑡_𝕊𝑢³_𝑥𝜎𝑑𝑥=−2_𝕊𝑢⁴_𝑥3()𝑑𝑥−3−𝜎2_𝕊𝑢²_𝑥𝐺∗𝑢²+𝑢²_𝑥2−9()𝑑𝑥𝜎−14_𝕊𝑢²_𝑥𝐺∗𝑢²_𝑥−1𝑑𝑥2_𝕊𝑢²_𝑥𝐺∗𝜋𝜌²𝑑𝑥+3_𝕊𝑢²_𝑥3−𝜎2𝑢²+12𝜋𝜌²𝜎𝑑𝑥≤−2_𝕊𝑢⁴_𝑥3(𝑑𝑥+3−𝜎)1−𝐶₀2_𝕊𝑢²_𝑥𝑢²+3𝑑𝑥2_𝕊𝑢²_𝑥𝜋𝜌²𝑑𝑥. (69) We also denote that 𝜑(𝑡)=_𝕊𝑢³_𝑥𝑑𝑥. (70) It follows that 𝑑𝜑(𝑡)𝜎𝑑𝑡≤−2𝐸₁(0)𝜑²(𝑡)+3(3−𝜎)1−𝐶₀2𝐶₁𝐸²₁(0)+32𝑀^∗2𝐸₁(0), (71) where we note 3(3−𝜎)(1−𝐶₀)/2>0. Similar to case (1), it is easy to see that under the corresponding condition (2) of our theorem and the previous arguments, blow-up phenomenon occurs. We complete the proof.

As we know, the key issue for partial differential equations lies in the estimates. In the following results, we apply different strategies to derive suitable bounds for solution, then blow-up phenomenon occurs while some special initial values are involved. Precisely, we show the following.

Theorem 12. Suppose that 𝑋₀=(𝑢₀,𝜋(𝜌₀))∈𝐻^𝑠×𝐻^𝑠−1/ℝ,𝑠≥5/2, 𝑢₀ does not vanish identically. If ∫_𝕊𝑢₀(𝑥)𝑑𝑥=0, and one of the following conditions is satisfied (1)()𝜎∈3sinh1/26+sinh(1/2),3,𝐸₁(0)>𝐶₂,(2)()𝜎∈0,3sinh1/2(),𝜑(0)6+sinh1/2<−3−𝜎−34𝜎()𝐸2sinh1/2³¹(0)+3𝜎𝑀^∗2𝐸²₁(0), (72) for some constant 𝐶₂ and function 𝜑(𝑡) is defined as previously, then the corresponding solution to initial data 𝑋₀ of (7) blows up in finite time.

Proof. Differentiating both sides of the first equation of (7) with respect to variable 𝑥, we obtain 𝑢_𝑥𝑡+𝜎𝑢²_𝑥+𝑢𝑢_𝑥𝑥+𝜕²_𝑥𝐺∗3−𝜎2𝑢²+𝜎2𝑢²_𝑥+12𝜋𝜌²=0. (73) Applying the relation 𝜕²_𝑥(𝐺∗𝑓)=𝐺∗𝑓−𝑓 to (73) gives 𝑢_𝑥𝑡+𝜎2𝑢²_𝑥+𝜎𝑢𝑢_𝑥𝑥+𝐺∗3−𝜎2𝑢²+𝜎2𝑢²_𝑥+12𝜋𝜌²−3−𝜎2𝑢²−12𝜋𝜌²=0. (74) Multiplying (74) with 𝑢²_𝑥 and integrating by parts subsequently, we obtain 𝑑𝑑𝑡_𝕊𝑢³_𝑥𝜎𝑑𝑥=−2_𝕊𝑢⁴_𝑥3()𝑑𝑥+3−𝜎2_𝕊𝑢²_𝑥𝑢²+3𝑑𝑥2_𝕊𝑢²_𝑥𝜋𝜌²𝑑𝑥−3_𝕊𝑢²_𝑥𝐺∗3−𝜎2𝑢²+𝜎2𝑢²_𝑥+12𝜋𝜌²𝑑𝑥, (75) where we have used the following identity: _𝕊𝑢²_𝑥𝑢𝑢_𝑥𝑥1𝑑𝑥=−3_𝕊𝑢⁴_𝑥𝑑𝑥. (76) On the other hand, we know that ∫_𝕊𝑢(𝑥,𝑡)𝑑𝑥=0 in view of the hypothesis, and the following inequality holds 1()(𝑥)≤()2sinh1/2≤𝐺cosh1/2(),2sinh1/2 (77) where cosh(1/2)2sinh(1/2)≈1.082>1. (78) Using Lemma 10 and (77), we obtain 𝑑𝑑𝑡_𝕊𝑢³_𝑥𝜎𝑑𝑥≤−2_𝕊𝑢⁴_𝑥𝑑𝑥+3−𝜎8_𝕊𝑢²_𝑥𝑑𝑥²+32_𝕊𝑢²_𝑥𝜋𝜌²−𝑑𝑥3𝜎2_𝕊𝑢²_𝑥𝐺∗𝑢²_𝑥𝜎𝑑𝑥≤−2_𝕊𝑢⁴_𝑥𝑑𝑥+3−𝜎8_𝕊𝑢²_𝑥𝑑𝑥²+32_𝕊𝑢²_𝑥𝜋𝜌²𝑑𝑥−3𝜎4sinh(1/2)_𝕊𝑢²_𝑥𝑑𝑥²𝜎=−2_𝕊𝑢⁴_𝑥𝑑𝑥−3𝜎()−4sinh1/23−𝜎8_𝕊𝑢²_𝑥𝑑𝑥²+32_𝕊𝑢²_𝑥𝜋𝜌²𝑑𝑥. (79) In view of (60), we obtain that 𝑑𝑑𝑡_𝕊𝑢³_𝑥𝜎𝑑𝑥≤−2_𝕊𝑢⁴_𝑥𝑑𝑥−3𝜎()−4sinh1/23−𝜎8×_𝕊𝑢²_𝑥𝑑𝑥²+32𝑀^∗2𝐸₁(0). (80) If 3sinh(1/2)/(6+sinh(1/2))<𝜎<3, it is easy to know that 3𝜎()−4sinh1/23−𝜎8>0. (81) If there exists a constant 𝐶₂ such that the initial energy 𝐸₁(0)>𝐶₂, then there is some 𝛿>0 such that 32𝑀^∗2𝐸₁(0)≤𝛿_𝕊𝑢²_𝑥𝑑𝑥²,3𝜎()−4sinh1/23−𝜎8−𝛿>0. (82) On the other hand, we have by Lemma 10_𝕊𝑢²_𝑥𝑑𝑥≥12‖𝑢‖13^2𝐻1(𝕊). (83) Therefore, the previous inequality yields 𝑑𝑑𝑡_𝕊𝑢³_𝑥𝜎𝑑𝑥≤−2_𝕊𝑢⁴_𝑥−𝑑𝑥1441693𝜎()−4sinh1/23−𝜎8−𝛿‖𝑢‖²_𝐻1(𝕊). (84) In view of Hölder's inequality, there holds _𝕊𝑢⁴_𝑥𝑑𝑥≥_𝕊𝑢³_𝑥𝑑𝑥^4/3. (85) For simplicity of notations, we denote by 𝜑(𝑡) and 𝜇 the following quantities: _𝕊𝑢³_𝑥𝑑𝑥,1441693𝜎()−4sinh1/23−𝜎8−𝛿, (86) respectively. Therefore we have 𝑑𝜑(𝑡)𝜎𝑑𝑡≤−2𝜑^4/3(𝑡)−𝜇‖𝑢‖⁴_𝐻1(𝕊). (87) First, we can easily get 𝜑(𝑡)≤𝜑(0)−𝜇‖𝑢‖⁴_𝐻1(𝕊)𝑡, and it is not difficult to find that there exists a 𝑡₀≥0 such that 𝜑(𝑡₀)<0. Then for 𝑡>𝑡₀, we get 𝑑𝜑(𝑡)𝜎𝑑𝑡≤−2𝜑^4/3(𝑡),with𝜑𝑡₀<0. (88) Solving this inequality yields 𝜑(𝑡)≤𝜑^−1/3𝑡₀+𝜎6𝑡−𝑡₀⁻³, (89) which approaches −∞ as 𝑡 arrives at 𝑇₀=−(6/𝜎)𝜑^−1/3(𝑡₀)+𝑡₀; that is, there exists a time 𝑇≤−(6/𝜎)𝜑^−1/3(𝑡₀)+𝑡₀ such that lim_{𝑡→𝑇𝕊}𝑢³_𝑥𝑑𝑥=−∞. (90) Therefore, it follows that lim_𝑡→𝑇inf_𝑥∈ℝ𝑢_𝑥(𝑥,𝑡)=−∞. (91) Then it contradicts the assumption 𝑢_𝑥(𝑥,𝑡)>−𝑀^∗. By blow-up scenario, we know that the solution must blow up in finite time.
If 0<𝜎≤3sinh(1/2)/(6+sinh(1/2)), then we have 𝑑𝑑𝑡_𝕊𝑢³_𝑥𝜎𝑑𝑥≤−2_𝕊𝑢⁴_𝑥+𝑑𝑥3−𝜎8−3𝜎()×4sinh1/2_𝕊𝑢²_𝑥𝑑𝑥²+32𝑀^∗2𝐸₁(0), (92) where 3−𝜎8−3𝜎4sinh(1/2)≥0. (93) We also use 𝜑(𝑡) as previously to get 𝑑𝜑(𝑡)𝜎𝑑𝑡≤−2𝐸₁(0)𝜑²(𝑡)+3−𝜎8−3𝜎()𝐸4sinh1/2²¹(0)+32𝑀^∗2𝐸₁(0). (94) Similar arguments to (64) in Theorem 11 and condition (2) guarantee the finite time blow-up of solution to (7).

The zero mean of 𝑢(𝑥,𝑡) in the previous theorem can be substituted by 𝐸₂(0)=0; blow-up still occurs with the aid of different estimate from (83).

Theorem 13. Assume that 𝑋₀=(𝑢₀,𝜋(𝜌₀))∈𝐻^𝑠×𝐻^𝑠−1/ℝ, 𝑠≥5/2, 𝑢₀ does not vanish identically, ∫_𝕊(𝑢³₀+𝜎𝑢₀𝑢²_0𝑥+𝑢₀𝜋(𝜌₀)²)𝑑𝑥=0, if one of the following conditions is satisfied: (1)()𝜎∈3sinh1/22+sinh(1/2),3,𝐸₁(0)>𝐶₃,(2)()𝜎∈0,3sinh1/2(),𝜑(0)2+sinh1/2<−3(3−𝜎)−34𝜎()𝐸2sinh1/2³¹(0)+3𝜎𝑀^∗2𝐸²₁(0), (95) for some constant 𝐶₃, then the corresponding solution to (7) blows up in finite time.

Proof. By assumption and the invariance property of 𝐸₂, we have _𝕊𝑢𝑢²+𝜎𝑢²_𝑥+𝜋𝜌²=𝑑𝑥_𝕊𝑢³+𝜎𝑢𝑢²_𝑥+𝑢𝜋𝜌²𝑑𝑥=0. (96) Therefore, 𝑢(𝑥,𝑡) must change sign, so there exists at least one zero point on 𝕊. Then for each 𝑡∈[0,𝑇), suppose that there is a 𝜉_𝑡∈[0,1] such that 𝑢(𝜉_𝑡,𝑡)=0, for 𝑥∈𝕊 we have 𝑢²(𝑥,𝑡)=^𝑥_𝜉𝑡𝑢_𝑥𝑑𝑥²≤𝑥−𝜉_𝑡^𝑥_𝜉𝑡𝑢²_𝑥𝑑𝑥,𝑥∈𝜉_𝑡,𝜉_𝑡+12. (97) Thus, the previous relation and an integration by parts yield ^𝜉𝑡_𝜉^+1/2_𝑡𝑢²𝑢²_𝑥𝑑𝑥≤^𝜉𝑡_𝜉^+1/2_𝑡𝑥−𝜉_𝑡𝑢²_𝑥^𝑥_𝜉𝑡𝑢²_𝑥=𝑑𝑧𝑑𝑥_{𝜉𝑡𝜉+1/2𝑡}𝑥−𝜉_𝑡^𝑥_𝜉𝑡𝑢²_𝑥𝑑𝑧𝑑^𝑥_𝜉𝑡𝑢²_𝑥=1𝑑𝑧4^𝜉𝑡_𝜉^+1/2_𝑡𝑢²_𝑥𝑑𝑧²−12^𝜉𝑡_𝜉^+1/2_𝑡^𝑥_𝜉𝑡𝑢²_𝑥𝑑𝑧²≤1𝑑𝑥4^𝜉𝑡_𝜉^+1/2_𝑡𝑢²_𝑥𝑑𝑧². (98) Doing a similar estimate on [𝜉_𝑡+1/2,𝜉_𝑡+1], we obtain _𝕊𝑢²𝑢²_𝑥1𝑑𝑥≤4_𝕊𝑢²_𝑥𝑑𝑥². (99) In view of (97), we also have ‖𝑢(𝑥,𝑡)‖²_𝐿^∞_(𝕊)≤12_𝕊𝑢²_𝑥𝑑𝑥. (100) Let 𝜑(𝑡)=_𝕊𝑢³_𝑥𝑑𝑥,𝑡≥0, (101) we obtain 𝑑𝜑(𝑡)𝜎𝑑𝑡=−2_𝕊𝑢⁴_𝑥+3()𝑑𝑥3−𝜎2_𝕊𝑢²_𝑥𝑢²3𝑑𝑥+2_𝕊𝑢²_𝑥𝜋𝜌²𝑑𝑥−3_𝕊𝑢²_𝑥𝐺∗3−𝜎2𝑢²+𝜎2𝑢²_𝑥+12𝜋𝜌²𝜎𝑑𝑥≤−2_𝕊𝑢⁴_𝑥1𝑑𝑥−43𝜎()−3()sinh1/23−𝜎2×_𝕊𝑢²_𝑥𝑑𝑥²+32_𝕊𝑢²_𝑥𝜋𝜌²𝑑𝑥, (102) where we use the fact 1/2sinh(1/2)≤𝐺(𝑥)≤cosh(1/2)/2sinh(1/2), then _𝕊𝑢²_𝑥𝐺∗𝑢²_𝑥𝑑𝑥≤𝐺∗𝑢²_{𝑥𝐿∞(𝕊)}≤12sinh(1/2)_𝕊𝑢²_𝑥𝑑𝑥². (103) For 𝜎∈(3sinh(1/2)/(2+sinh(1/2)),3), we have 3𝜎()−3()sinh1/23−𝜎2≥0. (104) If there is a constant 𝐶₃ such that the initial energy 𝐸₁(0)>𝐶₃, there exists some 𝛿>0 such that 32𝑀^∗2𝐸₁(0)≤𝛿4_𝕊𝑢²_𝑥𝑑𝑥²,3𝜎()−3()sinh1/23−𝜎2−𝛿>0. (105) On the other hand, we have ∫_𝕊𝑢²_𝑥𝑑𝑥≥(2/3)‖𝑢‖²_𝐻1(𝕊). Therefore, the previous inequality yields 𝑑𝜑(𝑡)𝜎𝑑𝑡≤−2_𝕊𝑢⁴_𝑥1𝑑𝑥−93𝜎()−3()sinh1/23−𝜎2×‖𝑢‖−𝛿_4𝐻1(𝕊)𝜎≤−2𝜑^4/3(𝑡)−193𝜎()−3()sinh1/23−𝜎2×‖𝑢‖−𝛿_4𝐻1(𝕊). (106) The remaining part is very close to Theorem 12, so we omit it. So the solution must blow up in finite time.
For 𝜎∈(0,3sinh(1/2)/(2+sinh(1/2))], then 𝑑𝜑(𝑡)𝜎𝑑𝑡≤−2𝐸₁(0)𝜑²(𝑡)+3(3−𝜎)8−3𝜎()𝐸4sinh1/2²¹(0)+32𝑀^∗2𝐸₁(0), (107) where 3(3−𝜎)/8−3𝜎/4sinh(1/2)≥0. Note that if the initial quantity satisfies 𝜑(0)<−3(3−𝜎)−34𝜎()𝐸2sinh1/2³¹(0)+3𝜎𝑀^∗2𝐸²₁(0), (108) then condition (2) can conclude that the solution to (7) goes to −∞ in finite time. This completes the proof.

Theorem 14. Suppose that 𝑋₀=(𝑢₀,𝜋(𝜌₀))∈𝐻^𝑠×𝐻^𝑠−1/ℝ,𝑠≥5/2, 𝑋=(𝑢,𝜋(𝜌)) is the solution to system (7) with the initial data 𝑋₀. If there is some point 𝑥₀∈𝕊 such that one of the following conditions is satisfied: (1)𝑢^′₀𝑥₀<−3−𝜎−2𝜎𝐶₀𝜎C₁𝐸₁(0)(2)𝑢,0<𝜎<1,_′0𝑥₀<−3−𝜎𝜎1−𝐶₀𝐶₁𝐸₁(0),1≤𝜎<3, (109) where the constants 𝐶₀ and 𝐶₁ are given in Lemmas 7 and 8, 𝐸₁(0) is the initial value of 𝐸₁, then the solution 𝑋 must blow up in finite time.

Proof. Differentiating the first equation in system (7) with respect to 𝑥 and noticing that 𝜕²_𝑥(𝐺∗𝑓)=𝐺∗𝑓−𝑓, we have 𝑢_𝑥𝑡+𝜎2𝑢²_𝑥+𝜎𝑢𝑢_𝑥𝑥+𝐺∗3−𝜎2𝑢²+𝜎2𝑢²_𝑥+12𝜋𝜌²−3−𝜎2𝑢²−12𝜋𝜌²=0. (110) When 0<𝜎<1, this equation, in combination with (11), yields 𝑑𝑢𝑑𝑡^𝑥𝑞₂()=𝑢𝑥,𝑡,𝑡_𝑥𝑡+𝜎𝑢𝑢_𝑥𝑥𝑞₂=−𝜎,𝑡2𝑢²_𝑥+3−𝜎2𝑢²+12𝜋𝜌²−𝐺∗3−𝜎2𝑢²+𝜎2𝑢²_𝑥+12𝜋𝜌²𝑞₂≤−𝜎,𝑡2𝑢²_𝑥+3−𝜎2𝑢²+12𝜋𝜌²−𝜎𝐺∗𝑢²+𝑢²_𝑥2−3−3𝜎2𝐺∗𝑢²𝑞₂,𝑡. (111) Note that _𝕊𝜋𝜌𝑑𝑥=0. (112) We can deduce that there exists at least one point 𝑥₀ such that 𝜋(𝜌(𝑞₂(𝑥₀,𝑡),𝑡))=0 for 𝑡∈[0,𝑇). Let us consider this problem at (𝑞₂(𝑥₀,𝑡),𝑡). For convenience, we denote 𝑢_𝑥(𝑞₂(𝑥₀,𝑡),𝑡)=𝑚(𝑡). Then we have 𝑑𝑚(𝑡)𝜎𝑑𝑡≤−2𝑚²(𝑡)+3−𝜎2𝑢²−𝜎𝐺∗𝑢²+𝑢²_𝑥2𝜎≤−2𝑚²(𝑡)+3−𝜎−2𝜎𝐶₀2𝐶₁𝐸₁(0). (113) Using the notation 𝜚=(3−𝜎−2𝜎𝐶₀/𝜎)𝐶₁𝐸₁(0), where 3−𝜎−2𝜎𝐶₀>0, we have 𝑑𝑚(𝑡)𝜎𝑑𝑡≤−2𝑚²(𝑡)−𝜚. (114) In view of the initial condition, it is not difficult to obtain 𝑑𝑚(𝑡)≤𝜎()𝑑𝑡𝛿−12𝑚²(𝑡), (115) with 0<𝛿<1 determined by 𝛿𝑚²(0)=𝜚. Then, by using the standard arguments for this type of inequality and our hypothesis, it is easy to conclude that the lifespan of the solution is finite; that is, blow-up phenomenon occurs.
When 1≤𝜎<3, then 𝑑𝑚(𝑡)𝜎𝑑𝑡≤−2𝑢²_𝑥+3−𝜎2𝑢²−3−𝜎2𝐺∗𝑢²+𝑢²_𝑥2−3𝜎−34𝐺∗𝑢²_𝑥𝜎≤−2𝑢²_𝑥+3−𝜎21−𝐶₀𝑢²𝜎≤−2𝑚²(𝑡)+3−𝜎21−𝐶₀𝐶₁𝐸₁(0), (116) where (3−𝜎)(1−𝐶₀)>0. Similar to the arguments about (113), under the condition (2), the corresponding solution 𝑋 blows up in finite time. We complete the proof.

Theorem 15. Suppose that 𝑋₀=(𝑢₀,𝜋(𝜌₀))∈𝐻^𝑠×𝐻^𝑠−1/ℝ,𝑠≥5/2, 𝑋=(𝑢,𝜋(𝜌)) is the solution to system (7) with the initial data 𝑋₀. If there holds one of the following conditions for some point 𝑥₀∈𝕊(1)𝑢^′₀𝑥₀<−3−𝜎−2𝜎𝐶₀𝜎𝐾(0)(2)𝑢,0<𝜎<1,_′0𝑥₀<−3−𝜎𝜎1−𝐶₀𝐾(0),1≤𝜎<3, (117) where the constant 𝐶₀ is the best constant given by Lemma 7 and 𝐾(0)=_𝕊𝑢₀(𝑥)𝑑𝑥²+_𝕊𝑢²₀+𝑢²_0𝑥+𝜋𝜌₀²𝑑𝑥. (118) Let 𝑇 be the maximal existence time of the corresponding solution to (7) with the initial data 𝑋₀. Then 𝑇 is finite.

Proof. We easily know that ∫_𝕊𝑢(𝑥,𝑡)𝑑𝑥 is also an invariant with respect to time. The result follows by using Lemma 9 that ‖𝑢(𝑥)‖²_𝐿^∞_(𝕊)≤_𝕊𝑢(𝑥)𝑑𝑥²+‖𝑢(𝑥)‖²_𝐻1(𝕊)<_𝕊𝑢₀(𝑥)𝑑𝑥²+𝐸₀=𝐾₀, (119) in (113) instead of ‖𝑢(𝑥)‖²_𝐿^∞_(𝕊)≤𝐶₁‖𝑢(𝑥)‖²_𝐻1(𝕊)≤𝐶₁𝐸₁(0).

In the following, as a byproduct, we examine the blow-up rate while the solution blows up in finite time.

Theorem 16. Assume that 𝑋₀=(𝑢₀,𝜋(𝜌₀))∈𝐻^𝑠×𝐻^𝑠−1/ℝ,𝑠≥5/2, 𝑋=(𝑢,𝜋(𝜌)) is the corresponding solution with the initial data 𝑋₀. If there holds the condition of Theorem 14, then we have the following description: lim_𝑡→𝑇()𝑚(𝑡)2𝑇−𝑡=−𝜎, (120) where 𝜎>0 and 𝑚(𝑡) is defined in Theorem 14.

Proof. The conclusion follows from the theory of ordinary differential equations to inequality (113). Indeed, when 0<𝜎<1, by (113) we have 𝑑𝑚(𝑡)𝜎𝑑𝑡≤−2𝑚²(𝑡)+3−𝜎−2𝜎𝐶₀2𝐶₁𝐸₁(0). (121) In view of Lemma 8 and the conservation of 𝐸₁, there holds for all 𝑡∈[0,𝑇) that 𝑑𝑚(𝑡)+𝜎𝑑𝑡2𝑚²(𝑡)≤𝜎𝜚2. (122) It follows that −𝜎𝜚2≤𝑑𝑚(𝑡)+𝜎𝑑𝑡2𝑚²(𝑡)≤𝜎𝜚2,a.e.on(0,𝑇). (123) Since lim_𝑡→𝑇𝑚(𝑡)=−∞ by Theorem 14, it implies that for any 𝜀∈(0,𝜎/2) there exists a 𝑡₀ such that 𝑚²(𝑡)>𝜎𝜚/2𝜀 for all 𝑡∈[𝑡₀,𝑇). Therefore, −𝑚²(𝑡)(𝑡)𝜀≤𝑑𝑚+𝜎𝑑𝑡2𝑚²(𝑡)≤𝑚²(𝑡)𝜀, (124) that is, −𝜎21−𝜀≤𝑚²(𝑡)𝑑𝑚(𝑡)𝜎𝑑𝑡≤−2+𝜀. (125) Direct integration from 𝑡 to 𝑇 gives −𝜎21−𝜀≤()𝑚(𝑡)𝜎𝑇−𝑡≤−2+𝜀, (126) and the arbitrariness of 𝜀 leads to our result.
When 1≤𝜎<3, by (116) we obtain 𝑑𝑚(𝑡)𝜎𝑑𝑡≤−2𝑚²(𝑡)+𝜎𝜔2, (127) where 𝜔=((3−𝜎)/𝜎)(1−𝐶₀)𝐶₁𝐸₁(0). Similar to the arguments for 0<𝜎<1, we can get the same result.

Acknowledgments

This work was partially supported by Natural Science Foundation of China (Grant no. 11226172), Zhejiang Provincial Natural Science Foundation of China (Grant nos. LQ12A01009 and LY12A01014).