Wave-Breaking Criterion for the Generalized Weakly Dissipative Periodic Two-Component Hunter-Saxton System

Abstract

This paper studies the wave-breaking criterion for the generalized weakly dissipative two-component Hunter-Saxton system in the periodic setting. We get local well-posedness for the generalized weakly dissipative two-component Hunter-Saxton system. We study a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles.

1. Introduction

In recent years, the Hunter-Saxton equation [1] 𝑢_𝑡𝑥𝑥+2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥=0 (1) models the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal. In Hunter and Saxton [1], 𝑥 is the space variable in a reference frame moving with the linearized wave velocity, 𝑡 is a slow-time variable, and 𝑢(𝑡,𝑥) is a measure of the average orientation of the medium locally around 𝑥 at time 𝑡. In order to be more precise, the orientation of the molecules is described by the field of unit vectors (cos𝑢(𝑡,𝑥), sin𝑢(𝑡,𝑥)) [2]. The Hunter-Saxton equation also arises in a different physical context as the high-frequency limit [3, 4] of the Camassa-Holm equation for shallow water waves [5, 6] and a reexpression of the geodesic flow on the diffeomorphism group of the circle [7] with a bi-Hamiltonian structure [1, 8] which is completely integrable [4, 9]. Hunter and Saxton [1] explored the initial value problem for the Hunter and Saxton equation on the line (nonperiodic case) and on the unit circle 𝑆=𝑅/𝑍 by using the method of characteristics, while Yin [2] studied it by using the Kato semigroup method. In addition, the two classes of admissible weak solutions, dissipative and conservative solutions, and their stability were investigated in [10–12]. Lenells [13] confirmed that the Hunter-Saxton equation also describes the geodesic flows on the quotient space of the infinite-dimensional group 𝐷^𝑠(𝑆) modulo the subgroup of rotations  Rot(𝑆).

The Camassa-Holm equation admits many integrable multicomponent generalizations. So many authors studied the two-component Camassa-Holm system [14, 15]. Inspired by this, recently, the researchers have made a study of the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting as follows: 𝑢_𝑡𝑥𝑥+2𝜎𝑢_𝑥𝑢_𝑥𝑥+𝜎𝑢𝑢_𝑥𝑥𝑥−𝜌𝜌_𝑥+𝐴𝑢_𝑥𝜌=0,𝑡>0,𝑥∈𝑅,_𝑡+𝜌𝑢_𝑥𝑢()()()(),𝑢()=0,𝑡>0,𝑥∈𝑅,𝑡,𝑥+1=𝑢𝑡,𝑥,𝜌𝑡,𝑥+1=𝜌𝑡,𝑥𝑡≥0,𝑥∈𝑅,0,𝑥=𝑢₀(𝑥),𝜌(0,𝑥)=𝜌₀(𝑥),𝑥∈𝑅. (2)

The authors of [16] have explored the particular choice of the parameter 𝜎=1. The authors of [17] have further studied the wave breaking and global existence for the system for the parameter 𝜎∈R to determine a wave-breaking criterion for strong solutions by using the localization analysis in the transport equation theory.

In general, avoiding energy dissipation mechanisms in a real world is not so easy. Wu and Yin [18, 19] have investigated the blow-up phenomena and the blow-up rate of the strong solutions of the weakly dissipative CH equation and DP equation. Inspired by the results mentioned above, we are going to discuss the initial value problem associated with the generalized weakly dissipative periodic two-component Hunter-Saxton system 𝑢_𝑡𝑥𝑥+2𝜎𝑢_𝑥𝑢_𝑥𝑥+𝜎𝑢𝑢_𝑥𝑥𝑥−𝜌𝜌_𝑥+𝐴𝑢_𝑥+𝜆𝑢−𝑢_𝑥𝑥𝜌=0,𝑡>0,𝑥∈𝑅,_𝑡+𝜌𝑢_𝑥𝑢()()()(),𝑢()=0,𝑡>0,𝑥∈𝑅,𝑡,𝑥+1=𝑢𝑡,𝑥,𝜌𝑡,𝑥+1=𝜌𝑡,𝑥𝑡≥0,𝑥∈𝑅,0,𝑥=𝑢₀(𝑥),𝜌(0,𝑥)=𝜌₀(𝑥),𝑥∈𝑅, (3) where 𝜎∈𝑅 is the new free parameter and  𝐴≥0, 𝜆<0.

Our major results of this paper are Theorems 11 and 12 (wave-breaking criterion). The remainder of the paper is organized as follows. Section 2 establishes the local well-posedness for (3) with the initial data in 𝐻^𝑠×𝐻^𝑠−1, 𝑠≥2. Section 3 deals with the wave breaking of this new system. Theorem 11, using transport equation theory, states a wave-breaking criterion which says that the wave breaking only depends on the slope of 𝑢, not the slope of 𝜌. Theorem 12 improves the blow-up criterion with a more precise condition.

Notation 1. Throughout this paper, 𝑆=𝑅/𝑍 will denote the unit circle. By 𝐻^𝑠, 𝑠≥0, we will represent the Sobolev spaces of equivalence classes of functions defined on the unit circle 𝑆 which have square-integrable distributional derivatives up to order 𝑠. The 𝐻^𝑠-norm will be designated by ‖⋅‖_𝐻^𝑠, and the norm of a vector 𝑣∈𝐻^𝑠×𝐻^𝑠−1 will be written as ‖𝑣‖_𝐻^𝑠_×𝐻^𝑠−1. Also, the Lebesgue spaces of order 𝑝∈[1,∞] will be denoted by 𝐿^𝑝(𝑆), and the norm of their elements will be denoted by ‖𝑓‖_{𝐿𝑝(𝑆)}. Finally, if 𝑝=2, we agree on the convention ‖⋅‖_𝐿²_(𝑆)=‖⋅‖.

2. Preliminaries

In this part, we will establish the local well-posedness for the Cauchy problem of system (3) by using Kato’s theory. To pursue our goal, we give the results we wanted in brief.

We now provide the framework in which we will reformulate (3). To do this, we observe that we can write the first equation of (3) in the following integrated form: 𝑢_𝑡𝑥+𝜎2𝑢²_𝑥+𝜎𝑢𝑢_𝑥𝑥−12𝜌²+𝐴𝑢+𝜆𝜕_𝑥⁻¹𝑢−𝜆𝑢_𝑥=𝑔(𝑡), (4) where 𝜕_𝑥⁻¹𝑓(𝑥)=∫^𝑥₀𝑓(𝑦)𝑑𝑦 and 𝑔(𝑡) is determined by the periodicity of 𝑢 to be 𝑔(𝑡)=−_𝑆𝜎2𝑢²_𝑥+12𝜌²−𝐴𝑢𝑑𝑥. (5)

Integrating both sides of (4) with respect to variable  𝑥, we get 𝑢_𝑡+𝜎𝑢𝑢_𝑥=𝜕_𝑥⁻¹𝜎2𝑢²_𝑥+12𝜌²−𝐴𝑢+𝑔+𝜆𝑢_𝑥−𝜆𝜕_𝑥⁻¹𝑢+ℎ(𝑡), (6) where ℎ(𝑡)∶[0,∞)→𝑅 is an arbitrary continuous function. Therefore, (3) can be written in the “transport” form as follows: 𝑢_𝑡+𝜎𝑢𝑢_𝑥=𝜕_𝑥⁻¹𝜎2𝑢²_𝑥+12𝜌²−𝐴𝑢−𝜆𝜕_𝑥⁻¹𝑢+𝜆𝑢_𝑥(𝑡)𝜌+𝑔+ℎ,𝑡>0,𝑥∈𝑅,_𝑡+𝑢𝜌_𝑥=−𝑢_𝑥𝑢()()()(),𝑢()𝜌,𝑡>0,𝑥∈𝑅,𝑡,𝑥+1=𝑢𝑡,𝑥,𝜌𝑡,𝑥+1=𝜌𝑡,𝑥𝑡≥0,𝑥∈𝑅,0,𝑥=𝑢₀(𝑥),𝜌(0,𝑥)=𝜌₀(𝑥),𝑥∈𝑅, (7) where ℎ(𝑡)∶[0,∞)→𝑅 is an arbitrary continuous function.

Next, we apply Kato’s theory to establish the local well-posedness for the system (3). Consider the abstract quasi-linear evolution equation 𝑑𝑣(𝑣)(𝑣)(0)𝑑𝑡+𝐴𝑣=𝑓,𝑡≥0,𝑣=𝑣₀. (8)

Proposition 1 (see [20]). Given the evolution equation (8), assume that the Kato conditions hold. For a fixed 𝑣₀∈𝑌, there is a maximal 𝑇>0 depending only on ‖𝑣₀‖_𝑌 and a unique solution 𝑣 to the abstract quasi-linear evolution equation (8) such that 𝑣=𝑣⋅,𝑣₀∈𝐶([0,𝑇);𝑌)∩𝐶¹([0,𝑇);𝑋). (9)
Moreover, the map 𝑣₀→𝑣(⋅,𝑣₀) is continuous from 𝑌 to 𝐶([0,𝑇);𝑌)∩𝐶¹([0,𝑇);𝑋). (10) One may follow the similar argument as in [17] to obtain the following local well-posedness for (3).

Theorem 2. Given any 𝑋₀=^𝑢0_𝜌0∈𝐻^𝑠×𝐻^𝑠−1, 𝑠≥2,   there exist a maximal  𝑇=𝑇(𝜎,𝐴;‖𝑋₀‖_{𝐻𝑠×𝐻𝑠−1})>0 and a unique solution 𝑋=^𝑢_𝜌 to (3) such that 𝑋=𝑋⋅,𝑋₀∈𝐶[0,𝑇);𝐻^𝑠(𝑆)×𝐻^s−1(𝑆)∩𝐶¹[0,𝑇);𝐻^𝑠−1(𝑆)×𝐻^𝑠−2(𝑆). (11)
Moreover, the solution depends continuously on the initial data, that is, the mapping 𝑋₀→𝑋(⋅,𝑋₀)∶𝐻^𝑠×𝐻^𝑠−1→𝐶([0,𝑇);𝐻^𝑠(𝑆)×𝐻^𝑠−1(𝑆))∩𝐶¹([0,𝑇);and𝐻^𝑠−1(𝑆)×𝐻^𝑠−2(𝑆))  is continuous, and the maximal existence time 𝑇 can be chosen independently of the Sobolev order 𝑠.
Now, discuss the initial value problem for the Lagrangian flow map as follows: 𝜕𝜑()[),𝜑()𝜕𝑡=𝑢𝑡,𝜑𝑡,𝑥,𝑡∈0,𝑇0,𝑥=𝑥,𝑥∈𝑅, (12) where 𝑢 is the first component of the solution 𝑋 to (3). Using classical results from ordinary differential equations, one can acquire the following result on 𝜑 which is of vital importance in the proof of the blow-up scenarios.

Lemma 3 (see [17]). Let 𝑢∈𝐶([0,𝑇);𝐻^𝑠)∩𝐶¹([0,𝑇);𝐻^𝑠−1), 𝑠≥2. Then, initial value problem (12) admits a unique solution  𝜑∈𝐶¹([0,𝑇)×𝑅,𝑅). Moreover, {𝜑(𝑡,⋅)}_{𝑡∈[0,𝑇)} is increasing diffeomorphism of 𝑅 with 𝜑_𝑥(𝑡,𝑥)=𝑒^{∫𝑇0𝑢𝑥(𝜏,𝜑(𝜏,𝑥))𝑑𝜏}>0,(𝑡,𝑥)∈[0,𝑇)×𝑅. (13)

Remark 4. Since 𝜑(𝑡,⋅)∶𝑅→𝑅 is a diffeomorphism of the linear for every 𝑡∈[0,𝑇), the 𝐿^∞-norm of any function 𝑣(𝑡,⋅)∈𝐿^∞, 𝑡∈[0,𝑇) is preserved under the family of diffeomorphisms 𝜑(𝑡,⋅) with 𝑡∈[0,𝑇), that is, ‖𝑣(𝑡,⋅)‖_𝐿^∞_(𝑆)=𝑣𝑡,𝜑(𝑡,⋅)_{𝐿∞(𝑆)},𝑡∈[0,𝑇). (14)
Similarly, we have inf_𝑥∈𝑆𝑣(𝑡,𝑥)=inf_𝑥∈𝑆𝑣𝑡,𝜑(𝑡,𝑥),𝑡∈[0,𝑇),sup_𝑥∈𝑆𝑣(𝑡,𝑥)=sup_𝑥∈𝑆𝑣𝑡,𝜑(𝑡,𝑥),𝑡∈[0,𝑇). (15)

Lemma 5. Let 𝑋₀=^𝑢0_𝜌0∈𝐻^𝑠×𝐻^𝑠−1, 𝑠≥2, and let 𝑇 be the maximal existence time of the solution 𝑋=^𝑢_𝜌 to (3) with initial data 𝑋₀. Then, for all 𝑡∈[0,𝑇), we have the following results: _𝑆𝜌(𝑡,𝑥)𝑑𝑥=_𝑆𝜌₀(𝑡)𝑑𝑥,_𝑆𝑢²_𝑥(𝑡,𝑥)+𝜌²(𝑡,𝑥)𝑑𝑥≤_𝑆𝑢²_0,𝑥(𝑥)+𝜌²₀(𝑥)𝑑𝑥≜𝐸₀. (16)(17)

Proof. On the one hand, integrating the second equation in (3) by parts and using the periodicity of 𝑢 and 𝜌, we acquire 𝑑𝑑𝑡_𝑆𝜌𝑑𝑥=−_𝑆𝑢𝜌_𝑥𝑑𝑥=0. (18)
On the other hand, multiplying (4) by 𝑢_𝑥 and integrating by parts, considering the periodicity of 𝑢, we obtain 𝑑𝑑𝑡_𝑆𝑢²_𝑥𝑑𝑥=−2_𝑆𝑢𝜌𝜌_𝑥𝑑𝑥+2𝜆_𝑆𝑢²𝑑𝑥+2𝜆_𝑆𝑢²_𝑥𝑑𝑥. (19)
Multiplying the second equation in (3) by 𝜌 and integrating by parts, we have 𝑑𝑑𝑡_𝑆𝜌²𝑑𝑥=2_𝑆𝑢𝜌𝜌_𝑥𝑑𝑥. (20)
Adding the above two equations, we get 𝑑𝑑𝑡_𝑆𝑢²_𝑥+𝜌²𝑑𝑥=2𝜆‖𝑢‖²_𝐻2,𝜆<0. (21)
We acquire _𝑆𝑢²_𝑥(𝑡,𝑥)+𝜌²(𝑡,𝑥)𝑑𝑥≤_𝑆𝑢²_0,𝑥(𝑥)+𝜌²₀(𝑥)𝑑𝑥≜𝐸₀. (22)
This completes the proof of Lemma 5.

Lemma 6. Let 𝑋₀=^𝑢0_𝜌0∈𝐻^𝑠×𝐻^𝑠−1, 𝑠≥2, and let 𝑇 be the maximal existence time of the solution 𝑋=^𝑢_𝜌 to (3) with initial data 𝑋₀. Then, for all 𝑡∈[0,𝑇), we have the following results: _𝑆𝑢²(𝑡,𝑥)𝑑𝑥≤𝑒^𝐶2𝑡_𝑆𝑢²₀(𝑥)𝑑𝑥+1, (23) where 𝐶₁=max(|𝜎|−𝜆,1)𝐸₀+sup_{𝑡∈[0,∞)}|ℎ(𝑡)|>0, 𝐶₂=𝐶₁+4𝐴−2𝜆.

Proof. By computing directly, we have 𝜕_𝑥⁻¹𝜎2𝑢²_𝑥+12𝜌²−𝐴𝑢+𝑔+𝜆𝑢_𝑥−𝜆𝜕_𝑥⁻¹𝑢(𝑡≤+ℎ)_𝑥0𝜎2𝑢²_𝑥+12𝜌²−𝐴𝑢+𝑔+𝜆𝑢_𝑥−𝜆𝜕_𝑥⁻¹𝑢|ℎ(𝑡≤1𝑑𝑥+)|2max(|𝜎|−𝜆,1)𝐸₀+𝑔(𝑡+|ℎ(𝑡+𝐴|)))|(|−𝜆¹₀|𝑢|𝜎|)𝐸𝑑𝑥−𝜆≤max(|−𝜆,1₀+|ℎ(𝑡)|+(2|𝐴|−𝜆)_𝑆|𝑢|𝑑𝑥−𝜆∶=𝐶₁+(2|𝐴|−𝜆)_𝑆|𝑢|𝑑𝑥−𝜆, (24) where 𝐶₁=max(|𝜎|−𝜆,1)𝐸₀+sup_{𝑡∈[0,∞)}|ℎ(𝑡)|>0 and 𝑔(𝑡)=−_𝑆𝜎2𝑢²_𝑥+12𝜌²≤1−𝐴𝑢𝑑𝑥2max(|𝜎|,1)𝐸₀+𝐴_𝑆|𝑢|𝑑𝑥. (25)
Multiplying (6) by 𝑢 and integrating with respect to 𝑥, using the periodicity of 𝑢 and (24), we obtain 12𝑑𝑑𝑡_𝑆𝑢²()=𝑡,𝑥𝑑𝑥_𝑆𝑢𝑢_𝑡𝑑𝑥=−𝜎_𝑆𝑢_𝑥𝑢²+𝑑𝑥_𝑆𝑢𝜕_𝑥⁻¹𝜎2𝑢²_𝑥+12𝜌²−𝐴𝑢+𝜆𝑢_𝑥−𝜆𝜕_𝑥⁻¹𝑢+𝑔+ℎ(𝑡)𝑑𝑥=_𝑆𝑢𝜕_𝑥⁻¹𝜎2𝑢²_𝑥+12𝜌²−𝐴𝑢+𝜆𝑢_𝑥−𝜆𝜕_𝑥⁻¹(𝑡≤𝐶𝑢+𝑔+ℎ)𝑑𝑥₁+(2|𝐴|−𝜆)_𝑆|𝑢|𝑑𝑥−𝜆_𝑆|𝑢|𝑑𝑥≤𝐶_1𝑆|𝑢|𝑑𝑥+(2|𝐴|−𝜆)_𝑆|𝑢|𝑑𝑥²−𝜆_𝑆𝑢²≤𝐶𝑑𝑥₁2+(2|𝐴|−2𝜆)_𝑆𝑢²𝐶𝑑𝑥+¹−𝜆2=𝐶₂2_𝑆𝑢²𝐶𝑑𝑥+¹−𝜆2, (26) where 𝐶₂=𝐶₁+4𝐴−2𝜆; note that 𝐶₂>𝐶₁.
By Gronwall's inequality, we get _𝑆𝑢²(𝑡,𝑥)𝑑𝑥≤𝑒^𝐶2𝑡_𝑆𝑢²₀(𝑥)𝐶𝑑𝑥+¹−𝜆𝐶₂−𝐶₁−𝜆𝐶₂≤𝑒^𝐶2𝑡_𝑆𝑢²₀(𝑥)𝑑𝑥+1. (27)
This completes the proof of Lemma 6.

Lemma 7. Assume that 𝑢₀∈𝐻^𝑠(𝑆), 𝑠≥2, 𝑢₀≠0, and that the corresponding solution 𝑢(𝑡,𝑥) of (3) has a zero point for any time 𝑡≥0. Then, for all 𝑡∈[0,𝑇) we have _𝑆𝑢²(𝑡,𝑥)𝑑𝑥≤_𝑆𝑢²_𝑥(𝑡,𝑥)𝑑𝑥≤𝐸₀. (28)

Proof. By assumption, there is 𝑥₀∈[0,1] such that 𝑢(𝑡,𝑥₀)=0 for each 𝑡∈[0,𝑇).
Then, for 𝑥∈𝑆, by holder equality, we have 𝑢²(𝑡,𝑥)=^𝑥_𝑥0𝑢_𝑥𝑑𝑥²≤𝑥−𝑥₀^𝑥_𝑥0𝑢²_𝑥𝑥𝑑𝑥,𝑥∈₀,𝑥₀+12. (29)
This implies sup_𝑥∈𝑆𝑢²(𝑡,𝑥)≤(1/2)∫_𝑆𝑢²_𝑥𝑑𝑥_𝑆𝑢²(𝑡,𝑥)𝑑𝑥≤sup_𝑥∈𝑆𝑢²()≤1𝑡,𝑥2_𝑆𝑢²_𝑥≤𝑑𝑥_𝑆𝑢²_𝑥+𝜌²𝑑𝑥≤𝐸₀. (30)

3. Wave-Breaking Criteria

In this section, by using transport equation theory, we obtain the wave-breaking criteria for solutions to (3). We first recall the following propositions.

Proposition 8 (1D Moser-type estimates). The following estimates hold:(a)For 𝑠≥0, 𝑓𝑔_{𝐻𝑠(𝑅)}≤𝐶𝑓_{𝐿∞(𝑅)}𝑔_{𝐻𝑠(𝑅)}+𝑓_{𝐻𝑠(𝑅)}𝑔_{𝐿∞(𝑅)}. (31)(b)For 𝑠>0, 𝑓𝜕_𝑥𝑔_{𝐻𝑠(𝑅)}≤𝐶𝑓_{𝐿∞(𝑅)}𝜕_𝑥𝑔_{𝐻𝑠(𝑅)}+𝑓_{𝐻𝑠+1(𝑅)}𝑔_{𝐿∞(𝑅)}. (32)(c)For 𝑠₁≤1/2, 𝑠₂>1/2,  𝑠₁+𝑠₂>0, f𝑔_{𝐻𝑠1(𝑅)}≤𝐶𝑓_{𝐻𝑠1(𝑅)}𝑔_{𝐻𝑠2(𝑅)}, (33) where 𝐶^′𝑠 are constants that are independent of 𝑓 and 𝑔.

Proposition 9 (see [21]). Suppose that 𝑠>−𝑑/2. Let 𝑣 be a vector field such that ∇𝑣 belongs to 𝐿¹([0,𝑇];𝐻^𝑠−1) if 𝑠>1+𝑑/2 or to 𝐿¹([0,𝑇];𝐻^𝑑/2∩𝐿^∞), otherwise. Suppose also that 𝑓₀∈𝐻^𝑠, 𝐹∈𝐿¹([0,𝑇];𝐻^𝑠) and that 𝑓∈𝐿^∞([0,𝑇];𝐻^𝑠)∩𝐶([0,𝑇];𝑆^′)  solves the 𝑑-dimensional linear transport equations (𝑇𝜕)^𝑡𝑓𝑓+𝑣⋅∇𝑓=𝐹,_𝑡=0=𝑓₀. (34)
Then 𝑓∈𝐶([0,𝑇];𝐻^𝑠). More precisely, there exists a constant 𝐶 depending only on 𝑠, 𝑝, and 𝑑 such that the following statements hold:(1)If 𝑠≠1+𝑑/2, 𝑓_𝐻𝑠≤𝑓_0𝐻𝑠+^𝑡₀‖𝐹(𝜏)‖_𝐻^𝑠𝑑𝜏+𝐶^𝑡₀𝑉^′(𝜏)𝑓(𝜏)_𝐻𝑠𝑑𝜏, (35)or 𝑓_𝐻𝑠≤𝑒^{𝐶𝑉(𝑡)}𝑓_0𝐻𝑠+^𝑡₀𝑒^{−𝐶𝑉(𝜏)}‖𝐹(𝜏)‖_𝐻^𝑠𝑑𝜏, (36) with 𝑉(𝑡)=∫^𝑡₀‖∇𝑣(𝜏)‖_𝐻^𝑑/2_∩𝐿^∞𝑑𝜏 if 𝑠<1+𝑑/2 and  𝑉(𝑡)=∫^𝑡₀‖∇𝑣(𝜏)‖_𝐻^𝑠−1𝑑𝜏 else.(2)   If 𝑓=𝑣, then for all 𝑠>0, estimates (35) and (36) hold with𝑉(𝑡)=^𝑡₀𝜕_𝑥𝑢(𝜏)_𝐿∞𝑑𝜏. (37)

Proposition 10 (see [21]). Let 0<𝑠<1. Suppose that 𝑓₀∈𝐻^𝑠, 𝑔∈𝐿¹([0,𝑇];𝐻^𝑠)𝑣, 𝜕_𝑥𝑣∈𝐿¹([0,𝑇];𝐿^∞) and that 𝑓∈𝐿^∞([0,𝑇];𝐻^𝑠)∩𝐶([0,𝑇];𝑆^′) solves the 1-dimensional linear transport equation (𝑇𝜕)^𝑡𝑓𝑓+𝑣⋅∇𝑓=𝑔,_𝑡=0=𝑓₀. (38)
Then 𝑓∈𝐶([0,𝑇];𝐻^𝑠). More precisely, there exists a constant 𝐶 depending only on 𝑠 such that the following statements hold: 𝑓_𝐻𝑠≤𝑓_0𝐻𝑠+𝐶^𝑡₀𝑔(𝜏)_𝐻𝑠𝑑𝜏+𝐶^𝑡₀𝑉^′(𝜏)𝑓(𝜏)_𝐻𝑠𝑑𝜏 (39) or 𝑓_𝐻𝑠≤𝑒^{𝐶𝑉(𝑡)}𝑓_0𝐻𝑠+𝐶^𝑡₀𝑔(𝜏)_𝐻𝑠𝑑𝜏, (40) with 𝑉(𝑡)=∫^𝑡₀(‖𝑣(𝜏)‖_𝐿^∞+‖𝜕_𝑥𝑣(𝜏)‖_𝐿∞)𝑑𝜏.

The above proposition was proved in [8] using Littlewood-Paley analysis for the transport equation and Moser-type estimates. Using this result and performing the same argument, as in [17], we can obtain the following blow-up criterion.

Theorem 11. Let 𝑋₀=^𝑢0_𝜌0∈𝐻^𝑠×𝐻^𝑠−1 with  𝑠≥2, and 𝑋=^𝑢_𝜌 be the corresponding solution to (3). Assume that 𝑇>0 is the maximal time of existence. Then 𝑇<∞⟹^𝑇₀𝜕_𝑥𝑢(𝜏)_𝐿∞𝑑𝜏=∞. (41)

Our next result describes the necessary and sufficient condition for the blow-up of solutions to (3).

Theorem 12. Suppose that 𝜎∈𝑅⧵{0}. Let 𝑋₀=^𝑢0_𝜌0∈𝐻^𝑠×𝐻^𝑠−1, with 𝑠≥2, and let 𝑇 be the maximal existence time of the solution 𝑋=^𝑢_𝜌 to (3) with initial data 𝑋₀. Then, the solution blows up in finite time if and only if liminf_{𝑡→𝑇−}inf_𝑥∈𝑆𝜎𝑢_𝑥(𝑡,𝑥)=−∞. (42)

The approach one takes here is the method of characteristics. Applying the following lemma, we may carry out the estimates along the characteristics 𝜑(𝑡,𝑥) which captures sup_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥) and inf_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥).

Lemma 13 (see [22]). Let 𝑇>0 and let 𝑣∈𝐶¹([0,𝑇];𝐻²(𝑅)). Then, for every 𝑡∈[0,𝑇), there exists at least one point 𝜉(𝑡)∈𝑅 with 𝑚(𝑡)∶=inf_𝑥∈𝑆𝑣_𝑥(𝑡,𝑥)=𝑣_𝑥(𝑡,𝜉(𝑡)), (43) and the function 𝑚(𝑡) is almost everywhere differentiable on (0,𝑇) with 𝑑𝑚(𝑡)𝑑𝑡=𝑣_𝑡𝑥(𝑡,𝜉(𝑡)),𝑎.𝑒.𝑜𝑛(0,𝑇). (44)

Lemma 14. Let 𝑋₀=^𝑢0_𝜌0∈𝐻^𝑠×𝐻^𝑠−1 with 𝑠≥2, and let 𝑇 be the maximal existence time of the solution 𝑋=^𝑢_𝜌 to (3) with initial data 𝑋₀. Then one has the following:(1)𝜎≠0sup_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥)≤𝑢_{0,𝑥𝐿∞(𝑆)}+𝜌₀²_{𝐿∞(𝑆)}+𝐾²₁(𝑇)𝜎+𝜆²𝜎²+𝜆𝜎(𝜎>0)inf_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥)≥−𝑢_{0,𝑥𝐿∞(𝑆)}−𝜆²𝜎²−𝐾²₂(𝑇)𝜎+𝜆𝜎(𝜎<0). (45)(46)(2)𝜎=0sup_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥)≤sup_𝑥∈𝑆𝑢_0,𝑥(𝑥)+12sup_𝑥∈𝑆𝜌²₀(𝑥)+𝐾²₁(𝑇)𝑒^𝜆𝑡−1𝜆inf_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥)≥inf_𝑥∈𝑆𝑢_0,𝑥(𝑥)+12inf_𝑥∈𝑆𝜌²₀(𝑥)−𝐾²₂(𝑇)𝑒^𝜆𝑡−1𝜆. (47)(48)
The constants above are defined as follows: 𝐾₁(𝑇)=𝐴2𝐴−𝜆+2𝐸₀+3𝐴−2𝜆2𝑒^𝐶2𝑇𝑢₀²_𝐿2(𝑆),𝐾+1₂(𝑇)=2𝐴−𝜆+𝐴+22𝐸₀+3𝐴−2𝜆2𝑒^𝐶2𝑇𝑢₀²_𝐿2(𝑆)+1. (49)(50)

Proof of Lemma 14. By Theorem 2 and a simple density argument, we show that the desired results are valid when  𝑠≥3, so we take 𝑠=3 in the proof.
Let 𝜎>0. Using Lemma 13 and the fact that sup_𝑥∈𝑆𝑣_𝑥(𝑡,𝑥)=−inf_𝑥∈𝑆−𝑣_𝑥(𝑡,𝑥). (51)
We can consider 𝑀(𝑡) and 𝛾(𝑡) as follows: 𝑀(𝑡)∶=𝑢_𝑥(𝑡,𝜉(𝑡))=sup_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥),𝑡∈[0,𝑇). (52)
Hence, 𝑢_𝑥𝑥(𝑡,𝜉(𝑡))=0,a.e.on𝑡∈[0,𝑇). (53)
Take the trajectory 𝜑(𝑡,𝑥) defined in (12). Then we know that 𝜑(𝑡,⋅)∶𝑅→𝑅 is a diffeomorphism for every 𝑡∈[0,𝑇). Therefore, there exists 𝑥₀(𝑡)∈𝑅 such that 𝜑𝑡,𝑥₀(𝑡)=𝜉(𝑡),𝑡∈[0,𝑇). (54)
Now, let 𝛾(𝑡)=𝜌𝑡,𝜑𝑡,𝑥₀,𝑡∈[0,𝑇). (55)
Therefore, along the trajectory 𝜑(𝑡,𝑥₀), (4) and the second equation of (3) become 𝑀^′(𝑡)𝜎=−2𝑀²(𝑡)(𝑡)+1+𝜆𝑀2𝛾²(𝑡)+𝑓𝑡,𝜑𝑡,𝑥₀𝛾^′(𝑡)=−𝛾𝑀,a.e.𝑡∈[0,𝑇), (56) where the notation denotes the derivative with respect to 𝑡 and 𝑓 represents the function 𝑓=−𝐴𝑢−𝜆𝜕_𝑥⁻¹𝑢+𝑔(𝑡)=−𝐴𝑢−𝜆𝜕_𝑥⁻¹𝑢−_𝑆𝜎2𝑢²_𝑥+12𝜌²−𝐴𝑢𝑑𝑥. (57)
We first compute the upper and lower bounds for 𝑓 for later use in getting the blow-up result as follows: 𝑓=−𝐴𝑢−𝜆^𝑥₀𝜎𝑢𝑑𝑥−2_𝑆𝑢²_𝑥1𝑑𝑥−2_𝑆𝜌²𝑑𝑥+𝐴_𝑆𝑢𝑑𝑥≤−𝐴𝑢−𝜆^𝑥₀𝑢𝑑𝑥+𝐴_𝑆𝐴𝑢𝑑𝑥≤21+𝑢²+𝐴−𝜆21+_𝑆𝑢²𝑑𝑥. (58)
Since 𝑢²≤(1/2)∫_𝑆(𝑢²+𝑢²_𝑥)𝑑𝑥, (17), we obtain the upper bound for 𝑓𝐴𝑓≤211+2_𝑆𝑢²+𝑢²_𝑥𝑑𝑥+𝐴−𝜆21+_𝑆𝑢²𝜆𝑑𝑥≤𝐴−2+𝐴4_𝑆𝜌²+𝑢²_𝑥𝑑𝑥+3𝐴−2𝜆4_𝑆𝑢²𝜆𝑑𝑥≤𝐴−2+𝐴4𝐸₀+3𝐴−2𝜆4𝑒^𝐶2𝑡_𝑆𝑢²₀(𝑥)𝜆𝑑𝑥+1≤𝐴−2+𝐴4𝐸₀+3𝐴−2𝜆4×𝑒^𝐶2𝑇𝑢₀²_𝐿2(𝑆)1+1∶=2𝐾²₁(𝑇). (59) Now we turn to the lower bound of 𝑓. Using previous arguments, we get −𝑓=𝐴𝑢+𝜆^𝑥₀𝜎𝑢𝑑𝑥+2_𝑆𝑢²_𝑥1𝑑𝑥+2_𝑆𝜌²𝑑𝑥−𝐴_𝑆𝑢𝑑𝑥≤𝐴1+𝑢²2+max(|𝜎|,1)2_𝑆𝜌²+𝑢²_𝑥+𝑑𝑥𝐴−𝜆21+_𝑆𝑢²𝜆𝑑𝑥≤𝐴−2+𝐴+2max(|𝜎|,1)4_𝑆𝜌²+𝑢²_𝑥+𝑑𝑥3𝐴−2𝜆4_𝑆𝑢²𝜆𝑑𝑥≤𝐴−2+𝐴+2max(|𝜎|,1)4𝐸₀+3𝐴−2𝜆4𝑒^𝐶2𝑇𝑢₀²_𝐿2(𝑆)+1. (60)
When 𝜎<0, we have a finer estimate 𝜆−𝑓≤𝐴−2+𝐴+24_𝑆𝜌²+𝑢²_𝑥𝑑𝑥+3𝐴−2𝜆4_𝑆𝑢²𝜆𝑑𝑥≤𝐴−2+𝐴+24𝐸₀+3𝐴−2𝜆4×𝑒^𝐶2𝑇𝑢₀²_𝐿2(𝑆)1+1∶=2𝐾²₂(𝑇). (61)
Combining (59) and (60), we obtain 𝑓𝜆≤𝐴−2+𝐴+2max(|𝜎|,1)4𝐸₀+3𝐴−2𝜆4𝑒^𝐶2𝑇𝑢₀²_𝐿2(𝑆)1+1∶=2𝐾²₃(𝑇). (62)
Since 𝑠≥3, we have 𝑢∈𝐶¹₀(𝑆). Therefore, sup_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥)≥0,inf_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥)≤0,𝑡∈[0,𝑇). (63)
Hence, 𝑀(𝑡)>0 for 𝑡∈[0,𝑇). From the second equation of (55), we obtain 𝛾(𝑡)=𝛾(0)𝑒^{−∫𝑡0𝑀(𝜏)𝑑𝜏}. (64)
Hence, 𝜌𝑡,𝜑𝑡,𝑥₀=𝛾(𝑡)≤𝛾(0)≤𝜌_{0𝐿∞(𝑆)}. (65)
For any given 𝑥∈𝑆, define 𝑝₁(𝑡)=𝑀(𝑡)−𝑢_{0,𝑥𝐿∞(𝑆)}−𝜌₀²_{𝐿∞(𝑆)}+𝐾²₁(𝑇)𝜎+𝜆²𝜎²−𝜆𝜎(𝜎>0). (66)
Observing that 𝑝₁(𝑡) is a 𝐶¹-differentiable function on [0,𝑇) and satisfies 𝑝₁(0)=𝑀(0)−𝑢_{0,𝑥𝐿∞(𝑆)}−𝜌₀²_{𝐿∞(𝑆)}+𝐾²₁(𝑇)𝜎+𝜆²𝜎²−𝜆𝜎≤𝑀(0)−𝑢_{0,𝑥𝐿∞(𝑆)}. (67)
We now claim that 𝑝₁(𝑡)≤0𝑡∈[0,𝑇).
Assume the contrary that there is 𝑡₀∈[0,𝑇) such that 𝑝₁(𝑡₀)>0.
Let 𝑡₁=max{𝑡<𝑡₀∶𝑝₁(𝑡)=0}. Then 𝑝₁(𝑡₁)=0 and 𝑝^′₁(𝑡₁)≥0, or equivalently, 𝑀𝑡₁=𝑢_{0,𝑥𝐿∞(𝑆)}+𝜌₀²_{𝐿∞(𝑆)}+𝐾²₁(𝑇)𝜎+𝜆²𝜎²+𝜆𝜎 (68) and 𝑀^′(𝑡₁)≥0 a.e. 𝑡∈[0,𝑇). On the other hand, we have 𝑀^′𝑡₁𝜎=−2𝑀²𝑡₁+𝜆𝑀𝑡₁+12𝛾²𝑡₁+𝑓𝑡₁,𝜑𝑡₁,𝑥₀[)𝜎a.e.𝑡∈0,𝑇≤−2𝑀𝑡₁−𝜆𝜎²+12𝜌₀²_{𝐿∞(𝑆)}+𝜆²+𝐾2𝜎₂₁(𝑇)2<0, (69) which is a contradiction. Therefore, 𝑝₁(𝑡)≤0 for all 𝑡∈[0,𝑇). Since 𝑥 is arbitrarily chosen, we obtain (45).
To derive (46) in the case of 𝜎<0, we consider 𝑀(𝑡) and 𝛾(𝑡) as in Lemma 13: 𝑀(𝑡)∶=𝑢_𝑥(𝑡,𝜁(𝑡))=inf_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥)𝑡∈[0,𝑇). (70)
Hence, 𝑢_𝑥𝑥(𝑡,𝜁(𝑡))=0a.e.𝑡∈[0,𝑇). (71)
Using previous arguments, we take the characteristic 𝜑(𝑡,𝑥) defined in (13) and choose 𝑥₁(𝑡)∈𝑅 such that 𝜑𝑡,𝑥₁(𝑡)=𝜁(𝑡). (72)
Let 𝛾(𝑡)=𝜌𝑡,𝜑(𝑡,𝑥),𝑡∈[0,𝑇). (73)
Hence, along the trajectory 𝑀^′(𝑡)=𝜆𝑀(𝑡) +(1/2)𝛾²(𝑡) +𝑓(𝑡,𝜑(𝑡,𝑥₀)) ≥𝜆𝑀(0)+(1/2)𝛾²(0) +(1/2)𝐾²₂(𝑇), (4) and the second equation of (3) become 𝑀^′(𝑡)𝜎=−2𝑀²(𝑡)𝑀(𝑡)+1+𝜆2𝛾²(𝑡)+𝑓𝑡,𝜑𝑡,𝑥₀𝛾^′(𝑡)=−𝛾𝑀,a.e.𝑡∈[0,𝑇). (74)
Define 𝑝₂(𝑡)=𝑀(𝑡)+𝑢_{0,𝑥𝐿∞(𝑆)}+𝜆²𝜎²−𝐾²₂(𝑇)𝜎−𝜆𝜎(𝜎<0). (75)
For any given 𝑥∈𝑆, Note that 𝑝₂(𝑡) is also 𝐶¹-differentiable function on [0,𝑇) and satisfies 𝑝₂(0)=𝑀(0)+𝑢_{0,𝑥𝐿∞(𝑆)}+𝜆²𝜎²−𝐾²₂(𝑇)𝜎−𝜆𝜎≥𝑀(0)+𝑢_{0,𝑥𝐿∞(𝑆)}≥0. (76)
We now claim that 𝑝₂(𝑡)≥0, for any 𝑡∈[0,𝑇).
Suppose not, then there is 𝑡̃∈[0,𝑇) such that 𝑝₂(𝑡₀)<0. Define 𝑡₂=max𝑡<𝑡₀∶𝑝₂(𝑡)=0. (77)
Then, 𝑝₂(𝑡₂)=0 and 𝑝^′₂(𝑡₂)<0, or equivalently, 𝑀𝑡₂=−𝑢_{0,𝑥𝐿∞(𝑆)}−𝜆²𝜎²−𝐾²₂(𝑇)𝜎+𝜆𝜎 (78) and 𝑀^′(𝑡₂)≤0 a.e. 𝑡∈[0,𝑇). However, we have 𝑀^′𝑡₂𝜎=−2𝑀²𝑡₂+𝜆𝑀𝑡₂+12𝛾²𝑡₂+𝑓𝑡₂,𝜑𝑡₂,𝑥₀[)𝜎a.e.𝑡∈0,𝑇≥−2−𝑢_{0,𝑥𝐿∞(𝑆)}−𝜆²𝜎²−𝐾²₂(𝑇)𝜎²+𝜆²−12𝜎2𝐾²₂(𝑇)>0. (79)
Therefore, 𝑝₂(𝑡)≥0 for any 𝑡∈[0,𝑇). Since 𝑥 is chosen arbitrarily, we obtain (46).
Let 𝜎=0. Using previous arguments, (56) becomes 𝑀^′(𝑡)𝜎=−2𝑀²(𝑡)(𝑡)+1+𝜆𝑀2𝛾²(𝑡)+𝑓𝑡,𝜑𝑡,𝑥₀𝛾^′(𝑡)=−𝛾𝑀,a.e.𝑡∈[0,T), (80) where the notation denotes the derivative with respect to 𝑡 and 𝑓 represents the function 𝑓=−𝐴𝑢−𝜆𝜕_𝑥⁻¹𝑢+𝑔(𝑡)=−𝐴𝑢−𝜆𝜕_𝑥⁻¹𝑢−_𝑆12𝜌²−𝐴𝑢𝑑𝑥. (81)
We first compute the upper and lower bounds for 𝑓 for later use in getting the blow-up result: 𝑓=−𝐴𝑢−𝜆^𝑥₀1𝑢𝑑𝑥−2_𝑆𝜌²𝑑𝑥+𝐴_𝑆𝑢𝑑𝑥≤−𝐴𝑢−𝜆^𝑥₀𝑢𝑑𝑥+𝐴_𝑆𝐴𝑢𝑑𝑥≤21+𝑢²+𝐴−𝜆21+_𝑆𝑢²≤𝐴𝑑𝑥211+2_𝑆𝑢²+𝑢²_𝑥𝑑𝑥+𝐴−𝜆21+_𝑆𝑢²𝜆𝑑𝑥≤𝐴−2+𝐴4_𝑆𝜌²+𝑢²_𝑥𝑑𝑥+3𝐴−2𝜆4_𝑆𝑢²𝜆𝑑𝑥≤𝐴−2+𝐴4𝐸₀+3𝐴−2𝜆4𝑒^𝐶2𝑡_𝑆𝑢²₀(𝑥)𝜆𝑑𝑥+1≤𝐴−2+𝐴4𝐸₀+3𝐴−2𝜆4𝑒^𝐶2𝑇𝑢₀²_𝐿2(𝑆)+1. (82)
Now, we turn to the lower bound of 𝑓: −𝑓=𝐴𝑢+𝜆^𝑥₀1𝑢𝑑𝑥+2_𝑆𝜌²𝑑𝑥−𝐴_𝑆𝑢𝑑𝑥≤𝐴1+𝑢²2+12_𝑆𝜌²𝑑𝑥+𝐴−𝜆21+_𝑆𝑢²𝜆𝑑𝑥≤𝐴−2+𝐴+24_𝑆𝜌²+𝑢²_𝑥𝑑𝑥+3𝐴−2𝜆4_𝑆𝑢²𝜆𝑑𝑥≤𝐴−2+𝐴+24𝐸₀+3𝐴−2𝜆4𝑒^𝐶2𝑇𝑢₀²_𝐿2(𝑆)+1. (83)
Combining (82) and (83), we obtain 𝑓𝜆≤𝐴−2+𝐴+24𝐸₀+3𝐴−2𝜆4𝑒^𝐶2𝑇𝑢₀²_𝐿2(𝑆)+1. (84)
We know 𝑀(𝑡)>0 for 𝑡∈[0,𝑇). From the second equation of (81), we obtain that 𝛾(𝑡)=𝛾(0)𝑒^{−∫𝑡0𝑀(𝜏)𝑑𝜏}. (85)
Hence, 𝜌𝑡,𝜑𝑡,𝑥₀=𝛾(𝑡)≤𝛾(0). (86)
Therefore, we have 𝑀^′(𝑡)(𝑡)+1=𝜆𝑀2𝛾²(𝑡)+𝑓𝑡,𝜑𝑡,𝑥₀(0)+1≤𝜆𝑀2𝛾²(0)+12𝐾²₁(𝑇),𝑀^′(𝑡)(𝑡)≤1−𝜆𝑀2sup_𝑥∈𝑆𝜌²₀(𝑥)+𝐾²₁(𝑇)^a._e.𝑡∈[0,𝑇). (87)(88)
Integrating (88) on [0,𝑡], we prove (47) as follows: 𝑀(𝑡)≤sup_𝑥∈𝑆𝑢_0,𝑥(𝑥)+12sup_𝑥∈𝑆𝜌²₀(𝑥)+𝐾²₁(𝑇𝑒)^𝜆𝑡−1𝜆. (89)
To obtain a lower bound for inf_𝑥∈𝑆𝑢_𝑥(𝑡,𝑥), we use the same argument.
Since  𝜎=0, (80) becomes 𝑀^′(𝑡)𝑀(𝑡)+1=𝜆2𝛾²(𝑡)+𝑓𝑡,𝜑𝑡,𝑥₁,𝛾^′(𝑡)=−𝛾𝑀,a.e.𝑡∈[0,𝑇). (90)
Because of 𝑀(𝑡)<0, we get from the second equation of (90) that 𝛾(𝑡)=𝛾(0)𝑒^{−∫𝑡0𝑀(𝜏)𝑑𝜏}. (91)
This means that 𝜌𝑡,𝜑𝑡,𝑥₁=𝛾(𝑡)≥𝛾(0). (92)
Then, 𝑀^′(𝑡)𝑀(𝑡)+1=𝜆2𝛾²(𝑡)+𝑓𝑡,𝜑𝑡,𝑥₀𝑀(0)+1≥𝜆2𝛾²(0)+12𝐾²₂(𝑇),𝑀^′(𝑡)𝑀(𝑡)≥1−𝜆2inf_𝑥∈𝑆𝜌²₀(𝑥)−𝐾²₂(𝑇)a.e.𝑡∈[0,𝑇). (93)(94)
Integrating (94) on [0,𝑡], we prove (48). This completes the proof of Lemma 14.

Lemma 15. Suppose that 𝜎∈𝑅⧵{0}. Suppose 𝑋₀=^𝑢0_𝜌0∈𝐻^𝑠×𝐻^𝑠−1 with 𝑠≥2, and let 𝑇 be the maximal existence time of the solution 𝑋=^𝑢_𝜌 to (3) with initial data 𝑋₀. Then we have 𝜌𝑡,𝜑(𝑡,𝑥)𝜑_𝑥(𝑡,𝑥)=𝜌₀(𝑥),(𝑡,𝑥)∈[0,𝑇]×𝑆. (95)
Moreover, if there exists 𝑀>0 such that inf_{()∈[]𝑡,𝑥0,𝑇×𝑆}𝜎𝑢_𝑥(𝑡,𝑥)≥−𝑀,(𝑡,𝑥)∈[0,𝑇]×𝑆. (96)
Then 𝜌(𝑡,⋅)_{𝐿∞(𝑆)}=𝜌𝑡,𝜑(𝑡,⋅)_{𝐿∞(𝑆)}≤𝑒^{𝑀𝑇/𝜎}𝜌_{0𝐿∞(𝑆)}()𝜌(𝜎>0𝑡,⋅)_{𝐿∞(𝑆)}=𝜌𝑡,𝜑(𝑡,⋅)_{𝐿∞(𝑆)}≤𝑒^{𝑁𝑇/𝜎}𝜌_{0𝐿∞(𝑆)}(𝜎<0), (97)(98) where 𝑁=‖𝑢_0,𝑥‖_{𝐿∞(𝑆)}+𝜆²/𝜎²−𝐾²₂(𝑇)/𝜎−𝜆/𝜎 and 𝐾₂(𝑇) is given in (50).

Proof. Differentiating the left hand side of (95) with respect to 𝑡, in view of the relations (12) and (3), we obtain 𝑑𝜌()𝜑𝑑𝑡𝑡,𝜑𝑡,𝑥_𝑥(=𝜌𝑡,𝑥)_𝑡𝑡,𝜑+𝜌_𝑥𝑡,𝜑𝜑_𝑡(𝑡,𝑥)×𝜑_𝑥(𝑡,𝑥)+𝜌𝑡,𝜑𝜑_𝑥𝑡()=𝜌𝑡,𝑥_𝑡𝑡,𝜑+𝜌_𝑥𝑡,𝜑𝜑_𝑡(𝑡,𝑥)𝜑_𝑥()𝑢𝑡,𝑥+𝜌𝑡,𝜑_𝑥𝑡,𝜑𝜑_𝑥()=𝜌𝑡,𝑥_𝑡𝑡,𝜑+𝜌_𝑥𝑡,𝜑𝜑_𝑡(𝑡,𝑥)+𝜌𝑡,𝜑𝑢_𝑥𝑡,𝜑×𝜑_𝑥(𝑡,𝑥)=0. (99)
This completes the proof of (95). In view of the assumption (96) and 𝜎>0, we obtain 𝑢(𝑡,𝑥)≥−(𝑀/𝜎)(𝑡,𝑥)∈[0,𝑇]×𝑆.
By Lemma 3 and (95), we have 𝜌(𝑡,⋅)_𝐿∞=𝜌𝑡,𝜑(𝑡,⋅)_𝐿∞=𝑒^{−∫𝑡0𝑢𝑥(𝜏,⋅)𝑑𝜏}𝜌₀(⋅)_𝐿∞≤𝑒^{𝑀𝑇/𝜎}𝜌₀(⋅)_𝐿∞. (100)
To obtain (98), we use a similar argument as before. Using (13) and the lower bound for 𝑢_𝑥(𝑡,𝑥) in (46), it follows that 𝜌(𝑡,⋅)_𝐿∞=𝜌𝑡,𝜑(𝑡,⋅)_𝐿∞=𝑒^{−∫𝑡0𝑢𝑥(𝜏,⋅)𝑑𝜏}𝜌₀(⋅)_𝐿∞≤𝑒^{𝑁𝑇/𝜎}𝜌₀(⋅)_𝐿∞, (101) which proves (98). This completes the proof of Lemma 15.

Proof of Theorem 12. Suppose that 𝑇<∞ and that (42) is not valid. Then, there is some positive number 𝑀>0 such that 𝜎𝑢_𝑥(𝑡,𝑥)≥−𝑀,(𝑡,𝑥)∈[0,𝑇]×𝑆. (102)
It now follows from Lemma 14 that |𝑢_𝑥(𝑡,𝑥)|≤𝐶, where 𝐶=𝐶(𝐴,𝑀,𝜎,𝐸₀,𝜆,‖𝑢₀‖,𝑇). Therefore, Theorem 11 implies that the maximal existence time 𝑇=∞, which contradicts with the assumption that 𝑇<∞.
Conversely, the Sobolev embedding theorem 𝐻^𝑠(𝑆)→𝐿^∞(𝑆) with 𝑠>1/2 implies that if (70) holds, the corresponding solution blows up in finite time, which completes the proof of Theorem 12.

Acknowledgments

The authors would like to thank the referee for comments and suggestions. This work is supported by the National Nature Science Foundation of China (no. 11171135), the Nature Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), and the high-level talented person special subsidizes of Jiangsu University (no. 05JDG047).