Global Existence and Blowup Analysis to Single-Species Bacillus System with Free Boundary

Abstract

This paper is concerned with a reaction-diffusion equation which describes the dynamics of single bacillus population with free boundary. The local existence and uniqueness of the solution are first obtained by using the contraction mapping theorem. Then we exhibit an energy condition, involving the initial data, under which the solution blows up in finite time. Finally we examine the long time behavior of global solutions; the global fast solution and slow solution are given. Our results show that blowup occurs if the death rate is small and the initial value is large enough. If the initial value is small the solution is global and fast, which decays at an exponential rate while there is a global slow solution provided that the death rate is small and the initial value is suitably large.

1. Introduction

As we know, mathematical aspects of biological population have been considered widely. Most of the authors have studied growth and diffusions of biological population in a homogeneous or heterogeneous fixed environment [1, 2], and the nonlinear differential equations are described such as Logistic equation and Fisher equation.

In this paper, we consider the following single bacillus population model: 𝑢_𝑡−𝑑Δ𝑢=𝐾𝑎𝑢²−𝑏𝑢, (1.1) which was first proposed by Verhulst see [3]. Parameters 𝑎,𝑏,𝑑 and 𝐾 are positive constants. Ecologically, 𝑎 represents the net birth rate, 𝑏 is the death rate, 𝑑 denotes the diffusion coefficient, and 𝐾 measures the living resource for bacillus. In [4], Jin et al. considered the model and established a time-dependent dynamic basis to quantitatively clarify the biological wave behavior of the popular growth and propagation.

The present paper aims to investigate the parabolic equation with a moving boundary in one-dimensional space.

As in [5], assumed that the amount of the species flowing across the free boundary is increasing with respect to the moving length, the condition on the interface (free boundary) is 𝑠^′(𝑡)=−𝜇𝑢_𝑥(𝑠(𝑡),𝑡) by using the Taylor expansion. Here 𝜇 is a positive constant and measures the ability of the bacillus disperse in a new area. The free boundary is regarded as the moving front, the detailed biological implication see Section 6 of [6] for the logistic model; the authors also compared their results in biological terms with some documented ecological observations there. In this way, we have the following problem for 𝑢(𝑥,𝑡) and a free boundary 𝑥=𝑠(𝑡) such that𝑢_𝑡−𝑑𝑢_𝑥𝑥=𝐾𝑎𝑢²(𝑡)𝑢(𝑠(𝑡))𝑢−𝑏𝑢,0<𝑥<𝑠,0<𝑡<𝑇,,𝑡=0,0<𝑡<𝑇,_𝑥()𝑠(0)0,𝑡=0,0<𝑡<𝑇,=𝑠₀𝑢()>0,𝑥,0=𝑢₀(𝑥)(0),𝑠⩾0,0⩽𝑥⩽𝑠_′(𝑡)=−𝜇𝑢_𝑥(𝑠(𝑡),𝑡),0<𝑡<𝑇, (1.2) where the condition 𝑢_𝑥(0,𝑡)=0 indicates that the habitat is semiunbounded domain and there is no migration cross the left boundary.

When 𝐾𝑎=𝑏=0, the problem is reduced to one phase Stefan problem, which accounts for phase transitions between solid and fluid states such as the melting of ice in contact with water [7]. Stefan problems have been studied by many authors. For example, the weak solution was considered by Oleĭnik in [8], and the existence of a classical solution was given by Kinderlehrer and Nirenberg in [9]. For the two-phase Stefan problem, the local classical solution was obtained in [10, 11], and the global classical solution was given by Borodin in [12].

The free boundary problems have been investigated in many areas, for example, the decrease of oxygen in a muscle in the vicinity of a clotted bloodvessel [13], the etching problem [14], the combustion process [15], the American option pricing problem [16, 17], chemical vapor deposition in hot wall reactor [18], image processing [19], wound healing [20], tumor growth [21–24] and the dynamics of population [5, 25, 26].

In this paper, we consider the free boundary problem (1.2) and focus on studying the blowup behavior of the solution and asymptotic behavior of the global solutions. We will give sufficient conditions to ensure the existence of fast solution and slow solution. Here if 𝑇=+∞, we say the solution exists globally whereas if the solution ceases to exist for some finite time, that is, 𝑇<+∞ and lim_𝑡→𝑇‖𝑢(𝑥,𝑡)‖_𝐿^∞_{([0,𝑠(𝑡)]×[0,𝑡])}→+∞, we say that the solution blows up. If 𝑇=∞ and lim_𝑡→∞𝑠(𝑡)<∞, the solution is called fast solution since that the solution decays uniformly to 0 at an exponential rate, while if 𝑇=∞ and lim_𝑡→∞𝑠(𝑡)=∞, it is called slow solution, see [27, 28] in detail.

The remainder of this paper is organized as follows. In Section 2, local existence and uniqueness will be given. Section 3 deals with the result of blowup behavior by constructing an energy condition. Section 4 is devoted to long time behaviors of global solutions, including the existence of global fast solution and slow solution.

2. Local Existence and Uniqueness

In this section, we prove the following local existence and uniqueness of the solution to (1.2) by contraction mapping principle.

Theorem 2.1. For any given 𝑢₀ satisfying 𝑢₀∈𝐶^1+𝛼([0,𝑠₀]) with 𝛼∈(0,1), 𝑢^′₀(0)=𝑢₀(𝑠₀)=0 and 𝑢₀>0 in [0,𝑠₀), there is a 𝑇>0 such that problem (1.2) admits a unique solution (𝑢,𝑠)∈𝐶^{1+𝛼,(1+𝛼)/2}([0,𝑠(𝑡)]×[0,𝑇])×𝐶^1+𝛼/2([0,𝑇]). (2.1) Furthermore, ‖𝑢(𝑥,𝑡)‖_𝐶^{1+𝛼,(1+𝛼)/2}_{([0,𝑠(𝑡)]×[0,𝑇])}+‖𝑠(𝑡)‖_𝐶^1+𝛼/2_([0,𝑇])⩽𝐶, (2.2) where 𝐶 and 𝑇 depend only on 𝛼,𝑠₀ and ‖𝑢₀‖_𝐶^1+𝛼_([0,𝑠0]).

Proof. We first make a change of variable to straighten the free boundary. Let 𝑥𝜉=𝑠(𝑡),𝑢(𝑥,𝑡)=𝑣(𝜉,𝑡). (2.3) Then the problem (1.2) is reduced to 𝑣_𝑡−𝑠^′(𝑡)𝑠(𝑡)𝜉𝑣_𝜉−𝑑𝑠²(𝑡)𝑣_𝜉𝜉=𝐾𝑎𝑣²𝑣()𝑣−𝑏𝑣,0<𝜉<1,0<𝑡<𝑇,1,𝑡=0,0<𝑡<𝑇,_𝜉()𝑠(0)0,𝑡=0,0<𝑡<𝑇,=𝑠₀𝑣()>0,𝜉,0=𝑣₀(𝜉)∶=𝑢₀𝑠₀𝜉𝑠⩾0,0⩽𝜉⩽1,_′(𝑡)𝜇=−𝑠(𝑡)𝑣_𝜉(1,𝑡),0<𝑡<𝑇. (2.4) This transformation changes the free boundary 𝑥=𝑠(𝑡) to the fixed line 𝜉=1 at the expense of making the equation more complicated. In the first equation of (2.4), the coefficients contain the unknown 𝑠(𝑡).
Now we denote 𝑠^∗=−𝜇𝑣^′₀(1) and set 𝑆_𝑇=𝑠∈𝐶¹[0,𝑇]∶𝑠(0)=𝑠₀,𝑠^′(0)𝑠₀=𝑠^∗,0⩽𝑠^′(𝑡)𝑠(𝑡)⩽𝑠^∗,𝑈+1_𝑇=𝑣∈𝐶([0,1]×[0,𝑇])∶𝑣(𝜉,0)=𝑣₀(𝜉),𝑣−𝑣_{0𝐶([0,1]×[0,𝑇])}⩽1. (2.5) It is easy to see that Σ_𝑇∶=𝑈_𝑇×𝑆_𝑇 is a complete metric space with the metric 𝒟𝑣₁,𝑠₁,𝑣₂,𝑠₂=𝑣₁−𝑣_{2𝐶([0,1]×[0,𝑇])}+𝑠^′₁𝑠₁−𝑠^′₂𝑠_{2𝐶([0,𝑇])}. (2.6)
Next applying standard 𝐿^𝑝 theory and the Sobolev imbedding theorem (see [29]), we then find that for any (𝑣,𝑠)∈Σ_𝑇, the following initial boundary value problem: 𝑣_𝑡−𝑠^′(𝑡)𝑠(𝑡)𝜉𝑣_𝜉−𝑑𝑠²(𝑡)𝑣_𝜉𝜉=𝐾𝑎𝑣²𝑣()𝑣−𝑏𝑣,0<𝜉<1,0<𝑡<𝑇,1,𝑡=0,0<𝑡<𝑇,_𝜉()𝑣()0,𝑡=0,0<𝑡<𝑇,𝜉,0=𝑣₀(𝜉)⩾0,0⩽𝜉⩽1 (2.7) admits a unique solution 𝑣∈𝐶^{1+𝛼,(1+𝛼)/2}([0,1]×[0,𝑇]) and ‖𝑣‖_𝐶^{1+𝛼,(1+𝛼)/2}_{([0,1]×[0,𝑇])}⩽𝐶‖𝑣‖_𝑊^2,1,𝑝_{([0,1]×[0,𝑇])}⩽𝐶₁, (2.8) where 𝑝=3/(1−𝛼), 𝐶₁ is a constant dependent on 𝛼,𝑠₀ and ‖𝑢₀‖_𝐶^1+𝛼_[0,𝑠0].
Defining 𝑠̃ by using the last equation of (2.4) 𝑠²̃(𝑡)=𝑠²₀−2^𝑡₀𝜇𝑣_𝜉(1,𝜏)d𝜏, (2.9) we have 𝑠^′̃(𝑡)𝑠̃(𝑡)=−𝜇𝑣_𝜉(1,𝑡),𝑠̃(0)=𝑠₀,𝑠^′̃(0)𝑠₀=−𝜇𝑣_𝜉(1,0)=𝑠^∗, (2.10) and hence 𝑠′̃∈𝐶^𝛼/2([0,𝑇]) with 𝑠^′̃(𝑡)𝑠̃(𝑡)_{𝐶𝛼/2([0,𝑇])}⩽𝐶₂∶=𝜇𝐶₁. (2.11)
Define map ℱ: Σ_𝑇→𝐶([0,1]×[0,𝑇])×𝐶¹[0,𝑇] by ℱ((𝑣(𝜉,𝑡),𝑠(𝑡)))=(𝑣(𝜉,𝑡),𝑠̃(𝑡)). (2.12) It is clear that (𝑣,𝑠)∈Σ_𝑇 is a fixed point of ℱ if and only if it solves (2.4).
By (2.10) and (2.11), we have 𝑠^′̃(𝑡)𝑠(𝑡)−𝑠^′̃(0)𝑠(0)_𝐶[0,𝑇]⩽𝑠^′̃(𝑡)𝑠(𝑡)_{𝐶𝛼/2[0,𝑇]}𝑇^𝛼/2⩽𝐶₂𝑇^𝛼/2,𝑣(𝜉,𝑡)−𝑣₀(𝜉)_{𝐶([0,1]×[0,𝑇])}⩽𝑣−𝑣_{0𝐶0,(1+𝛼)/2([0,1]×[0,𝑇])}𝑇^(1+𝛼)/2⩽𝐶₁𝑇^(1+𝛼)/2. (2.13) Therefore if we take 𝑇⩽min{(𝐶₂)^−2/𝛼,𝐶₁^{−2/(1+𝛼)}}, then ℱ maps Σ_𝑇 into itself.
To prove that ℱ is a contraction mapping on Σ_𝑇 for 𝑇>0 sufficiently small, we take (𝑣_𝑖,𝑠_𝑖)∈Σ_𝑇(𝑖=1,2) and denote (𝑣_𝑖,𝑠_𝑖)=ℱ(𝑣_𝑖,𝑠_𝑖). Then it follows form (2.8) and (2.11) that 𝑣_{𝑖𝐶1+𝛼,(1+𝛼)/2([0,1]×[0,𝑇])}⩽𝐶𝑣_{𝑖𝑊2,1,𝑝([0,1]×[0,𝑇])}⩽𝐶₁,𝑠^′_𝑖(𝑡)𝑠_𝑖(𝑡)_{𝐶𝛼/2[0,𝑇]}⩽𝐶₂. (2.14)
By setting 𝑉=𝑣₁−𝑣₂ it follows that 𝑉(𝜉,𝑡) satisfies 𝑉_𝑡−𝑠^′₁𝑠₁𝜉𝑉_𝜉−𝑑𝑠²₁𝑉_𝜉𝜉−𝐾𝑎𝑣₁+𝑣₂=𝑠𝑉+𝑏𝑉_′1𝑠₁−𝑠^′₂𝑠₂𝜉𝑣_2,𝜉+1𝑠²₁−1𝑠²₂𝑑𝑣_2,𝜉𝜉𝑉(),0<𝜉<1,0<𝑡<𝑇,1,𝑡=0,𝑉_𝜉()𝑉()0,𝑡=0,0<𝑡<𝑇,𝜉,0=0,0⩽𝜉⩽1. (2.15)
Using the 𝑊^2,1,𝑝 estimates for parabolic equations and Sobolev’s imbedding yields 𝑣₁−𝑣_{2𝐶1+𝛼,(1+𝛼)/2([0,1]×[0,𝑇])}⩽𝐶₃𝑣₁−𝑣_{2𝐶([0,1]×[0,𝑇])}+𝑠₁−𝑠_{2𝐶1[0,𝑇]}, (2.16) where 𝐶₃ is independent of 𝑇. Taking the difference of the equations for 𝑠^′₁𝑠₁,𝑠^′₂𝑠₂ results in 𝑠^′₁(𝑡)𝑠₁(𝑡)−𝑠^′₂(𝑡)𝑠₂(𝑡)_{𝐶𝛼/2[0,𝑇]}⩽𝜇𝑣_1,𝜉(𝜉,𝑡)−𝑣_2,𝜉(𝜉,𝑡)_{𝐶0,𝛼/2([0,1]×[0,𝑇])}. (2.17) Combining inequalities (2.16) and (2.17), we obtain 𝑣₁(𝜉,𝑡)−𝑣₂(𝜉,𝑡)_{𝐶1+𝛼,(1+𝛼)/2([0,1]×[0,𝑇])}+𝑠^′₁(𝑡)𝑠₁(𝑡)−𝑠^′₂(𝑡)𝑠₂(𝑡)_{𝐶𝛼/2[0,𝑇]}⩽𝐶₄𝑣₁(𝜉,𝑡)−𝑣₂(𝜉,𝑡)_{𝐶([0,1]×[0,𝑇])}+𝑠₁(𝑡)−𝑠₂(𝑡)_{𝐶1[0,𝑇]}, (2.18) where 𝐶₄ is independent of 𝑇. Using the property of norm 𝑠^′₁(𝑡)𝑠₁(𝑡)−𝑠^′₂(𝑡)𝑠₂(𝑡)⩾𝑠^′₁(𝑡)−𝑠^′₂(𝑡)𝑠₁(𝑡)−𝑠^′₂(𝑡)𝑠₁(𝑡)−𝑠₂(𝑡), (2.19) and the fact 𝑠₁(0)=𝑠₂(0),𝑠_𝑖(𝑡)⩾𝑠₀ and 𝑠^′₂(𝑡)⩽(𝑠^∗+1)/𝑠₀ give that 𝑠^′₁−𝑠^′_{2𝐶[0,𝑇]}⩽1𝑠₀𝑠^′₁s₁−𝑠^′₂𝑠_{2𝐶[0,𝑇]}+𝑠^∗+1𝑠²₀𝑇𝑠^′₁−𝑠^′_{2𝐶[0,𝑇]}⩽2𝑠₀𝑠^′₁(𝑡)𝑠₁(𝑡)−𝑠^′₂(𝑡)𝑠₂(𝑡)_𝐶[0,𝑇] (2.20) if 𝑇⩽𝑠²₀/(2𝑠^∗+2). Further, 𝑠₁−𝑠_{2𝐶1[0,𝑇]}⩽(1+𝑇)𝑠^′₁−𝑠^′_{2𝐶[0,𝑇]}⩽2+2𝑇𝑠₀𝑠^′₁𝑠₁−𝑠^′₂𝑠_{2𝐶[0,𝑇]}. (2.21) Hence, for 𝑇∶=min1,𝐶₂^−2/𝛼,𝐶₁^{−2/(1+𝛼)},𝑠²₀2𝑠^∗,+22𝐶₄41+𝑠₀^−2/𝛼, (2.22) we have 𝑣₁(𝜉,𝑡)−𝑣₂(𝜉,𝑡)_{𝐶([0,1]×[0,𝑇])}+𝑠^′₁𝑠₁(𝑡)−𝑠^′₂(𝑡)𝑠₂(𝑡)_𝐶[0,𝑇]⩽𝑇^(1+𝛼)/2𝑣₁−𝑣_{2𝐶(1+𝛼)/2,1+𝛼([0,1]×[0,𝑇])}+𝑇^𝛼/2𝑠^′₁𝑠₁(𝑡)−𝑠^′₂(𝑡)𝑠₂(𝑡)_{𝐶𝛼/2[0,𝑇]}⩽𝐶₄𝑇^𝛼/2𝑣₁−𝑣_{2𝐶([0,1]×[0,𝑇])}+𝑠₁−𝑠_{2𝐶1[0,𝑇]}(2.18)⩽𝐶₄𝑇^𝛼/2𝑣₁−𝑣_{2𝐶([0,1]×[0,𝑇])}+2+2𝑇𝑠₀𝑠^′₁𝑠₁−𝑠^′₂𝑠_{2𝐶[0,𝑇]})⩽1(2.212𝑣₁−𝑣_{2𝐶([0,1]×[0,𝑇])}+𝑠^′₁𝑠₁−𝑠^′₂𝑠_{2𝐶[0,𝑇]}. (2.23) Thus for this 𝑇, ℱ is a contraction. Now using the contraction mapping theorem gives the conclusion that there is a (𝑣(𝜉,𝑡),𝑠(𝑡)) in Σ_𝑇 such that ℱ(𝑣(𝜉,𝑡),𝑠(𝑡))=(𝑣(𝜉,𝑡),𝑠(𝑡)). In other words, (𝑣(𝜉,𝑡),𝑠(𝑡)) is the solution of the problem (2.4) and therefore (𝑢(𝑥,𝑡),𝑠(𝑡)) is the solution of the problem (1.2). Moreover, by using the Schauder estimates, we have additional regularity of the solution, 𝑠(𝑡)∈𝐶^1+𝛼/2[0,𝑇] and 𝑢∈𝐶^{2+𝛼,1+𝛼/2}((0,𝑠(𝑡))×(0,𝑇]). Thus (𝑢(𝑥,𝑡),𝑠(𝑡)) is the classical solution of the problem (1.2).

Now we give the monotone behavior of the free boundary 𝑠(𝑡).

Theorem 2.2. The free boundary for the problem (1.2) is strictly monotone increasing, that is, for any solution in (0,𝑇], one has 𝑠^′(𝑡)>0for0<𝑡⩽𝑇. (2.24)

Proof. Using the Hopf lemma to the equation (1.2) yields that 𝑢_𝑥(𝑠(𝑡),𝑡)<0for0<𝑡⩽𝑇. (2.25) Thus, combining this inequality with the Stefan condition gives the result.

Remark 2.3. If the initial function 𝑢₀ is smooth and satisfies the consistency condition: −𝑑𝑢₀^′′𝑠₀+𝜇𝑢^′₀𝑠₀𝑢^′₀𝑠₀=𝑢₀𝑠₀𝐾𝑎𝑢₀𝑠₀−𝑏, (2.26) then the solution (𝑢,𝑠)∈𝐶^{2+𝛼,1+𝛼/2}([0,𝑠(𝑡)]×[0,𝑇])×𝐶^1+(1+𝛼)/2([0,𝑇]).

3. Finite Time Blowup

In this section we discuss the blowup behavior. First we present the following lemma.

Lemma 3.1. The solution of the problem (1.2) exists and is unique, and it can be extended to [0,𝑇_max) where 𝑇_max⩽∞. Moreover, if 𝑇_max<∞, one has limsup_{𝑡→𝑇max}‖𝑢(𝑥,𝑡)‖_𝐿^∞_{([0,𝑠(𝑡)]×[0,𝑡])}=∞. (3.1)

Proof. It follows from the uniqueness and Zorn’s lemma that there is a number 𝑇_max such that [0,𝑇_max) is the maximal time interval in which the solution exists. In order to prove the present lemma, it suffices to show that, when 𝑇_max<∞, limsup_{𝑡→𝑇max}‖𝑢‖_𝐿^∞_{([0,𝑠(𝑡)]×[0,𝑡])}=∞. (3.2)
In what follows we use the contradiction argument. Assume that 𝑇_max<∞ and ‖𝑢‖_𝐿^∞_{([0,𝑠(𝑡)]×[0,𝑡])}<∞. Since 𝑠′(𝑡) is bounded in [0,𝑇_max) by Theorem 2.1, using a bootstrap argument and Schauder's estimate yields a priori bound of ‖𝑢(𝑡,𝑥)‖_𝐶^1+𝛼_{[0,𝑠(𝑡)]} for all 𝑡∈[0,𝑇_max). Let the bound be 𝑀^∗. It follows from the proof of Theorem 2.1 that there exists a 𝜏>0 depending only on 𝑀^∗ such that the solution of the problem (1.2) with the initial time 𝑇_max−𝜏/2 can be extended uniquely to the time 𝑇_max−𝜏/2+𝜏, which contradicts the assumption. This completes the proof.

In order to investigate the behavior of the free boundary, we introduce the energy of the solution 𝑢 at 𝑡 by 𝐸(𝑡)=₀^𝑠(𝑡)𝑑2𝑢_𝑥²−𝐾𝑎3𝑢³+𝑏2𝑢²(𝑥,𝑡)d𝑥, (3.3) and its 𝐿¹-norm by |𝑢(𝑡)|₁=∫₀^𝑠(𝑡)𝑢(𝑥,𝑡)d𝑥. Then we have the following lemma.

Lemma 3.2. Let 𝑢 be the solution of the problem (1.2). Then one has the relations d𝐸d𝑡=−₀^𝑠(𝑡)𝑢²_𝑡()𝑑𝑥,𝑡d𝑥−2𝜇²𝑠^′3(𝑡), (3.4)|𝑢(𝑡)|₁−𝑢₀₁=𝑑𝜇𝑠₀−𝑠(𝑡)+^𝑡₀₀^𝑠(𝜏)𝐾𝑎𝑢²−𝑏𝑢(𝑥,𝜏)𝑑𝑥𝑑𝜏. (3.5)

Proof. It is easy to see that d𝐸=d𝑡₀^𝑠(𝑡)𝑑𝑢_𝑥𝑢_𝑥𝑡−𝐾𝑎𝑢²𝑢_𝑡+𝑏𝑢𝑢_𝑡(𝑥,𝑡)d𝑥+𝑠^′(𝑡)×𝑑2𝑢_𝑥²(𝑠(𝑡),𝑡)−𝐾𝑎3𝑢³(𝑠(𝑡))+𝑏,𝑡2𝑢²(𝑠(𝑡),𝑡). (3.6) Integrating by parts and using 𝑢_𝑥(0,𝑡)=0 yield ₀^𝑠(𝑡)𝑢_𝑥𝑢_𝑥𝑡d𝑥=−₀^𝑠(𝑡)𝑢_𝑥𝑥𝑢_𝑡𝑑𝑥+𝑢_𝑥𝑢_𝑡(𝑠(𝑡),𝑡). (3.7) Differentiating the second equation of (1.2) with respect to 𝑡, we have d(𝑢(𝑠(𝑡)d𝑡,𝑡))=𝑠^′(𝑡)𝑢_𝑥(𝑠(𝑡),𝑡)+𝑢_𝑡(𝑠(𝑡),𝑡)=0, (3.8) which implies that 𝑢_𝑥𝑢_𝑡(𝑠(𝑡),𝑡)=−𝑠^′(𝑡)𝑢²_𝑥(𝑠(𝑡))𝑠,𝑡=−^′3(𝑡)𝜇². (3.9) By substitution, we get d𝐸d𝑡=−₀^𝑠(𝑡)𝑑𝑢_𝑥𝑥𝑢_𝑡+𝐾𝑎𝑢²𝑢_𝑡−𝑏𝑢𝑢_𝑡()𝑑𝑥,𝑡d𝑥−𝜇²𝑠^′3(𝑡)+𝑑2𝜇²𝑠^′3(𝑡)=−₀^𝑠(𝑡)𝑢²_𝑡()𝑑𝑥,𝑡d𝑥−2𝜇²𝑠^′3(𝑡), (3.10) that is (3.4).
Now we show (3.5). It is obvious that dd𝑡₀^𝑠(𝑡)𝑢(𝑥,𝑡)d𝑥=₀^𝑠(𝑡)𝑢_𝑡(𝑥,𝑡)d𝑥+𝑠^′(𝑡)𝑢(𝑠(𝑡))=,𝑡_0𝑠(𝑡)𝑑𝑢_𝑥𝑥(𝑥,𝑡)d𝑥+₀^𝑠(𝑡)𝐾𝑎𝑢²()𝑑−𝑏𝑢𝑥,𝑡d𝑥=−𝜇𝑠^′(𝑡)+₀^𝑠(𝑡)𝐾𝑎𝑢²−𝑏𝑢(𝑥,𝑡)d𝑥. (3.11) Integrating the equation above, we get |𝑢(𝑡)|₁−𝑢₀₁=𝑑𝜇𝑠₀−𝑠(𝑡)+^𝑡₀₀^𝑠(𝑡)𝐾𝑎𝑢²−𝑏𝑢(𝑥,𝜏)d𝑥d𝜏. (3.12) This completes the proof.

Lemma 3.3. Assume 𝑇_max=∞, and let 𝐴=∫^∞₀𝑠′³(𝑡)d𝑡. If 𝑏<𝜋𝑑³/8(2𝑑𝑠₀+𝜇|𝑢₀|₁)², then one has 𝐴⩾𝜇³𝑑³𝜋²|𝑢₀|³₁/64(2𝑑𝑠₀+𝜇|𝑢₀|₁)⁴.

Proof. We see the following auxiliary free boundary problem: 𝑣_𝑡−𝑑𝑣_𝑥𝑥(𝑡)𝑣()=−𝑏𝑣,0<𝑥<ℎ,0<𝑡<∞,𝑥,0=𝑢₀(𝑥),0⩽𝑥⩽𝑠₀,ℎ(0)=𝑠₀,𝑣(ℎ(𝑡),𝑡)=𝑣_𝑥()ℎ0,𝑡=0,0<𝑡<∞,_′(𝑡)=−𝜇𝑣_𝑥(ℎ(𝑡),𝑡),0<𝑡<∞. (3.13) By the same argument as in Theorem 2.1, the solution of the above problem exists for all 𝑡>0 since the solution is bounded. Moreover, one can deduce from the maximum principle that 𝑢⩾𝑣⩾0 and 𝑠(𝑡)⩾ℎ(𝑡)⩾𝑠₀ on (0,𝑇_max). Similarly as in Lemma 3.2, denoting |𝑣(𝑡)|₁=∫₀^ℎ(𝑡)𝑣(𝑡,𝑥)d𝑥, we easily obtains ℎ(𝑡)−𝑠₀=𝜇𝑑𝑢₀₁−|𝑣(𝑡)|₁−𝜇𝑏𝑑^𝑡₀|𝑣(𝜏)|₁d𝜏. (3.14) Using Hölder's inequality and the fact that 𝑠(𝑡)⩾ℎ(𝑡)⩾𝑠₀ yields that for all 𝑡⩾0, ^𝑡₀𝑠^′(𝜏)d𝜏⩽^𝑡₀𝑠^′(𝜏)³d𝜏^1/3𝑡₀d𝜏^2/3⩽𝐴^1/3𝑡^2/3, (3.15) so we have 𝐴⩾𝑡^−2𝑡₀𝑠^′(𝜏)d𝜏³=𝑡⁻²𝑠(𝑡)−𝑠₀³⩾𝑡⁻²𝜇𝑑𝑢₀₁−|𝑣(𝑡)|₁−𝜇𝑏𝑑^𝑡₀|𝑣(𝜏)|₁d𝜏³. (3.16)
On the other hand, by the maximum principle, we have 𝑣⩽𝑤, where 𝑤 is the solution of the following Cauchy problem: 𝑤_𝑡−𝑑𝑤_𝑥𝑥𝑤()==0,−∞<𝑥<∞,𝑡>0,𝑥,0𝑢₀(𝑥)=𝑢₀(|𝑥|),−𝑠₀⩽𝑥⩽𝑠₀,ℝ0,𝑥∈−𝑠₀,𝑠₀. (3.17) By the 𝐿¹−𝐿^∞ estimate for the heat equation, we have ‖𝑣(𝑡)‖_∞⩽‖𝑤(𝑡)‖_∞⩽(4𝑑𝜋𝑡)^−1/2|𝑤(0)|₁=(𝑑𝜋𝑡)^−1/2𝑢₀₁, (3.18) hence, by (3.14), |𝑣(𝑡)|₁⩽ℎ(𝑡)‖𝑣(𝑡)‖_∞⩽ℎ(𝑡)(𝑑𝜋𝑡)^−1/2𝑢₀₁=𝑠₀+𝜇𝑑𝑢₀₁−𝜇𝑑|𝑣(𝑡)|₁−𝜇𝑏𝑑^𝑡₀|𝑣(𝑡)|₁𝑑𝜏(𝑑𝜋𝑡)^−1/2𝑢₀₁⩽𝑠₀+𝜇𝑑𝑢₀₁−𝜇𝑑|𝑣(𝑡)|₁(𝑑𝜋𝑡)^−1/2𝑢₀₁. (3.19) Therefore, we have |𝑣(𝑡₀)|₁⩽|𝑢₀|₁/2 for 𝑡₀=(𝑑𝜋)⁻¹(2𝑠₀+(𝜇/𝑑)|𝑢₀|₁)². If 𝑏<𝜋𝑑³/8(2𝑑𝑠₀+𝜇|𝑢₀|₁)², we get 𝑏∫^𝑡0₀|𝑣(𝜏)|₁d𝜏⩽(1/4)|𝑢₀|₁. Taking 𝑡=𝑡₀ in the inequality (3.16) yields the desired estimate.

Theorem 3.4. Let 𝑢 be the solution of the problem (1.2), if 𝑏<𝜋𝑑³/8(2𝑑𝑠₀+𝜇|𝑢₀|₁)², then one has 𝑇_max<∞ whenever 𝐸(0)<𝜇𝑑⁴𝜋²𝑢₀³₁1282𝑑𝑠₀+𝜇𝑢₀₁⁴. (3.20)

Proof. As in [28], define the function 𝐹(𝑡)=^𝑡₀₀^𝑠(𝜏)𝑢²(𝑥,𝜏)d𝑥d𝜏. (3.21) Direct calculations show that 𝐹′(𝑡)=∫₀^𝑠(𝑡)𝑢²(𝑥,𝑡)𝑑𝑥 and 𝐹^′′(𝑡)=₀^𝑠(𝑡)2𝑢𝑢_𝑡(𝑥,𝑡)d𝑥+𝑠^′(𝑡)𝑢²(𝑠(𝑡),𝑡)=₀^𝑠(𝑡)2𝑢𝑢_𝑡(𝑥,𝑡)d𝑥=2₀^𝑠(𝑡)𝑑𝑢𝑢_𝑥𝑥+𝑢²(𝐾𝑎𝑢−𝑏)(𝑥,𝑡)d𝑥=2₀^𝑠(𝑡)𝑢²(𝐾𝑎𝑢−𝑏)−𝑑𝑢²_𝑥(𝑥,𝑡)d𝑥+2𝑑𝑢𝑢_𝑥(𝑥,𝑡)₀^𝑠(𝑡)=−6𝐸(𝑡)+₀^𝑠(𝑡)𝑑𝑢²_𝑥+𝑏𝑢²(𝑥,𝑡)d𝑥. (3.22) It follows from the identity (3.4) that 𝐹^′′(𝑡)=6^𝑡₀₀^𝑠(𝜏)𝑢²_𝑡(𝑥,𝜏)d𝑥d𝜏+3𝑑𝜇^2𝑡₀𝑠^′3(𝜏)(0)+𝑑𝜏−6𝐸_0𝑠(𝑡)𝑑𝑢²_𝑥+𝑏𝑢²(𝑥,𝑡)d𝑥. (3.23) Now assume 𝑇_max=∞ by contradiction. The assumption (3.20), together with Lemma 3.3, implies that 𝐸(0)<𝑑2𝜇^2𝑡₀𝑠^′3(𝜏)d𝜏 (3.24) for all 𝑡⩾𝑡₀ sufficiently large, we then have 𝐹^′′(𝑡)>6^𝑡₀₀^𝑠(𝜏)𝑢²_𝑡(𝑥,𝜏)d𝑥d𝜏,𝑡⩾𝑡₀. (3.25) Applying the Cauchy-Schwarz inequality yields 𝐹(𝑡)𝐹^′′(𝑡)⩾6^𝑡₀₀^𝑠(𝜏)𝑢²d𝑥d𝜏^𝑡₀₀^𝑠(𝜏)𝑢²_𝑡d𝑥d𝜏⩾6^𝑡₀₀^𝑠(𝜏)𝑢𝑢_𝑡d𝑥d𝜏²=32𝐹^′(𝑡)−𝐹^′(0)² (3.26) since 𝐹^′′(𝜏)=∫₀^𝑠(𝜏)2𝑢𝑢_𝑡(𝑥,𝜏)d𝑥 by (3.22).
On the other hand, (3.25) implies that 𝐹^′(𝑡)⩾𝐹^′𝑡₀+1=^𝑠(𝑡0₀⁺¹⁾𝑢²𝑥,𝑡₀+1d𝑥>0,𝑡⩾𝑡₀+1, (3.27) so that lim_𝑡→∞𝐹(𝑡)=∞. We then obtain 𝐹(𝑡)𝐹^′′(𝑡)⩾54𝐹^′2(𝑡),𝑡⩾𝑡₁ (3.28) for some large 𝑡₁>𝑡₀+1.
Defining 𝐺(𝑡)=𝐹^−1/4(𝑡) for 𝑡⩾𝑡₁, it follows that 𝐺^′(𝑡)1=−4𝐹^′(𝑡)𝐹^−5/4(𝑡)<0,𝑡⩾𝑡₁,𝐺^′′(𝑡)1=−4𝐹^−9/4𝐹𝐹^′′−54𝐹^′2(𝑡)⩽0,𝑡⩾𝑡₁. (3.29) This implies that 𝐺 is concave, decreasing, and positive for 𝑡⩾𝑡₁, which is impossible. The contradiction shows that 𝑇_max<∞, which gives the blowup result.

Remark 3.5. The above theorem shows that the solution of the free boundary problem (1.2) blows up if the death rate (𝑏) is sufficiently small and the initial datum (𝑢₀) is sufficiently large.

4. Global Fast Solution and Slow Solution

In this section, we study the asymptotic behavior of the global solutions of (1.2). We first give the following existence of fast solution.

Theorem 4.1 (fast solution). Let 𝑢 be a solution of problem (1.2). If 𝑢₀ is small in the following sense: 𝑢_0∞⩽𝑑132min𝐾𝑎𝑠²₀,1𝜇, (4.1) then 𝑇_max=∞. Moreover, 𝑠_∞<∞ and there exist real numbers 𝐶, 𝛽>0 depending on 𝑢₀ such that ‖𝑢(𝑡)‖_∞⩽𝐶𝑒^−𝛽𝑡,𝑡⩾0. (4.2)

Proof. It suffices to construct a suitable global supersolution. Inspired by [30], we define 𝜗(𝑡)=2𝑠₀2−𝑒^−𝛾𝑡,𝑡⩾0,𝑉𝑦=1−𝑦²𝑣(),0⩽𝑦⩽1,𝑥,𝑡=𝜀𝑒^−𝛽𝑡𝑉𝑥𝜗(𝑡),0⩽𝑥⩽𝜗(𝑡),𝑡⩾0, (4.3) where 𝛾,𝛽 and 𝜀>0 to be chosen later.
An easy computation yields 𝑣_𝑡−𝑑𝑣_𝑥𝑥−𝑣(𝐾𝑎𝑣−𝑏)=𝜀𝑒^−𝛽𝑡−𝛽𝑉−𝑥𝜗^′𝜗⁻²𝑉^′−𝑑𝜗⁻²𝑉^′′−𝑉𝐾𝑎𝜀𝑒^−𝛽𝑡𝑉−𝑏⩾𝜀𝑒^−𝛽𝑡𝑉𝑑−𝛽+8𝑠²₀−𝐾𝑎𝜀, (4.4) for all 𝑡>0 and 0<𝑥<𝜗(𝑡).
On the other hand, we have 𝜗′(𝑡)=2𝛾𝑠₀𝑒^−𝛾𝑡>0 and −𝑣_𝑥(𝜗(𝑡),𝑡)=2𝜀𝑒^−𝛽𝑡/𝜗(𝑡). Setting 𝛾=𝛽=𝑑/16𝑠²₀, and 𝜀⩽𝜀₀=min{𝑑/16𝐾𝑎𝑠²₀,𝑑/16𝜇}, it follows that 𝑣_𝑡−𝑑𝑣_𝑥𝑥()(𝑡)𝜗−𝑣𝐾𝑎𝑣−𝑏⩾0,0<𝑥<𝜗,𝑡>0,_′(𝑡)>−𝜇𝑣_𝑥(𝜗(𝑡))𝑣(𝜗(𝑡)),𝑡,𝑡>0,,𝑡=𝑣_𝑥()𝜗(0)0,t=0,𝑡>0,=2𝑠₀>𝑠₀. (4.5) Assume that ‖𝑢₀‖_∞⩽min{𝑑/32𝐾𝑎𝑠²₀,𝑑/32𝜇} and choose 𝜀=2‖𝑢₀‖_∞, we also get 𝑢₀(𝑥)<𝑣(𝑥,0) for 0⩽𝑥⩽𝑠₀.
By using the maximum principle, one then sees that 𝑠(𝑡)<𝜗(𝑡) and 𝑢(𝑥,𝑡)<𝑣(𝑥,𝑡) for 0⩽𝑥⩽𝑠(𝑡), as long as 𝑢 exists. In particular, it follows from the continuation property (3.1) that 𝑢 exists globally. The proof is complete.

Remark 4.2. The above result shows that the free boundary converges to a finite limit and that the solution 𝑢(𝑡) decays uniformly to 0 at an exponential rate. Compared to the case (see Theorem 4.5), the free boundary grows up to infinity and the decay rate of the solution is at most polynomial, the former solution is therefore called fast solution.

Before we give the existence result of slow solution, we need the following uniform a priori estimate for all global solutions of problem (1.2).

Proposition 4.3. Let 𝑢 be a solution of the problem (1.2) with 𝑇_max=∞. Then there is a constant 𝐶=𝐶(‖𝑢₀‖_𝐶^1+𝛼,𝑠₀,1/𝑠₀)>0, such that sup_𝑡⩾0‖𝑢(𝑥,𝑡)‖_𝐿^∞_{(0,𝑠(𝑡))}⩽𝐶, (4.6) where 𝐶 remains bounded for ‖𝑢₀‖_𝐶^1+𝛼, 𝑠₀, and 1/𝑠₀ bounded.

Proof. First from the local theory for problem (1.2), for each 𝑀>1 there exists 𝜎>0 such that, if ‖𝑢₀‖_𝐶^1+𝛼<𝑀 and 1/𝑀<𝑠₀<𝑀, then ‖𝑢(𝑥,𝑡)‖_𝐿^∞<2𝑀 on [0,𝜎].
Assume that the result is false. Then there exists a 𝑀>0 and a sequence of global solutions (𝑢_𝑛,𝑠_𝑛) of (1.2), such that 1𝑀<𝑠_𝑛(0)<𝑀,𝑢_𝑛(𝑥,0)_{𝐶1+𝛼[0,𝑠0]}<𝑀,sup_𝑡⩾0𝑢_𝑛(𝑥,𝑡)_{𝐿∞(0,𝑠𝑛(𝑡))}⟶∞as𝑛⟶∞. (4.7) For all large 𝑛 there exist 𝑡_𝑛⩾𝜎 and 𝑥_𝑛∈[0,𝑠_𝑛(𝑡_𝑛)) such that sup_𝑡⩾0𝑢_𝑛(𝑥,𝑡)_{𝐿∞(0,𝑠𝑛(𝑡))}=𝑢_𝑛𝑥_𝑛,𝑡_𝑛≜𝜚_𝑛. (4.8) We define 𝜆_𝑛=𝜚_𝑛^−1/2, then it is evident to see that 𝜆_𝑛→0 as 𝑛→∞. We extend 𝑢_𝑛(⋅,𝑡) by 0 on (𝑠_𝑛(𝑡),∞) and define the rescaled function 𝑣_𝑛𝑦,𝜏=𝜆²_𝑛𝑢_𝑛𝑥_𝑛+𝜆_𝑛𝑦,𝑡_𝑛+𝜆²_𝑛𝜏, (4.9) for (𝑦,𝜏)∈𝐷_𝑛={(𝑦,𝜏)∶−𝜆_𝑛⁻¹𝑥_𝑛⩽𝑦<∞and−𝜆_𝑛⁻²𝑡_𝑛⩽𝜏⩽0}. Also, we denote 𝑦₁=−𝜆_𝑛⁻¹𝑥_𝑛,𝑦₂(𝜏)=𝜆_𝑛⁻¹𝑠𝑡_𝑛+𝜆²_𝑛𝜏−𝑥_𝑛,𝐷_𝑛=𝑦,𝜏∶𝑦₁⩽𝑦<𝑦₂(𝜏)and−𝜆_𝑛⁻²𝑡_𝑛⩽𝜏⩽0, (4.10) which corresponds to the domain {𝑥<𝑠(𝑡)}. The function 𝑣_𝑛 satisfies 𝑣_𝑛(0,0)=1,0⩽𝑣_𝑛⩽1 and 𝜕_𝜏𝑣_𝑛−𝑑𝜕²_𝑦𝑣_𝑛=𝑣_𝑛𝐾𝑎𝑣_𝑛−𝑏_𝑛,𝑦,𝜏∈𝐷_𝑛, (4.11) where 𝑏_𝑛=𝜆²_𝑛𝑏. Note that 𝑏_𝑛→0 as 𝑛→+∞. Similarly as Lemma 2.1 in [27], there exists a subsequence {𝑣_𝑛𝑘} of {𝑣_𝑛} such that {𝑣_𝑛𝑘} converges in 𝐿^𝑝_loc([0,+∞)×(−∞,0]) to a function 𝑤(𝑦,𝑡)∈𝐿^𝑝_loc([0,+∞)×(−∞,0]) and {𝑣_𝑛𝑘(𝑦,0)} converges in 𝐶_loc([0,+∞)) to a function 𝑧(𝑦)∈𝐶([0,+∞)), which satisfies 𝑧(0)=1. Moreover, similarly as Lemmas 2.2 and 2.3 in [27], 𝑤_𝑡=0 in 𝐷′((0,+∞)×(−∞,0)) and there is a function 𝑤(𝑦)⩾0 which is bounded, continuous on [0,∞) and satisfies that −𝑤_𝑦𝑦=𝐾𝑎𝑤², hence 𝑤 is concave. Therefore 𝑤≡0, which leads to a contradiction to the fact that 𝑤(0)=𝑧(0)=1. This completes the proof of Proposition 4.3.

The above proposition shows that all global solutions are uniformly bounded and the coming result implies that all global solutions decay uniformly to 0.

Proposition 4.4. Let 𝑢 be a solution of the problem (1.2) with 𝑇_max=∞. Then it holds that lim_𝑡→+∞‖𝑢(𝑥,𝑡)‖_𝐿^∞_{(0,𝑠(𝑡))}=0. (4.12)

Proof. Assume that 𝑙∶=limsup_𝑡→+∞‖𝑢(𝑥,𝑡)‖_𝐿^∞_{(0,𝑠(𝑡))}>0 by contradiction. It follows from Proposition 4.3 that 𝑙<+∞. Let 𝑡₀>0 be such that sup_[𝑡0,+∞)‖𝑢(𝑥,𝑡)‖_𝐿^∞_{(0,𝑠(𝑡))}⩽(3/2)𝑙. Then there exists a sequence 𝑡_𝑛→+∞ such that ‖𝑢(𝑥,𝑡_𝑛)‖_𝐿^∞_{(0,𝑠(𝑡𝑛))}⩾(3/4)𝑙.
Now pick 𝑥_𝑛∈[0,𝑠_𝑛(𝑡_𝑛)) such that 𝑢(𝑥,𝑡_𝑛)_{𝐿∞(0,𝑠(𝑡𝑛))}=𝑢𝑥_𝑛,𝑡_𝑛. (4.13) As above, we define 𝜆_𝑛=𝑢^−1/2(𝑥_𝑛,𝑡_𝑛) and then 0<𝜆_𝑛⩽(3𝑙/4)^−1/2. We extend 𝑢(⋅,𝑡) by 0 on (𝑠(𝑡),∞) and define the rescaled function 𝑣_𝑛𝑦,𝜏=𝜆²_𝑛𝑢𝑥_𝑛+𝜆_𝑛𝑦,𝑡_𝑛+𝜆²_𝑛𝜏 (4.14) for (𝑦,𝜏)∈𝐷_𝑛={(𝑦,𝜏)∶−𝜆_𝑛⁻¹𝑥_𝑛⩽𝑦<∞and𝜆_𝑛⁻²(𝑡₀−𝑡_𝑛)⩽𝜏⩽0}. Also, we denote 𝑦₁=−𝜆_𝑛⁻¹𝑥_𝑛,𝑦₂(𝜏)=𝜆_𝑛⁻¹𝑠𝑡_𝑛+𝜆²_𝑛𝜏−𝑥_𝑛,𝐷_𝑛=𝑦,𝜏∶𝑦₁⩽𝑦<𝑦₂(𝜏)and𝜆_𝑛⁻²𝑡₀−𝑡_𝑛⩽𝜏⩽0. (4.15) Therefore the function 𝑣_𝑛 satisfies 𝑣_𝑛(0,0)=1,0⩽𝑣_𝑛⩽2, lim_𝑦→+∞𝑣_𝑛(𝑦,𝜏)=0 and 𝜕_𝜏𝑣_𝑛−𝑑𝜕²_𝑦𝑣_𝑛=𝑣_𝑛𝐾𝑎𝑣_𝑛−𝑏_𝑛,𝑦,𝜏∈𝐷_𝑛, (4.16) where 𝑏_𝑛=𝜆²_𝑛𝑏. Note that 𝑏_𝑛⩽𝑏(3𝑙/4)⁻¹, therefore there exists a subsequence {𝑏_𝑛𝑘} and 𝑏^∗⩽𝑏(3𝑙/4)⁻¹ such that 𝑏_𝑛𝑘→𝑏^∗ as 𝑘→+∞. Similarly as Lemmas 2.1–2.3 in [27], we have obtained a function 𝑤(𝑦)⩾0, bounded and continuous on [0,∞) and satisfies that −𝑤_𝑦𝑦=𝐾𝑎𝑤²−𝑏^∗𝑤. Therefore 𝑤≡0 or 𝑤≡𝑏^∗/𝐾𝑎. If 𝑤≡0, this is a contradiction to the fact that 𝑤(0)=1 since 𝑣_𝑛(0,0)=1. If 𝑤≡𝑏^∗/𝐾𝑎, this is also a contradiction to the fact that lim_𝑦→+∞𝑤(𝑦)=0.

Theorem 4.5 (slow solution). Let 𝜙∈𝐶¹([0,𝑠₀]) satisfy 𝜙⩾0,𝜙≢0 with 𝜙_𝑥(0)=𝜙(𝑠₀)=0, and 𝑏 satisfy the same condition as in Theorem 3.4. Then there exists 𝜆>0 such that the solution of (1.2) with initial data 𝑢₀=𝜆𝜙 is a global slow solution, which satisfies that 𝑠_∞=∞ and 𝑠(𝑡)=𝑂𝑡^2/3as𝑡⟶∞. (4.17)

Proof. Denote the solution of (1.2) as 𝑢(𝑢₀;⋅) to emphasize the dependence on the initial function 𝑢₀ when necessary. So as to 𝑠(𝑡),𝑠_∞ and the maximal existence time 𝑇_max.
Similarly as in [27], define Σ=𝜆>0;𝑇_max𝜆𝜙=∞and𝑠_∞𝜆𝜙<∞. (4.18) According to the Theorem 4.1, Σ≠∅ since that the solution is global if 𝜆 is sufficiently small.
If 𝑏<𝜋𝑑³/8(2𝑑𝑠₀+𝜇|𝑢₀|₁)², then for 𝜆 sufficiently large, we have 𝐸(𝜆𝜙)<0, which implies 𝑇_max(𝜆𝜙)<∞. Therefore Σ is bounded.
Let 𝜆^∗=supΣ∈(0,∞),𝑣(𝑥,𝑡)=𝑢(𝜆^∗𝜙;𝑥,𝑡),𝜎(𝑡)=𝑠(𝜆^∗𝜙;𝑡) and 𝜏=𝑇_max(𝜆^∗𝜙).
We first claim that 𝜏=∞. In fact, by continuous dependence, when 𝜆→𝜆^∗, we know that 𝑢(𝜆𝜙;𝑥,𝑡) approaches 𝑣(𝑥,𝑡) in 𝐿^∞(0,∞) and 𝑠(𝜆𝜙;𝑡)→𝜎(𝑡) for each fixed 𝑡∈[0,𝜏). Since 𝑇_max(𝜆𝜙)=∞ for all 𝜆∈(0,𝜆^∗), it follows from Proposition 4.3 that ‖𝑣(𝑥,𝑡)‖_𝐿^∞_{(0,𝜎(𝑡))}⩽𝐶 for all 𝑡∈[0,𝜏). Therefore, 𝜏=∞ since nonglobal solutions must satisfy limsup_𝑡→𝑇‖𝑢(𝑥,𝑡)‖_𝐿^∞_{(0,𝑠(𝑡))}=∞.
Next we claim that 𝜎_∞=∞. Assume 𝜎_∞<∞ for contradiction. If follows from Proposition 4.4 that ‖𝑣(𝑥,𝑡)‖_𝐿^∞_{(0,𝜎(𝑡))}→0 as 𝑡→∞, which implies 𝑣𝑥,𝑡_{0𝐿∞(0,𝜎(𝑡0))}𝑑<min32𝐾𝑎𝜎²𝑡₀,𝑑32𝜇 (4.19) for some large 𝑡₀. By continuous dependence, we have 𝑢𝜆𝜙;𝑥,𝑡_{0𝐿∞(0,ℎ(𝑡0))}𝑑⩽min32𝐾𝑎𝑠²𝑡₀,𝑑32𝜇 (4.20) for 𝜆>𝜆^∗ sufficiently close to 𝜆^∗. But this implies that 𝑠_∞(𝜆𝜙)<∞ by Theorem 4.1, which is a contradiction with the definition of 𝜆^∗.
On the other hand, as a consequence of the blowup result of Theorem 3.4, we know that 𝑇_max=∞ implies 𝐸(𝑡)⩾0 for all 𝑡⩾0, hence by using (3.4) we have ^∞₀₀^𝑠(𝑡)𝑢²_𝑡()𝑑𝑥,𝑡𝑑𝑥𝑑𝑡+2𝜇^2∞₀𝑠^′3(𝑡)𝑑𝑡⩽𝐸(0)<∞. (4.21) The estimate (4.17) follows from Hölder’s inequality and (4.21), by writing 𝑠(𝑡)−𝑠₀=∫^𝑡₀𝑠′(𝜏)𝑑𝜏⩽(∫^𝑡₀(𝑠′(𝜏))³𝑑𝜏)^1/3𝑡^2/3. The proof is complete.

Acknowledgments

The work is partially supported by PRC Grants NSFC 11071209, NSFC 10801115.