Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 45, 1-10; http://www.math.u-szeged.hu/ejqtde/

Non-Simultaneous Blow-Up for a Reaction-Diffusion

System with Absorption and Coupled Boundary Flux^{∗}

Jun Zhou^{a,b}^{†} Chunlai Mu^{b}^{‡}

a. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China b. College of Mathematics and Statistics, Chongqing University, Chongqing, 400044, China

Abstract. This paper deals with non-simultaneous blow-up for a reactiondiffusion system with absorption and nonlinear boundary flux. We establish necessary and sufficient conditions for the occurrence of non-simultaneous blow-up with proper initial data.

Keywords. non-simultaneous blow-up; reaction-diffusion system; nonlinear absorption; nonlinear boundary flux; blow-up rate Mathematics Subject Classification (2000). 35K55; 35B33

1 Introduction

In this paper, we study non-simultaneous blow-up for the following reaction-diffusion system

\begin{matrix}u_{t} = u_{xx} − a_{1}e^{α_{1}u}, v_{t} = v_{xx} − a_{2}e^{β_{1}v}, & (x, t) ∈ (0, 1) × (0, T), \\ u_{x}(1, t) = e^{α_{2}u(1,t)+pv(1,t)}, v_{x}(1, t) = e^{qu(1,t)+β_{2}v(1,t)}, & t ∈ (0, T), \\ u_{x}(0, t) = 0, v_{x}(0, t) = 0, & t ∈ (0, T), \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ [0, 1], \end{matrix} (1.1)

where p, q, a_{i} > 0, α_{i}, β_{i} ≥ 0, i = 1, 2. The initial data satisfy u_{0}, v_{0} ≥ 0, u0^{′}, v0^{′} ≥ 0, u0^{′′} − a_{1}e^{α_{1}u_{0}}, v0^{′′} − a_{2}e^{β_{1}v_{0}} ≥ δ > 0, as well as the compatibility conditions on [0, 1]. By comparison principle, it follows that u_{t}, v_{t} > 0, u_{x}, v_{x} ≥ 0 and u, v ≥ 0 for (x, t) ∈ [0, 1] × [0, T).

The reaction-diffusion system (1.1) can be used to describe heat propagations in mixed solid media with nonlinear absorption and nonlinear boundary flux [1-3, 5, 9, 11, 16]. The nonlinear Neumann boundary values in (1.1), coupling the two heat equations, represent some cross-boundary flux.

∗J. Zhou is supported by the Fundamental Research Funds for the Central Universities (No.

XDJK2009C069) and C. L. Mu is supported by NNSF of China (No. 10771226).

†Corresponding author. E-mail: zhoujun math@hotmail.com ‡E-mail: chunlaimu@yahoo.com.cn

The problem of heat equations

u_{t} = Δu, v_{t} = Δv, (x, t) ∈ Ω × (0, t),  (1.2)

coupled via somewhat special nonlinear Neumann boundary conditions

\frac{∂u}{∂ν} = v^{p}, \frac{∂v}{∂ν} = u^{q}, (x, t) ∈ ∂Ω × (0, T),  (1.3)

was studied by Deng [7] and Lin and Xie [11], who showed that the solutions globally exist if pq ≤ 1 and may blow up in finite time if pq > 1 with the blow-up rates O ((T − t)^{−(p+1)/2(pq−1)}) and O ((T − t)^{−(q+1)/2(pq−1)}). Similarly, the blow-up rates for the corresponding scalar case of (1.2) and (1.3) was shown to be O ((T − t)^{−1/2(p−1)}) in [10].

The system (coupled via a variational boundary flux of exponential type)

\begin{matrix}u_{t} = Δu, v_{t} = Δv, & (x, t) ∈ Ω × (0, t), \\ \frac{∂u}{∂ν} = e^{pv}, \frac{∂v}{∂ν} = e^{qu}, & (x, t) ∈ ∂Ω × (0, t), \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ Ω, \end{matrix}  (1.4)

was studied by Deng [7], and has blow-up rates

−\frac{1}{2q} log c(T − t) ≤ max_{\overline{Ω}} u(·, t) ≤ −\frac{1}{2q} log C(T − t),

−\frac{1}{2p} log c(T − t) ≤ max_{\overline{Ω}} v(·, t) ≤ −\frac{1}{2p} log C(T − t),

(1.5)

for t ∈ (0, T). This is the special case with α_{i} = β_{i} = a_{i} = 0, i = 1, 2, in our system (1.1).

Zhao and Zheng [17] studied the following nonlinear parabolic system:

\begin{matrix}u_{t} = Δu, v_{t} = Δv, & (x, t) ∈ Ω × (0, t), \\ \frac{∂u}{∂ν} = e^{α_{2}u+pv}, \frac{∂v}{∂ν} = e^{qu+β_{2}v}, & (x, t) ∈ ∂Ω × (0, t), \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ Ω. \end{matrix}  (1.6)

The blow-up rates for (1.6) were shown to be

max_{\overline{Ω}} u(·, t) = O (log(T − t)^{α/2}) , max_{\overline{Ω}} v(·, t) = O (log(T − t)^{β/2}) ,  (1.7)

as t → T, where (α, β)^{T} is the only positive solution of

( \begin{matrix}α_{2} & p \\ q & β_{2} \end{matrix} ) ( \begin{matrix}α \\ β \end{matrix} ) = ( \begin{matrix}1 \\ 1 \end{matrix} ) ,

namely,

α = \frac{p − β_{2}}{pq − α_{2}β_{2}} , β = \frac{q − α_{2}}{pq − α_{2}β_{2}} .

Clearly the blow-up rate estimate (1.5) is just the special case of (1.7) with α_{2} = β_{2} = 0.

The phenomenon of non-simultaneous blow-up is researched extensively [see 4, 13-15]. Recently Zheng and Qiao [20] consider the non-simultaneous blow-up phenomenon of following reaction-diffusion problem

\begin{matrix}u_{t} = u_{xx} − λ_{1}u^{α_{1}}, v_{t} = v_{xx} − λ_{2}v^{β_{1}}, & (x, t) ∈ (0, 1) × (0, T), \\ u_{x}(1, t) = u^{α_{2}}v^{p}, v_{x}(1, t) = u^{q}v^{β_{2}}, & t ∈ (0, T), \\ u_{x}(0, t) = 0, v_{x}(0, t) = 0, & t ∈ (0, T), \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ [0, 1], \end{matrix}  (1.8)

and they get the following conclusions:

(1) If q < α_{2} − 1 with either α_{2} > μ or α_{2} = μ > 1, then there exists initial data (u_{0}, v_{0}) such that u blows up at a finite time T while v remains bounded.

(2) If u blows up at time T and v remains bounded up to that time, then q < α_{2} − 1 with either α_{2} > μ or α_{2} = μ > 1.

(3) Under the condition of (1), if in addition either (i) β_{2} ≤ 1, or (ii) 1 < β_{2} < γ and q < \frac{(α_{2}−1)(γ−β_{2})}{γ−1} hold, then any blow-up must be non-simultaneous, namely, u blows up at a finite time T while v remains bounded.

The critical exponents for the system (1.1) were studied in [18] by Zheng and Li, where the following characteristic algebraic system was introduced:

( \begin{matrix}α_{2} − \frac{1}{2}α_{1} & p \\ q & β_{2} − \frac{1}{2}β_{1} \end{matrix} ) ( \begin{matrix}τ_{1} \\ τ_{2} \end{matrix} ) = ( \begin{matrix}1 \\ 1 \end{matrix} ) ,  (1.9)

namely,

τ_{1} = \frac{p + \frac{1}{2}β_{1} − β_{2}}{pq − (\frac{1}{2}α_{1} − α_{2})(\frac{1}{2}β_{1} − β_{2})}, τ_{2} = \frac{q + \frac{1}{2}α_{1} − α_{2}}{pq − (\frac{1}{2}α_{1} − α_{2})(\frac{1}{2}β_{1} − β_{2})}.  (1.10)

To state their main result, first we give some information about eigenfunction for Laplace’s equation.

Let φ_{0} be the first eigenfunction of

φ′′ + λφ = 0 in (−1, 1); φ(−1) = φ(1) = 0,  (1.11)

with the first eigenvalue λ_{0}, normalized by ∥φ_{0}∥_{∞} = 1. It is well know that [6] φ_{0} > 0 in (−1, 1), and there are positive constants c_{i} (i=1,2,3,4) and ε_{0} such that

\begin{matrix}c_{1} ≤ φ0^{′}(−1), −φ0^{′}(1) ≤ c_{2} ≤ max_{[−1,1]} |φ0^{′}| = c_{4}, \\ |φ0^{′}| ≥ \frac{c_{1}}{2} on \{x ∈ (−1, 1) : dist(x, −1) ≤ ε_{0}\} ∪ \{x ∈ (−1, 1) : dist(x, 1) ≤ ε_{0}\}, \\ φ_{0} ≥ c_{3} on \{x ∈ (−1, 1) : dist(x, −1) ≥ ε_{0}\} ∩ \{x ∈ (−1, 1) : dist(x, 1) ≥ ε_{0}\}. \end{matrix} (1.12)

Now we can state the main result of [18] Proposition 1.1 (1) If 1/τ_{1} > 0 or 1/τ_{2} > 0, then the solutions of (1.1) blow up in finite time with large initial data.

(2) If 1/τ_{i} < 0, i=1,2, then the solutions of (1.1) are globally bounded.

(3) Assume that 1/τ_{1} = 1/τ_{2} = 0.

(i) If α_{2} > \frac{1}{2}α_{1} and β_{2} > \frac{1}{2}β_{1}, then the solutions of (1.1) blow up in finite time with large initial data.

(ii) If a_{1} ≥ 2^{α_{1}} (\frac{λ_{0}}{c_{1}} + \frac{3c_{4}^{2}}{c_{1}^{2}} ) , a_{2} ≥ 2^{β_{1}} (\frac{λ_{0}}{c_{1}} + \frac{3c_{4}^{2}}{c_{1}^{2}} ) with α_{2} < \frac{1}{2}α_{1}, β_{2} < \frac{1}{2}β_{1}, then the solutions of (1.1) are globally bounded.

(iii) If a_{1} ≤ min \{\frac{c_{1}^{2}M^{2}}{4α_{1}} , \frac{λ_{0}c_{3}^{2}M^{2}}{α_{1}} \} , a_{2} ≤ min \{\frac{c_{1}^{2}M^{2}}{4β_{1}} , \frac{λ_{0}c_{3}^{2}M^{2}}{β_{1}} \} with α_{2} < \frac{1}{2}α_{1}, β_{2} < \frac{1}{2}β_{1}, M = min{α_{1}/(2c_{2}), β_{1}/(2c_{2})}, then the solutions of (1.1) blow up in finite time for large initial data.

Intrigued by [18-20], we consider the non-simultaneous blow-up of (1.1). The main results of this paper are the following two Theorems for non-simultaneous blow-up. Without loss of generality, we only deal with the case where u blows up while v remains bounded.

Theorem 1.1 If q < α_{2} with either 2α_{2} > α_{1} or 2α_{2} = α_{1}, a_{1} ≤ \frac{α_{1}}{4} min\{c_{1}^{2}/(4c_{2}^{2}), λ_{0}c_{3}^{2}/c_{2}^{2}\}, then there exists initial data (u_{0}, v_{0}) such that u blows up at a finite time T while v remains bounded up to that time.

Theorem 1.2 If u blows up at a finite time T while v remains bounded up to that time, then q < α_{2} with either 2α_{2} > α_{1} or 2α_{2} = α_{1}, a_{1} ≤ \frac{α_{1}}{4} min\{c_{1}^{2}/(4c_{2}^{2}), λ_{0}c_{3}^{2}/c_{2}^{2}\}.

We will prove Theorem 1.1 and 1.2 in the next two sections.

2 Proof of Theorem 1.1

At first, we consider the scalar problem of the form

\begin{matrix}u_{t} = u_{xx} − a_{1}e^{α_{1}u}, & (x, t) ∈ (0, 1) × (0, T), \\ u_{x}(1, t) = e^{α_{2}u(1,t)}e^{ph(t)}, u_{x}(0, t) = 0, & t ∈ (0, T), \\ u(x, 0) = u_{0}(x), & x ∈ [0, 1], \end{matrix}  (2.1)

with a_{1}, α_{i} in (1.1), i=1,2 and h(t) continuous, non-decreasing, 0 ≤ h(t) ≤ K. Similarly to Theorem 3.2 in [19], we can prove the following Lemma, where and in the sequel C is used to represent positive constants independent of t, and may change from line to line.

Lemma 2.1 Let u be a solution of (2.1). Assume (i) 2α_{2} > α_{1} or (ii) 2α_{2} = α_{1} with a_{1} ≤ \frac{α_{1}}{4} min\{c_{1}^{2}/(4c_{2}^{2}), λ_{0}c_{3}^{2}/c_{2}^{2}\}. Then u blows up in a finite time T for sufficiently large initial value, and moreover

u(1, t) = max_{[0,1]} u(·, t) ≤ log C(T − t)^{− \frac{1}{2α_{2}}} , 0 < t < T.  (2.2)

Proof. Let w slove

\begin{matrix}w_{t} = w_{xx} − a_{1}e^{α_{1}w}, & (x, t) ∈ (0, 1) × (0, T), \\ w_{x}(1, t) = e^{α_{2}w(1,t)}, w_{x}(0, t) = 0, & t ∈ (0, T), \\ w(x, 0) = u_{0}(x), & x ∈ [0, 1]. \end{matrix}  (2.3)

Then, w ≤ u in (0, 1)×[0, T) by the comparison principle. Notice that the two assumptions 2α_{2} > α_{1} or 2α_{2} = α_{1} with a_{1} ≤ \frac{α_{1}}{4} min\{c_{1}^{2}/(4c_{2}^{2}), λ_{0}c_{3}^{2}/c_{2}^{2}\} are corresponding to the blow-up conditions by Proposition 1.1 with α_{1} = β_{1}, α_{2} = β_{2}, p = q = 0 and u_{0} = v_{0}, a_{1} = a_{2} in (1.1). So there exists initial data such that w blows up in finite time T′. Then u blows up in finite time t = T.

To establish the desired blow-up rate, we exploit the method used in [19]. From the assumptions on initial data, we know that w_{t} > 0 and w_{x} ≥ 0 for (x, t) ∈ [0, 1) × [0, T). Set J(x, t) = \sqrt{w_{t}} − εw_{x} for (x, t) ∈ (0, 1) × [0, T). Let ε be sufficiently small such that

J(x, 0) = \sqrt{w_{t}(x, 0)} − εw_{x}(x, 0) ≥ 0, x ∈ [0, 1],  (2.4)

a simple computation yields

\begin{matrix}J_{x}(1, t) − [(\frac{1}{2}α_{2}e^{α_{2}w} − εwt^{\frac{1}{2}} − ε^{2}e^{α_{2}w}) J] (1, t) \\ = ε (\frac{1}{2}α_{2}e^{2α_{2}w} − a_{1}e^{α_{1}w} − ε^{2}e^{2α_{2}w}) (1, t) ≥ 0, t ∈ (0, T), \end{matrix}  (2.5)

when (i) 2α_{2} > α_{1} or (ii) 2α_{2} = α_{1} with a_{1} ≤ \frac{α_{1}}{4} min\{c_{1}^{2}/(4c_{2}^{2}), λ_{0}c_{3}^{2}/c_{2}^{2}\}.

For (x, t) ∈ (0, 1) × [0, T), a simple computation shows

J_{t} − J_{xx} + \frac{1}{2}a_{1}α_{1}e^{α_{1}w}J = \frac{1}{4}w_{t}^{− \frac{3}{2}} w_{tx}^{2} + \frac{1}{2}εa_{1}α_{1}e^{α_{1}w}w_{x} ≥ 0.  (2.6)

By the comparison principle [12], we have J ≥ 0 and hence

w_{t}(1, t) ≥ ε^{2}w_{x}^{2}(1, t) = ε^{2}e^{2α_{2}w(1,t)}, t ∈ [0, T).  (2.7)

Integrating (2.7) from t to T, we get (2.2) immediately. □

Proof of Theorem 1.1. It suffices to choose initial data (u_{0}, v_{0}) such that u blows up while v remains bounded. At first, fix v_{0} ≥ 0 and take K = max_{[0,1]} v_{0} = v_{0}(1), N = \frac{1}{K} e^{2β_{2}K} + 3. Thus, w(x, t) which solves (2.3) is a subsolution of u. Since Proposition 1.1, there exists initial data u_{0} such that w blows up at a finite time T′. Now, for the fixed v_{0}, retake u_{0}(x) = w(x, T′ − ε), the u blows up in a finite time T ≤ ε.

If v remain bounded by v < 2K for t ∈ [0, T], the proof is complete.

Otherwise, Let t_{0} be the first time such that max_{[0,1]} v(·, t_{0}) = v(1, t_{0}) = 2K. Now, we introduce the following cut-off function:

ṽ(x, t) = \{ \begin{matrix}v(x, t), & (x, t) ∈ [0, 1] × [0, t_{0}], \\ 2K, & (x, t) ∈ [0, 1] × [t_{0}, T]. \end{matrix}  (2.8)

Corresponding, let ũ(x, t) solve

\begin{matrix}ũ_{t} = ũ_{xx} − a_{1}e^{α_{1}ũ}, & (x, t) ∈ (0, 1) × (0, T̃), \\ ũ_{x}(1, t) = e^{α_{2}ũ(1,t)}e^{pṽ(1,t)}, ũ_{x}(0, t) = 0, & t ∈ (0, T̃), \\ ũ(x, 0) = u_{0}(x), & x ∈ [0, 1], \end{matrix}  (2.9)

where T̃ is the blow-up time of ũ satisfying T̃ ≥ T. By Lemma 2.1,

ũ(1, t) = max_{[0,1]} ũ(·, t) ≤ log C(T̃ − t)^{− \frac{1}{2α_{2}}} , 0 < t < T̃.  (2.10)

Therefore,

u(1, t) = ũ(1, t) ≤ log C(T̃ − t)^{− \frac{1}{2α_{2}}} ≤ log C(T − t)^{− \frac{1}{2α_{2}}} , 0 < t ≤ t_{0}.  (2.11)

Let Γ(x, t) be the fundamental solution of the heat equation in [0, 1], namely

Γ(x, t) = \frac{1}{2\sqrt{πt}} exp \{\frac{−x^{2}}{4t} \} .  (2.12)

It is know that Γ satisfies (see [8])

\begin{matrix}∫_{0}^{1} Γ(x − y, t − z)dy ≤ 1, \\ ∫_{z}^{t} Γ(1, t − τ)\frac{1}{2(t − τ)}dτ ≤ C^{∗}\sqrt{t − z}, ∫_{z}^{t} Γ(0, t − τ)dτ = \frac{1}{\sqrt{π}} \sqrt{t − z}, \\ \frac{∂Γ}{∂ν_{y}} (x − y, t − τ) = \frac{x − y}{2(t − τ)}Γ(x − y, t − τ), x, y ∈ [0, 1], 0 ≤ z < t. \end{matrix}  (2.13)

By the Greeen’s identity with (1.1) for v,

\begin{matrix}v(x, t) = ∫_{0}^{1} Γ(x−y, t−z)v(y, z)dy + ∫_{z}^{t} ∫_{0}^{1} Γ(x−y, t−τ) (−a_{2}e^{β_{1}v(y,τ)}) dydτ \\ + ∫_{z}^{t} \frac{∂v}{∂x}(1, τ)Γ(x − 1, t − τ)dτ − ∫_{z}^{t} \frac{∂Γ}{∂ν_{y}} (x − 1, t − τ)v(1, τ)dτ \\ + ∫_{z}^{t} \frac{∂Γ}{∂ν_{y}} (x, t − τ)v(0, τ)dτ, \end{matrix} (2.14)

where 0 ≤ z < t < T, 0 < x < 1. With z = 0 and x → 1, it follows that

\begin{matrix}v(x, t) = ∫_{0}^{1} Γ(1−y, t)v(y, 0)dy + ∫_{0}^{t} ∫_{0}^{1} Γ(x−y, t−τ) (−a_{2}e^{β_{1}v(y,τ)}) dydτ \\ + ∫_{0}^{t} e^{qu(1,τ)+β_{2}v(1,τ)}Γ(0, t − τ)dτ + ∫_{0}^{t} v(0, τ)Γ(1, t − τ)\frac{1}{2(t − τ)}dτ. \end{matrix} (2.15)

By (2.11), we have furthermore

v(1, t_{0}) ≤ v_{0}(1) + C_{0}e^{β_{2}v(1,t_{0})} ∫_{0}^{t_{0}}(t_{0} − τ)^{− \frac{q}{2α_{2}} − \frac{1}{2}} dτ + C^{∗}\sqrt{t_{0}}v(1, t_{0}).  (2.16)

Since q < α_{2}, the integral term in (2.16) is smaller than 1/(NC_{0}) with \sqrt{t_{0}} ≤ \sqrt{T} ≤ 1/(NC^{∗}) if we choose u_{0} large to make T sufficiently small. This yields

\frac{N − 1}{N}v(1, t_{0}) ≤ v_{0}(1) + \frac{1}{N} e^{β_{2}v(1,t_{0})}.  (2.17)

Consequently,

\frac{2(N − 1)}{N}K ≤ K + \frac{1}{N} e^{2β_{2}K},  (2.18)

and hence

N ≤ \frac{1}{K} e^{2β_{2}K} + 2,  (2.19)

a contradiction. □

3 Proof of Theorem 1.2

We begin with a Lemma to prove Theorem 1.2.

Lemma 3.1 Let u be a solution of

\begin{matrix}u_{t} = u_{xx} − a_{1}e^{α_{1}u}, & (x, t) ∈ (0, 1) × (0, T), \\ u_{x}(1, t) ≤ Le^{α_{2}u(1,t)}, u_{x}(0, t) = 0, & t ∈ (0, T), \\ u(x, 0) = u_{0}(x), & x ∈ [0, 1], \end{matrix}  (3.1)

where a_{1} > 0, α_{i} ≥ 0, i=1,2 and L is a positive constant. If u blows up at a finite time, then either 2α_{2} > α_{1} or 2α_{2} = α_{1}, a_{1} ≤ \frac{α_{1}}{4} min\{c_{1}^{2}/(4c_{2}^{2}), λ_{0}c_{3}^{2}/c_{2}^{2}\}. Furthermore,

u(1, t) = max_{[0,1]} u(·, t) ≥ log C(T − t)^{− \frac{1}{2α_{2}}} , as t → T.  (3.2)

Proof. The blow-up of u implies either 2α_{2} > α_{1} or 2α_{2} = α_{1}, a_{1} ≤ \frac{α_{1}}{4} min\{c_{1}^{2}/(4c_{2}^{2}), λ_{0}c_{3}^{2}/c_{2}^{2}\} by Proposition 1.1.

By the Green’s identity, similarly to (2.14)

\begin{matrix}u(x, t) ≤ ∫_{0}^{1} Γ(x − y, t − z)u(y, z)dy + L ∫_{z}^{t} e^{α_{2}u(1,τ)}Γ(x − 1, t − τ)dτ \\ − ∫_{z}^{t} \frac{∂Γ}{∂ν_{y}} (x − 1, t − τ)u(1, τ)dτ + ∫_{z}^{t} \frac{∂Γ}{∂ν_{y}} (x, t − τ)u(0, τ)dτ, \end{matrix}  (3.3)

where 0 < z < t < T, 0 < x < 1. Let x → 1 with the jumping relations to obtain

\begin{matrix}\frac{1}{2}u(1, t) ≤ ∫_{0}^{1} Γ(1 − y, t − z)u(y, z)dy + L ∫_{z}^{t} e^{α_{2}u(1,τ)}Γ(0, t − τ)dτ \\ + ∫_{z}^{t} \frac{∂Γ}{∂ν_{y}} (1, t − τ)u(0, τ)dτ \\ ≤ u(1, z) + \frac{L}{\sqrt{π}} \sqrt{T − z}e^{α_{2}u(1,t)} + C^{∗}\sqrt{T − z}u(1, t). \end{matrix}  (3.4)

For any z ∈ (0, T) with C^{∗}\sqrt{T − z} ≤ 1/4, choose t ∈ (z, T) such that \frac{1}{4}u(1, t) − u(1, z) ≥ C_{0} > 0. Then

C_{0} ≤ \frac{L}{\sqrt{π}} \sqrt{T − t}e^{α_{2}u(1,t)},  (3.5)

which implies (3.2). □

Proof of Theorem 1.2. Since v ≤ K for (x, t) ∈ [0, 1] × [0, T), we have

\begin{matrix}u_{t} = u_{xx} − a_{1}e^{α_{1}u}, & (x, t) ∈ (0, 1) × (0, T), \\ u_{x}(1, t) ≤ e^{pK}e^{α_{2}u(1,t)}, u_{x}(0, t) = 0, & t ∈ (0, T), \\ u(x, 0) = u_{0}(x), & x ∈ [0, 1]. \end{matrix}  (3.6)

Then, we obtain from Lemma 3.1 that 2α_{2} > α_{1} or 2α_{2} = α_{1} with a_{1} ≤ \frac{α_{1}}{4} min \{\frac{c_{1}^{2}}{4c_{2}^{2}} , \frac{λ_{0}c_{3}^{2}}{c_{2}^{2}} \}, and moreover,

u(1, t) = max_{[0,1]} u(·, t) ≥ log C(T − t)^{− \frac{1}{2α_{2}}} , as t → T.  (3.7)

Next, let us show q < α_{2}. Due to (2.14), we have by letting x → 1 that

v(1, t) ≥ ∫_{z}^{t} e^{qu(1,τ)+β_{2}v(1,τ)}Γ(0, t − τ)dτ − a_{2} ∫_{z}^{t} ∫_{0}^{1} Γ(1 − y, t − τ)e^{β_{1}v(y,τ)}dydτ, (3.8)

and so,

v(1, t) ≥ C_{1} ∫_{z}^{t}(T − τ)^{− \frac{q}{2α_{2}} − \frac{1}{2}} dτ − a_{2}e^{β_{1}v(1,τ)}.  (3.9)

The boundedness of v(1, t) as t → T requires that q < α_{2}. □

Acknowledgements

The authors would like to thank the referee for the careful reading of this paper and for the valuable suggestions to improve the presentation and style of the paper.

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(Received June 10, 2010)