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For appropriate initial datau0,v0, there are solutions to 1.1 that blowup in a finite time T < ∞inL ∞-norm, that is, lim sup t → T u ·,t ∞+v ·,t ∞ However, we note that a priori, there

Volume 2007, Article ID 75258, 7 pages

doi:10.1155/2007/75258

Research Article

Simultaneous versus Nonsimultaneous Blowup for a System

of Heat Equations Coupled Boundary Flux

Mingshu Fan and Lili Du

Received 5 November 2006; Revised 18 January 2007; Accepted 23 March 2007

Recommended by Gary M Lieberman

This paper deals with a semilinear parabolic system in a bounded interval, completely coupled at the boundary with exponential type We characterize completely the range of parameters for which nonsimultaneous and simultaneous blowup occur

Copyright © 2007 M Fan and L Du This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

In this paper, we consider the positive blowup solution to the following parabolic prob-lem:

u t = u xx, v t = v xx, (x,t) ∈(0,L) ×(0,T),

− u x(0,t) = e p11u(0,t)+p12v(0,t), − v x(0,t) = e p21u(0,t)+p22v(0,t), t ∈(0,T),

(1.1)

where we assume the parametersp i j ≥0 (i, j =1, 2),p11+p22> 0 and p21+p12> 0 which

ensure that (1.1) completely coupled with the nontrivial nonlinear boundary flux The initial values u0(x), v0(x) are positive, nontrivial, bounded, and compatible with the

boundary data and smooth enough to guarantee thatu, v are regular.

The study of reaction-diffusion systems has received a great deal of interest in recent years and has been used to model, for example, heat transfer, population dynamics, and chemical reactions (see [1] and references therein) The parabolic system like (1.1) can

be used to describe, for example, heat propagations in mixed solid nonlinear media with nonlinear boundary flux The nonlinear Nuemann boundary values in (1.1), coupling

the two heat equations, represent some cross-boundary flux LetT denote the maximal

existence time for the solution (u, v) If it is infinite, we say that the solution is global.

For appropriate initial datau0,v0, there are solutions to (1.1) that blowup in a finite time

T < ∞inL ∞-norm, that is,

lim sup

t → T



u( ·,t)

∞+v( ·,t)

∞



However, we note that a priori, there is no reason for both componentsu and v should

go to infinity simultaneously at timeT In this paper, our first purpose is to show that

for some certain choice of parametersp i j, there are some initial data for which one of the components remains bounded, while the other blows up (we denote this phenomenon

as nonsimultaneous blowup), and for others both components blowup simultaneously Moreover, we give the complete classification of the simultaneous and nonsimultane-ous blowups by the parametersp i j Nonsimultaneous blowup phenomenon for the heat equations with nonlinear power-like-type boundary conditions was carried out in [2–4] Let us examine what is known in blowup for the heat equations with nonlinear boundary conditions before presenting our results In [5], Deng obtained the blowup rate maxΩu( ·,t) = O(log(T − t) −1/2p21), maxΩv( ·,t) = O(log(T − t) −1/2p12) for the following problem (withp11=0 andp22=0):

∂u

∂η = e p11u+p12v, ∂v

∂η = e p21u+p22v, (x,t) ∈ ∂Ω ×(0,T),

(1.3)

In [6], Zhao and Zheng considered the problem (1.3) with p21> p11and p12> p22 and obtained the blowup rates However, whenever there is blowup, both components be-come unbounded at the same time (see [6, Lemma 2.2]) That is,u blows up in L ∞-norm

at timeT if and only if v also does so Nonsimultaneous blowup is therefore not possible

in this case

In order to study the nonsimultaneous blowup phenomena for system (1.1), we need

to make further assumptions on the initial data:

u0,v0≥ δ1> 0, u 0(x),v 0(x) ≤0, u 0(x),v0(x) ≥ δ2> 0 for x ∈[0,L]. (1.4) Firstly, we give a set of parameters for which nonsimultaneous blowup indeed occurs

Theorem 1.1 There exists a pair of suitable initial data ( u0,v0) such that nonsimultaneous

blowup occurs if and only if p11> p21or p22> p12.

Corollary 1.2 If p11≤ p21and p22≤ p12, then u and v blowup at the same time for any pairs of initial data.

However, in this case, we do not exclude the possibility of exceptional solutions with simultaneous blowup In fact, when p11> p21 and p22> p12, this implies that each of the components may blowup by itself, then there exists a pair of initial data for which simultaneous blowup indeed occurs

Theorem 1.3 If p11> p21and p22> p12, both simultaneous and nonsimultaneous blowup may occur, provided that the initial data are chosen properly.

Theorem 1.4 (i) If p11> p21 and p22≤ p12, then there exists a finite time T, such that u

(ii) If p22> p12and p11≤ p21, then there exists a finite time T, such that v blows up at

T, while u remains bounded up to that time for every pair of initial data.

2 Proof of main results

Without loss of generality, we consider the casep11> p21, to show that there exists a pair

of initial data such thatu blows up at a finite time and v remains bounded up to this time

if and only ifp11> p21 The casep22> p12is handled in a completely analogous form In this paper, we usec and C to denote positive constants independent of t, which may be

different from line to line, even in the same line

Firstly, we give the estimate of blowup rate foru in the case u blows up while v

re-mains bounded, which plays an important role in the proof ofTheorem 1.1 We consider

e p12v(0,t)as a frozen coefficient and regard u as a blowup solution to the following auxiliary problem:

u t = u xx, (x,t) ∈(0,L) ×(0,T), − u x(0,t) = e p11u(0,t) h(t), t ∈(0,T),

whereu0satisfies (1.4) The functionh(t) ≥ δ > 0 is bounded, continuous and h (t) ≥0 The solutions of problem (2.1) blowup if p11> 0 (see [7]) First, we try to establish the upper blowup estimate

Lemma 2.1 If p11> 0 and u is a solution of ( 2.1 ), then there exists C0> 0 such that

x ∈[0,L] u( ·,t) ≤ − 1

initial data, we know thatu t > 0, u x ≥0, so we can chooseε small enough such that

J(x,0) = u t(x,0) − εu2

x(x,0) ≥0, x ∈[0,L],

− J x(0,t) −p11−2ε

h(t)e p11u(0,t) J(0,t)

= h (t)e p11u(0,t)+

p11−2ε

h3(t)e3p11u(0,t) ≥0, t ∈(0,T).

(2.3)

For (x,t) ∈(0,L) ×[0,T), a simple computation yields J t − J xx =2εu2

xx ≥0 DefineJ(x,t) =

we haveJ ≥0 Thus

u t(0,t) ≥ εu2

x(0,t) ≥ εδ2e2p11u(0,t), t ∈[0,t). (2.4)

In order to obtain that v is bounded when p11> p21, we introduce the following lemma, which has been proved in [2, Section 3]

Lemma 2.2 Consider the following system with K1> 0:

z t = z xx, (x,t) ∈(0,L) ×(0,T), − z x(0,t) = K1(T − t) − p21/2p11, t ∈(0,T),

(2.5)

If p21< p11, then there exists T small enough such that the solution of ( 2.5 ) verifies

0<t<T

z( ·,t)

∞ ≤v0(·)

for given ε > 0 and v0> 0 In particular, z is bounded.

Next, we consider the auxiliary problem

w t = w xx, (x,t) ∈(0,L) ×0,T0 

,

− w x(0,t) = C − p21/2p11

0 e p22w(0,t)(T − t) − p21/2p11, t ∈(0,T0),

w x(L,t) =0, t ∈0,T0



(2.7)

whereC0is defined in (2.2)

Lemma 2.3 Assume p11> p21, and let w solve ( 2.7 ), then for given ε and v0, w satisfies ( 2.6 ) provided that T is sufficiently small In particular, w is bounded.

0 e p22 ( v0 ∞+ε) ChooseT small enough that (2.6) holds, thenz is a supersolution of (2.7) By comparison

to make the blowup timeT satisfy (2.2) and (2.6), and we have

v t = v xx, (x,t) ∈(0,L) ×(0,T),

− v x(0,t) ≤ C − p21/2p11

0 e p22v(0,t)(T − t) − p21/2p11, t ∈(0,T),

(2.8)

By comparison principle,v ≤ w in (0,L) ×(0,T) Hence v is bounded.

Next, we assume thatu blows up in finite time T, while v remains bounded for (x,t) ∈

the lower blowup estimate of problem (2.1) firstly Let us defineM(t) = u( ·,t) ∞ =

ϕ M(y,s) = e u(ay,bs+t) − M(t), 0≤ y ≤ L

a,− t

wherea = e − p11M,b = e −2p11M Since p11> 0 and u blows up at T, then a,b 0 ast  T.

The functionϕ M satisfies 0≤ ϕ M ≤1, (ϕ M)s ≥0,ϕ M(0, 0)=1, and



ϕ M

s =ϕ M

y y − Aϕ M, (y,s) ∈



0,L a



×



− t

b, 0

,

−ϕ M

y(0,s) = ϕ p11 +1

ϕ M

y



L

a,s



=0, s ∈



− t

b, 0

, (2.10)

whereA = bu2

x(ay,bs + t) ≤ bu2

x(0,bs + t) = h2(bs + t) Noticing that h(bs + t) is bounded,

by Schauder estimate, we see thatϕ M is uniformly bounded inC2+α,1+α for someα > 0

(see [9]) Consequently, (ϕ M)s(0, 0)≤ C, which yields

x ∈[0,L] u( ·,t) ≥ − 1

2p11logC1(T − t), for 0< t < T, (2.11) whereC1is a positive constant

We suppose on the contrary that p11≤ p21, then from [2, Lemma 3.2], the solution

of (2.5) blows up atT Choose K1≤ C − p21/2p11

1 , whereC1is defined in (2.11), thenv is a

supersolution of problem (2.5), which contradicts the fact thatv remains bounded up to

the timeT Therefore, if u blows up while v remains bounded, then p11> p21 

Finally, we will prove that there are two regions of the parameters where nonsimulta-neous blowup occurs for any initial data Before proving this, we would like to give the blowup set of (1.1) provided that p11,p22> 0, which will play an important role in the

proof ofTheorem 1.4

Lemma 2.4 Under the assumptions of ( 1.4 ), then the point x = 0 is the only blowup point

of ( 1.1 ) provide that p11,p22> 0.

generality, we may assume that maxx ∈[0,L] u( ·,t) = u(0,t) → ∞, ast → T Assume on the

contrary thatu blows up at another point x ∗ > 0 as t → T, that is, limsup t → T u(x ∗,t) = ∞ Sinceu(x,t) is nonincreasing in x, limsup t → T u(x,t) = ∞for anyx ∈[0,x ∗] SetJ(x,t) =

u x+σ(L − x)e p11u, for (x,t) ∈[0,L] ×[0,T), where σ is a small constant to be determined.

Noticing thatu0is nontrivial, from the assumptions onu0(x) in (1.4), we haveu 0(x) <

0 provide thatx L and t ∈(0,T) We choose σ small enough such that

J(x,0) ≤ u 0(x) + σ(L − x)e p11 maxx ∈(0,L) u0 (x) ≤0, x ∈(0,L),

J(0,t) = − e p11u(0,t)+p12v(0,t)+σLe p11u(0,t) ≤ e p11u(0,t)(σL −1)≤0, t ∈(0,T),

(2.12)

On the other hand, a simple computation yields

J t − J xx =2p11σe p11u u x − p2

11σe p11u u2

x ≤0, for (x,t) ∈(0,L) ×(0,T). (2.13)

Application of the maximum principle to (2.12)-(2.13) ensures thatJ(x,t) ≤0, for (x,t) ∈

(0,L) ×(0,T) Namely, − e − p11u u x ≥ σ(L − x).

Integrating from 0 tox ∗yields 0< 0x ∗ σ(L − x)dx ≤(1/ p11)e − p11u(x ∗,t),t ∈(0,T) The

fact that lim supt → T u(x ∗,t) = ∞andp11> 0 lead to a contradiction Therefore, u blows

up at a single pointx =0, and so does the solution (u,v) of problem (1.1) 

simultane-ous blowup does not occur in this case Suppose on the contrary that there exist initial data (u0,v0) such that u and v blowup simultaneously Let us define M(t) = u(0,t) =

that

ϕ M(y,s) = e u(ay,bs+t) − M(t), ψ N(y,s) = e v(cy,ds+t) − N(t),

− t

b,− t

d

wherea2= b = e −(2p11 +1)M −2p12N,c2= d = e −2p22N −(2p21 +1)M The pair of function (ϕ M,ψ N) satisfies 0≤ ϕ M,ψ N ≤1,ϕ M(0, 0)= ψ N(0, 0)=1 and (ϕ M)s, (ψ N)s ≥0, and is the solution

of the parabolic problem



ϕ M

s =ϕ M

y y − Aϕ M, 

ψ N

s =ψ N

y y − Bψ N,

−ϕ M)(0,s) = e − M(t) ϕ p11 +1

M (0,s)ψ p12

N (0,s), −ψ N



(0,s) = e − M(t) ψ p22 +1

N (0,s)ϕ p21

M (0,s),

(2.15) whereA = bu2

x(ay,bs + t) ≤ bu2

x(0,bs + t) ≤ e −2M(t),B = dv2

x(ay,bs + t) ≤ dv2

x(0,bs + t) ≤

e −2M(t)

With the same idea of the proof ofTheorem 1.1, by the well-known Schauder esti-mates, it is easy to see that there exists a positive constantC such that for sufficiently large

M and N,



ϕ M

s(0, 0)≤ C, 

ψ N

Next, we claim that there exists a positive constantc such that for every pair of large

M, N,



ϕ M

To prove this claim, suppose on the contrary there should be a sequence { ϕ M j } such that (ϕ M j)s(0, 0)→0 asM j,N j → ∞ Asϕ M j is uniformly bounded inC2+α,1+α(see [9]), passing to a subsequence if necessary, we obtain a positive functionϕ such that ϕ M j →

ϕ y(0,s) =0 in (0, +∞)×(−∞, 0] We setw = ϕ sasw satisfies the heat equation, with the

boundary conditionw y(0,s) = w(0,0) =0 We conclude using Hopf ’s lemma thatw ≡0, that is,ϕ(y,s) does not depend on s and then ϕ(y) ≡1 Hence,u(ay,bs + t) ≡ M(t) for

all (y,s) ∈(0, +∞)×(−∞, 0] ast → T, which leads to a contradiction with the fact that

u of the system (1.1) possesses a single blowup point atx =0 provided thatp11> 0 (see

Lemma 2.4) Thus we arrive at inequality (2.17)

Inequalities (2.16) and (2.17) imply that ce2p12N ≤ e −2(p11 +1)M M (t), e −2p22N N (t) ≤

Ce2(p21 +1)M Noticing thatp11> p21andp22≤ p12, a direct computation yields

1

2

p21− p11

e2(p21− p11 )M(t) ≥

⎧

⎪

⎪

C

2

p12− p22

e2(p12− p22 )N(t)+C  forp22< p12,

(2.18)

whereC > 0 and C  are constants independent oft Obviously, they contradict the

as-sumption thatu and v blowup simultaneously.

Acknowledgment

The authors would like to thank the referees for the valuable comments and careful read-ing

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Mingshu Fan: Department of Mathematics, Jincheng College of Sichuan University,

Chengdu 611731, China

Email address:mingshufan@sohu.com

Lili Du: Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China;

School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China

Email addresses:du nick@sohu.com ; lldu@scut.edu.cn

u of the system (1.1) possesses a single... os, and J D Rossi, “Non -simultaneous blow-up for a quasilinear parabolic

system with reaction at the boundary, ” Communications on Pure and Applied Analysis, vol 4,... L Zhao and S Zheng, “Blow-up estimates for system of heat equations coupled via nonlinear

boundary flux,” Nonlinear Analysis, vol 54, no 2, pp 251–259, 2003.