Nonlinear Analysis 69 (2008) 24–34www.elsevier.com/locate/na

Uniform blow-up profiles and boundary layer for a parabolic systemwith localized sources

Jun Zhoua,∗, Chunlai Mub

a School of Mathematics and Statistics, Southwest University, Chongqing 400715, PR Chinab School of Mathematics and Physics, Chongqing University, Chongqing, 400044, PR China

Received 19 November 2006; accepted 25 April 2007

Abstract

This paper deals with the Dirichlet problem for a parabolic system with localized sources. We first obtain some sufficientconditions for blow-up in finite time, and then deal with the possibilities of simultaneous blow-up under suitable assumptions.Moreover, when simultaneous blow-up occurs, we also establish the uniform blow-up profiles in the interior and estimate theboundary layer.c© 2007 Elsevier Ltd. All rights reserved.

MSC: 35K57; 35K60; 35B40

Keywords: Diffusion system; Localized source; Simultaneous blow-up; Uniform blow-up profiles; Boundary layer

1. Introduction

In this paper, we study the blow-up properties of solutions to the following parabolic system coupled via localizednonlinear sources:

ut = ∆u + vq1(x(t), t)ep1u(x(t),t), (x, t) ∈ Ω × (0, T ),

vt = ∆v + u p2(x(t), t)eq2v(x(t),t), (x, t) ∈ Ω × (0, T ),

u(x, t) = v(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ),

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω ,

(1.1)

where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω and x(t) : R+→ Ω is Holder continuous. The initial

data u0(x), v0(x) ∈ C0(Ω) are nonnegative nontrivial functions. The parameters pi , qi (i = 1, 2) are nonnegativenumbers with p2q1 > 0.

The system (1.1) describes chemical reaction–diffusion process in which the nonlinear reaction in a dynamicalsystem takes place only at a single (or sometimes several) site(s). As an example, the influence of the defect structures

∗ Corresponding author.E-mail address: zhoujun math@hotmail.com (J. Zhou).

0362-546X/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2007.04.038

J. Zhou, C. Mu / Nonlinear Analysis 69 (2008) 24–34 25

on a catalytic surface can be modeled by a similar equation (see [1,12]). Similar phenomena are also frequentlyobserved in biological systems, for instance on chemically active membranes (see [4] and references therein). Theadditional motivation for this study comes from parabolic inverse problems and so-called nonclassical equations(see [2,3]). Using the methods used in [7,14] we know that (1.1) has a local nonnegative solution, and that thecomparison principle is true.

In recently years, a lot of efforts have been devoted to localized (nonlocal) problems, the blow-up properties ofsolution to the following problem with a single equation:

ut = ∆u + f (u(x(t), t)), (x, t) ∈ Ω × (0, T ),

u(x, 0) = u0(x) ≥ 0, x ∈ Ω ,(1.2)

with homogeneous Dirichlet conditions have been discussed by many authors (see [5,6,14–16] and the referencestherein). In particular, Souplet [15] proved that if f (u) = u p with p > 1, then

limt→T

(T − t)1/(p−1)u(x, t) = limt→T

(T − t)1/(p−1)‖u(·, t)‖∞ = (p − 1)−1/(p−1), (1.3)

uniformly on the compact subset of Ω , and if f (u) = eu , then

limt→T

| ln(T − t)|−1u(x, t) = limt→T

| ln(T − t)|−1‖u(·, t)‖∞ = 1, (1.4)

uniformly on the compact subset of Ω , where T is the blow-up time of u.As far as the system is concerned, let us introduce some specific models. Firstly, for when x(t) ≡ x0, where x0 is

a fixed point in Ω , in [10], Lin et al. studied the blow-up properties of solutions to the parabolic system

ut = ∆u + ev(x0,t), vt = ∆v + eu(x0,t), (x, t) ∈ Ω × (0, T ),

u(x, t) = v(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ),

u(x, 0) = u0(x) ≥ 0, v(x, 0) = v0(x) ≥ 0, x ∈ Ω .

(1.5)

They first proved that the solution (u, v) of (1.5) blows up in finite T . And for a special case Ω = B(0, R) and x0 = 0,they obtained the following blow-up rate estimates:

− ln(T − t) − v0(0) ≤ sup u(·, t) ≤ C(1 − ln(T − t)), t ∈ [0, T ),

− ln(T − t) − u0(0) ≤ sup v(·, t) ≤ C(1 − ln(T − t)), t ∈ [0, T ),(1.6)

for some constant C > 0 and all 0 < t < T .Furthermore, the following systems coupled with localized nonlinear source of exponent type:

ut = ∆u + ep1u(x0,t)+q1v(x0,t), (x, t) ∈ Ω × (0, T ),

vt = ∆v + ep2u(x0,t)+q2v(x0,t), (x, t) ∈ Ω × (0, T ),

u(x, t) = v(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ),

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω ,

(1.7)

and of power type:

ut = ∆u + u p1(x0, t)vq1(x0, t), (x, t) ∈ Ω × (0, T ),

vt = ∆v + u p2(x0, t)vq2(x0, t), (x, t) ∈ Ω × (0, T ),

u(x, t) = v(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ),

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω ,

(1.8)

were consider in [8,9] respectively. The parameters pi , qi (i = 1, 2) are nonnegative numbers with p2q1 > 0. Theyobtain the critical exponents and establish the uniform blow-up profiles in the interior.

26 J. Zhou, C. Mu / Nonlinear Analysis 69 (2008) 24–34

For the Neumann boundary version of the following problem:

ut = ∆u + f (v(x0, t)), vt = ∆v + g(u(x0, t)), (x, t) ∈ Ω × (0, T ),

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω ,(1.9)

we remark on a recent paper [17], in which Xiang et al. establish the blow-up criterion and estimated theblow-up profiles of blow-up solutions for f (v(x0, t)) = v p(x0, t), g(u(x0, t)) = uq(x0, t) or f (v(x0, t)) =

λepv(x0,t), g(u(x0, t)) = µequ(x0,t), respectively.Secondly, for system with moving localized sources, the study is not very extensive to our knowledge. We only

introduce a recent paper [18], in which, Xiang et al. study the following two types of diffusion systems:

ut = ∆u + λep1u(x(t),t)+q1v(x(t),t), (x, t) ∈ RN× (0, T ),

vt = ∆v + µep2u(x(t),t)+q2v(x(t),t), (x, t) ∈ RN× (0, T ),

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ RN ,

(1.10)

and

ut = ∆u + u p1(x(t), t)vq1(x(t), t), (x, t) ∈ RN× (0, T ),

vt = ∆v + u p2(x(t), t)vq2(x(t), t), (x, t) ∈ RN× (0, T ),

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ RN .

(1.11)

They first give the blow-up criterion, and then deal with the possibilities of simultaneous blow-up or nonsimultaneousblow-up under some suitable assumptions. Moreover, when simultaneous blow-up occurs, they also establish theprecise blow-up rate estimates.

Finally, we remark that in connection with the local parabolic systems of semilinear type

ut = ∆u + vq1 ep1u, vt = ∆v + u p2eq2v, (x, t) ∈ Ω × (0, T ), (1.12)

and of degenerate type

ut = ∆um+ vq1 ep1u, vt = ∆vn

+ u p2 eq2v, (x, t) ∈ Ω × (0, T ), (1.13)

with initial–boundary values, where the parameters pi , qi (i = 1, 2) are nonnegative numbers with p2q2 > 0 andm, n > 1, a lot of works have been done in the past few years on the blow-up of their solutions (see [11,13] and thereferences therein).

Motivated by the above cited papers, in this paper, we investigate the blow-up properties of the Dirichlet problem(1.1). Note that x(t) ≡ x0 in [8–10] is very important; it ensures that one may directly construct a supersolution orsubsolution. For system (1.1), we use some ideas of Souplet [15] and define a pair of functions (u, v), which are theintegrals of the reaction terms in time and depend only on the time variable t (see Section 2 for details).

First, we give the global existence (blow-up) properties of the nonnegative solution of problem (1.1).

Theorem 1.1. Assume that (u, v) is a nonnegative solution of problem (1.1);

(1) if p1 = q2 = 0 and p2q1 ≤ 1, then (u, v) exists globally;(2) if p1 = q2 = 0 and p2q1 > 1, then (u, v) blows up in finite time and the blow-up set is any compact subset of Ω ;(3) if p1 > 0 or q2 > 0, then (u, v) blows up in finite time and the blow-up set is any compact subset of Ω .

Let T be the maximal existence time to the solution (u, v) of system (1.1). (2) and (3) of Theorem 1.1 suggestT < ∞ and limt→T ‖u‖∞ + ‖v‖∞ = +∞. However, there is no reason for both components of the system to blowup simultaneously. Since we consider the uniform blow-up profiles and boundary layer for problem (1.1), it needs thatu and v blow up simultaneously. To this end, in the following theorem, we give a sufficient condition and a necessarycondition which lead u and v to blow up simultaneously.

Theorem 1.2. Assume (u, v) is a nonnegative solution of problem (1.1);

(1) if p1 = q2 = 0 and p2q1 > 1, then u and v blow up simultaneously;

J. Zhou, C. Mu / Nonlinear Analysis 69 (2008) 24–34 27

(2) suppose u and v blow up simultaneously, then p1 > 0 if and only if q2 > 0.

Theorem 1.3. Assume (u, v) is a nonnegative solution of problem (1.1), which blows up simultaneously in finite timeT . Then following statements hold uniformly on any compact subset of Ω :

(1) if p1 = q2 = 0 and p2q1 > 1, then limt→T u(x, t)(T − t)α = limt→T ‖u(·, t)‖∞(T − t)α = α1

p2q1−1 βq1

p2q1−1 ,

limt→T v(x, t)(T − t)β = limt→T ‖v(·, t)‖∞(T − t)β = β1

p2q1−1 αp2

p2q1−1 , where α = (q1 + 1)/(p2q1 − 1), β =

(p2 + 1)/(p2q1 − 1);(2) if p1 > 0 and q2 > 0, then limt→T u(x, t)| ln(T − t)|−1

= limt→T ‖u(·, t)‖∞| ln(T − t)|−1= 1/p1;

limt→T v(x, t)| ln(T − t)|−1= limt→T ‖v(·, t)‖∞| ln(T − t)|−1

= 1/q2.

Theorem 1.4. Assume (1) of Theorem 1.3 holds. Then for any given constant K > 0, there exist some constantC2 ≥ C1 > 0 and some 0 < t0 < T such that:

(i) C1d(x)

√T −t

‖u(·, t)‖∞ ≤ u(x, t) ≤ C2d(x)

√T −t

‖u(·, t)‖∞,

(ii) C1d(x)

√T −t

‖v(·, t)‖∞ ≤ v(x, t) ≤ C2d(x)

√T −t

‖v(·, t)‖∞ hold for all (x, t) ∈ Ω×[tK , T ) satisfying d(x) ≤ K√

T − t .

Theorem 1.5. Assume (2) of Theorem 1.3 holds.

(i) There exists some constant C > 0 and some t1 ∈ (0, T ) such that (u, v) satisfies u(x,t)‖u(·,t)‖∞

,v(x,t)

‖v(·,t)‖∞≤

C d(x)√

(T −t) ln(T −t), ∀(x, t) ∈ Ω × [t1, T ).

(ii) For any given constant K > 0, there exist some constant m(K ) > 0 and some 0 < tK = t (K ) < T , suchthat u(x, t), v(x, t) ≥ m(K )

d(x)√

T −tholds for all (x, t) ∈ Ω × [tK , T ) satisfying d(x) ≤ K

√T − t , where

d(x) = dist(x, ∂Ω).

Theorem 1.6. Assume (1) of Theorem 1.3 holds; if α > 1 and β > 1, then there exists some constant C > 0 andsome t0 ∈ (0, T ) such that

(1) u(x, t) ≥

(1 − C T −t

d2(x)

)‖u(·, t)‖∞, ∀(x, t) ∈ Ω × [t0, T ),

(2) v(x, t) ≥

(1 − C T −t

d2(x)

)‖v(·, t)‖∞, ∀(x, t) ∈ Ω × [t0, T ).

This paper is organized as follows. In the next section, we give some preliminaries, which are important to ourproofs. In Section 3, we consider the blow-up properties of problem (1.1) and prove Theorems 1.1 and 1.2. InSection 4, we consider the uniform blow-up profiles and boundary layer for problem (1.1) and prove Theorems 1.3–1.6.

2. Preliminaries

In this section, we give some preliminaries, which are useful for proving Theorems 1.2–1.6.For convenience, define

g1(t) = vq1(x(t), t)ep1u(x(t),t), Gi (t) =

∫ t

0gi (s)ds, i = 1, 2,

g2(t) = u p2(x(t), t)eq2v(x(t),t), Hi (t) =

∫ t

0Gi (s)ds, i = 1, 2.

(2.1)

The following lemma comes from Lemma 4.4, Theorem 4.1, and Lemma 4.5 of [15], and plays an important role inour remaining proofs.

Lemma 2.1. Suppose that u and v blow up simultaneously in finite time T . Then we have:

(1) there exists C1 ≥ 0 such that u(x, t) ≤ C1 + G1(t), v(x, t) ≤ C1 + G2(t) in Ω × [T/2, T );(2) the following arguments hold uniformly on any compact subset of Ω : limt→T

u(x,t)G1(t)

= limt→T‖u(·,t)‖∞

G1(t)=

1, limt→Tv(x,t)G2(t)

= limt→T‖v(·,t)‖∞

G2(t)= 1;

28 J. Zhou, C. Mu / Nonlinear Analysis 69 (2008) 24–34

(3) define Kρ :≡ y ∈ Ω : dist(y, ∂Ω) ≥ ρ > 0; then there exists a constant C2 > 0 such that for(x, t) ∈ Kρ × [T/2, T ), u(x, t) ≥ G1(t) −

C2ρN+1 (1 + H1(t)), v(x, t) ≥ G2(t) −

C2ρN+1 (1 + H2(t)).

Remark 2.1. If we only assume u blows up in finite time T , then the corresponding conclusions about u still hold.Similar conclusions are true for v.

For ease of writing in the following, we introduce two definitions.

Definition 2.1. Let f (t) and g(t) be two functions defined in [0, T ); if there exists some constant k1 ≥ k2 > 0 andsome t0 ∈ (0, T ) such that k2 f (t) ≤ g(t) ≤ k1 f (t), ∀t ∈ [t0, T ), we say f (t) is equipotent with g(t) as t → T , andwritten by f (t) w g(t).

Definition 2.2. Let f (t) and g(t) be two functions defined in [0, T ); if limt→Tf (t)g(t) = 1, we say we say f (t) is

equivalent to g(t), and written by f (t) ∼ g(t).

Obviously, the equipotent relation has the following properties:

(1) if f (t) w g(t), then f µ(t) w gµ(t) for all µ ∈ R;(2) if f (t) w g(t) and g(t) w h(t), then f (t) w h(t);(3) if f (t) w g(t) and r(t) w s(t), then r(t) f (t) w s(t)g(t);(4) if limt→T

f (t)g(t) = C > 0, then f (t) w g(t);

(5) if f (t) w g(t), then∫ t

a f (s)ds w∫ t

a g(s)ds for all a ∈ [0, T );(6) assume that f (t) → ∞ and g(t) → ∞ as t → T , or f (t) → 0 and g(t) → 0 as t → T ; if f (t) w g(t), then

ln f (t) ∼ ln g(t).

Lemma 2.2. Suppose that u and v blow up simultaneously in finite time T . Then we have: (1) if p1 > 0,then g1(t) = G ′

1(t) w Gq12 ep1G1 ; (2) if q2 > 0, then g2(t) = G ′

2(t) w G p21 eq2G2 ; (3) if p1 = 0, then

g1(t) = G ′

1(t) w Gq12 ; (4) if q2 = 0, then g2(t) = G ′

2(t) w G p21 .

Proof. We only prove Case (1); the other cases can be proved in a similar way. By the results (1) and (2) of Lemma 2.1,we have

0 ≤ g1(t) ≤ (C1 + G2(t))q1ep1(C1+G1(t)), lim

t→TGi (t) = ∞, i = 1, 2.

Thus, (C1 + G1(t)) ∼ G1(t) and (C1 + G2(t)) ∼ G2(t). Consequently, there exist some constant k1 > 0 andt1 ∈ (0, T ), such that

0 ≤ g1(t) ≤ k1Gq12 (t)ep1G1(t), ∀t ∈ [t1, T ). (2.2)

On the other hand, by virtue of (3) of Lemma 2.1, we know that for all (x, t) ∈ Kρ × [T/2, T ),

g1(t) ≥

G2(t) −

C2

ρN+1 (1 + H2(t))

q1

exp

p1G1(t) −p1C2

ρN+1 (1 + H1(t))

. (2.3)

Since Gi (t) (i = 1, 2) is nondecreasing, it follows that for any ε > 0,

Hi (t)

Gi (t)=

∫ t0 Gi (s)ds

Gi (t)=

∫ T −ε

0 Gi (s)ds +∫ t

T −εGi (s)ds

Gi (t)≤

∫ T −ε

0 Gi (s)ds

Gi (t)+ ε, i = 1, 2.

Taking into account that limt→T Gi (t) = ∞, we deduce that

Hi (t)

Gi (t)→ 0, as t → T, i = 1, 2. (2.4)

Then it follows from (2.3) and (2.4) that there exist some constant C > 0 and some t2 ∈ (0, T ), such that

g1(t) ≥ CGq12 (t) exp

p1G1(t) −

p1C2

ρN+1 (1 + H1(t))

, ∀(x, t) ∈ Kρ × [t2, T ). (2.5)

J. Zhou, C. Mu / Nonlinear Analysis 69 (2008) 24–34 29

In view of (2.4) and the fact limt→T G2(t) = ∞, we see that there exists t3 ∈ (t2, T ), such that

G2(t) ≥ 1, p1G1(t) −p1C2

ρN+1 (1 + H1(t)) ≥p1

2G1(t), ∀t ∈ [t3, T ),

and hence, G ′

1(t) = g1(t) ≥ Cep1G1(t)/2, ∀t ∈ [t3, T ). Therefore, G1(t) ≤2p1

| ln(T − t)| + ln(p1C) − ln 2, ∀t ∈

[t3, T ). Subsequently,

H1(t) =

∫ t

0G1(s)ds ≤ C, t ∈ [0, T ). (2.6)

Combining (2.5) and (2.6) asserts the existence of a constant k2 satisfying

g1(t) ≥ k2Gq12 (t)ep1G1(t), t ∈ [t3, T ). (2.7)

Setting t0 = maxt1, t3, then by virtue of (2.2) and (2.7), we get k2Gq12 (t)ep1G1(t) ≤ g1(t) ≤ k1Gq1

2 (t)ep1G1(t), i.e.g1(t) = G ′

1(t) w Gq12 (t)ep1G1(t). The proof of Lemma 2.2 is complete.

Lemma 2.3. Suppose that u and v blow up simultaneously in finite time T . If p1 > 0 and q2 > 0, then we haveG p2

1 e−p1G1(t) w Gq12 e−q2G2(t).

Proof. From Lemma 2.2, we have G ′

1(t) w Gq12 ep1G1(t) and G ′

2(t) w G p21 eq2G2(t), and hence

G ′

1(t)

G ′

2(t)w

Gq12 ep1G1(t)

G p21 eq2G2(t)

. (2.8)

Integrating (2.8) yields∫ T

te−p1G1(s)G p2

1 (s)G ′

1(s)ds w∫ T

te−q2G2(s)Gq1

2 G ′

2(s)ds. (2.9)

Since p1 > 0 and limt→T G1(t) = ∞, the integral∫ T

t e−p1G1(s)G p21 (s)G ′

1(s)ds is convergent and

limt→T∫ T

t e−p1G1(s)G p21 (s)G ′

1(s)ds = 0. Notice that

limt→T

∫ Tt e−p1G1(s)G p2

1 (s)G ′

1(s)ds

e−p1G1(t)G p21 (t)

= limz→∞

∫∞

z s p2 e−p1sds

z p2 e−p1z = limz→∞

−z p2e−p1z

p2z p2−1e−p1z − p1z p2 e−p1z=

1p1

,

namely,

p1

∫ T

te−p1G1(s)G p2

1 (s)G ′

1(s)ds ∼ e−p1G1(t)G p21 (t). (2.10)

Analogously, we have

q2

∫ T

te−q2G2(s)Gq1

2 (s)G ′

2(s)ds ∼ e−q2G2(t)Gq12 (t). (2.11)

Therefore, by joining (2.9) with (2.10) and (2.11), we reach the desired conclusion. The proof or Lemma 2.3 iscomplete.

3. Blow-up properties

In this section, we consider the global existence and blow-up properties for problem (1.1). We define

u(x, t) = u(t) =

∫ t

0vq1(x(s), s)ep1u(x(s),s)ds, u = u + C0, (3.1)

v(x, t) = v(t) =

∫ t

0u p2(x(s), s)eq2v(x(s),s)ds, v = v + C0, (3.2)

30 J. Zhou, C. Mu / Nonlinear Analysis 69 (2008) 24–34

where C0 = ‖u0‖∞ + ‖v0‖∞. It is clear that

ut − ∆u = ut − ∆u = vq1(x(t), t)ep1u(x(t),t)= ut − ∆u,

vt − ∆v = vt − ∆v = u p2(x(t), t)eq2v(x(t),t)= vt − ∆v,

0 = u|∂Ω = u|∂Ω ≤ u|∂Ω , 0 = v|∂Ω = v|∂Ω ≤ v|∂Ω .

Since u(x, 0) = 0 ≤ u0(x) ≤ u(x, 0) and v(x, 0) = 0 ≤ v0(x) ≤ v(x, 0), we have u(x, t) ≤ u(x, t) ≤ u(x, t) by themaximal principle as long as (u, v) exists. In particular, we obtain u(t) ≤ u(x(t), t) ≤ u(t) and v(t) ≤ v(x(t), t) ≤

v(t), which play important roles in the following proof.

Proof of Theorem 1.1. It follows from (u, v) ≥ (u, v) ≥ (u, v) that

ut = vq1(x(t), t)ep1u(x(t),t)≤ vq1(t)ep1u(t),

vt = u p2(x(t), t)eq2v(x(t),t)≤ u p2(t)eq2v(t),

(3.3)

and

ut = vq1(x(t), t)ep1u(x(t),t)≥ vq1(t)ep1u(t),

vt = u p2(x(t), t)eq2v(x(t),t)≥ u p2(t)eq2v(t).

(3.4)

Since the nonlinearities of (3.3) and (3.4) are not necessarily locally Lipschitz, it is not clear whether the comparisonprinciple is applicable in all cases. However, by u(0) = v(0) = C0 > 0, we may apply the comparison principle for(u, v) and (w, z) (for the construction of (w, z), see the following arguments).

Case (i) p1 = q2 = 0 and p2q1 < 1. Take w(x, t) = (C + t)p and z(x, t) = (C + t)q , where C > 0 is to be chosen,p = 2(q1 + 1)/(1 − p2q1), q = 2(p2 + 1)/(1 − p2q1). It follows that p, q > 0, p − qq1 = 2 and q − pp2 = 2. Aftera simple computation, we get

wt = p(C + t)p−1 > (C + t)p−2= zq1 , w|∂Ω ≥ u|∂Ω ,

zt = q(C + t)q−1 > (C + t)q−2= w p2 , z|∂Ω ≥ v|∂Ω ,

for C sufficiently large. If we further take C > 0 such that C p≥ u(0) = C0 and Cq

≥ v(0) = C0, we have(u, v) ≤ (w, z) for t ≥ 0 by (3.3) and (3.4) and the comparison principle. This shows that (u, v) exists globally.

Case (ii) p1 = q2 = 0 and p2q1 = 1. Given large C > C0, we take w(x, t) = Cept , z = Ceqt , where C is to bechosen later and q = q−1

1 p = p2 p. Then

wt = Cpept > Cq1eqq1t= zq1 , w|∂Ω ≥ u|∂Ω ,

zt = Cqeqt > C p2 epp2t= w p2 , z|∂Ω ≥ v|∂Ω ,

for p and C large enough. Therefore, arguments analogous to those for Case (i) imply that the solution of problem(1.1) is global.

Case (iii) p1 = q2 = 0 and p2q1 > 1. By the claim in the proof of Theorem 1.2, we have

ut ≥ vq1 ≥ cu(p2q1+q1)/(q1+1), t ∈ (T0, T ),

namely,

−q1 + 1

p2q1 − 1

(u(1−p2q1)/(q1+1)

)t≥ c, t ∈ (T0, T ).

Integrating this inequality from T0 to t , we obtain

u(1−p2q1)/(q1+1)(T0) − u(1−p2q1)/(q1+1)(t) ≥ c(t − T0).

Since p2q1 > 1, the above inequality cannot hold for all time. Therefore, (u, v) blows up in finite time, and so does(u, v). It follows from u ≤ u and v ≤ v that the blow-up set is any compact subset of Ω .

Case (iv) p1 > 0 or q2 > 0. Without loss of generality, we suppose T > 1 and p1 > 0. Using (3.4) and noticingthat v(t) is nondecreasing in t , we have u(t) ≥ vq1(1)ep1u(t), t > 1. Combining this inequality with u(1) > 0, wesee that (u, v) blows up in finite time. Therefore, the solution (u, v) of problem (1.1) also blows up in finite time. Theproof of Theorem 1.1 is complete.

J. Zhou, C. Mu / Nonlinear Analysis 69 (2008) 24–34 31

Proof of Theorem 1.2. (1) First we claim the following facts. If p1 = q2 = 0 and p2q1 > 1, then there existconstants C > c > 0 such that

cu p2+1≤ vq1+1

≤ Cu p2+1, cu p2+1≤ vq1+1

≤ Cu p2+1, t ∈ (T0, T ).

We prove that vq1+1≤ Cu p2+1 holds for all t ∈ (T0, T ). Let

J (t) = Cu p2+1

p2 + 2−

vq1+1

q1 + 1,

where C is a constant to be determined. Notice that

vt = vt ≤ u p2 = (u + C0)p2 .

By (3.3) and (3.4), we easily obtain

J ′(t) = Cu p2ut − vq1vt ≥ Cu p2vq1 − vq1(u + C0)p2 = Cu p2vq1

(1 −

1

C

(u + C0

u

)p2)

.

On the other hand, note that u and v are nondecreasing in t . Taking C with

C ≥ max(

u(T0) + C0

u(T0)

)p2

,

(p2 + 1q1 + 1

)vq1+1(T0)

u p2+1(T0)

,

we see that J ′(t) > 0 for all t ∈ (T0, T ) and J (T0) > 0, which implies J (t) > 0 for all t ∈ (T0, T ). Therefore, thereexists some constant C > 0 such that vq1+1

≤ Cu p2+1 holds for all t ∈ (T0, T ). The other inequalities can be provedby similar arguments.

Now we come to the proof of (1) of Theorem 1.2. The assumption of Theorem 1.2 ensures that (u, v) blows up infinite time. The simultaneous blow-up can be directly obtained by the claim.

(2) When p1 > 0, we assume on the contrary that q2 = 0. As at the beginning of the proof of Lemma 2.3, weobtain∫ T

te−p1G1(s)G p2

1 G ′

1(s)ds w∫ T

tGq1

2 G ′

2(s)ds.

Since limt→T Gi (t) = ∞ (i = 1, 2), the integral∫ T

t e−p1G1(s)G p21 G ′

1(s)ds is convergent, and the integral∫ Tt Gq1

2 G ′

2(s)ds is divergent. This is impossible. Therefore, q2 > 0. In a similar way, we can also prove that p1 > 0when q2 > 0. The proof Theorem 1.2 is complete.

4. Uniform blow-up profiles and boundary layer

In this section, we consider the uniform blow-up profiles and boundary layer of problem (1.1) and proveTheorems 1.3–1.6.

Proof of Theorem 1.3. (1) When p1 = q2 = 0 and p2q1 > 1. By the conclusions (3)–(4) of Lemma 2.2, we get

G p21 G ′

1(t) ∼ Gq12 G ′

2(t). (4.1)

Integrating (4.1), we have

(q1 + 1)G p2+11 (t) ∼ (p2 + 1)Gq1+1

2 (t). (4.2)

Then the conclusion of (3) of Lemma 2.2 together with (4.2) shows that

G ′

1(t) ∼ Gq12 (t) ∼

(q1 + 1p2 + 1

)q1/(q1+1)

Gq1(p2+1)

q1+1

1 (t). (4.3)

Notice that p2q1 > 1 and limt→T G1(t) = ∞; from (4.3) it follows that

G1(t) ∼

(q1 + 1

p2q1 − 1

) q1+1p2q1−1

(p2 + 1q1 + 1

) q1p2q1−1

(T − t)−

q1+1p2q1−1 ,

32 J. Zhou, C. Mu / Nonlinear Analysis 69 (2008) 24–34

i.e. G1(t) ∼ α1/(p2q1−1)βq1/(p2q1−1)(T − t)−α , where α and β are defined in Theorem 1.3. Applying the conclusion(2) of Lemma 2.1, we see that in any compact subset of Ω there holds u(x, t) ∼ α1/(p2q1−1)βq1/(p2q1−1)(T − t)−α .Similarly, we can get the estimate of v(x, t), that is v(x, t) ∼ β1/(p2q1−1)α p2/(p2q1−1)(T − t)−β .

(2) When p1 > 0 and q2 > 0. If u and v blow up simultaneously in finite time T , then the conclusions (1), (2) ofLemmas 2.2 and 2.3 show that

G ′

1 w Gq12 ep1G1 = ep1G1(Gq1

2 e−q2G2)eq2G2 w ep1G1(G p21 e−p1G1)eq2G2 = G p2

1 eq2G2 w G ′

2.

Integrating the above yields G1(t) w G2(t). Hence,

G ′

1(t) w Gq11 (t)ep1G(t). (4.4)

Integrating (4.4), we find that∫ T

tG−q1

1 (s)e−p1G1(s)G ′

1(s)ds w T − t. (4.5)

Note that∫ T

t G−q11 (s)e−p1G1(s)G ′

1(s)ds w G−q11 (t)e−p1G1(t); (4.5) shows that

G−q11 (t)e−p1G1(t) w T − t,

that is

−p1G1(t) − q1 ln G1(t) ∼ ln(T − t). (4.6)

Recalling the result (2) of Lemma 2.1, we find that

−p1u(x, t) − q1 ln u(x, t) ∼ ln(T − t) (4.7)

holds on any compact subset of Ω . Since −p1u(x, t) − q1 ln u(x, t) ∼ −p1u(x, t) hold uniformly on any compactsubset of Ω , we have −p1u(x, t) ∼ ln(T − t). Like with above process, we get the estimate for v, that is−q2v(x, t) ∼ ln(T − t). The proof of Theorem 1.3 is complete.

From the proof of Theorem 1.3, we can also obtain the following lemma.

Lemma 4.1. Under the assumptions of Theorem 1.3, the following statements hold:

(1) when p1 > 0 and q2 > 0, if u and v blow up simultaneously in finite time T , then

G ′

1(t) w1

T − t, G ′

2(t) w1

T − t;

(2) if p1 = q2 = 0 and p2q1 > 1, then

G ′

1(t) ∼ αp2q1

p2q1−1 βq1

p2q1−1 (T − t)−q1β , G ′

2(t) ∼ βp2q1

p2q1−1 αp2

p2q1−1 (T − t)−p2α.

In the rest of this paper, we investigate the boundary layer profiles of solutions to problem (1.1) and proveTheorems 1.4–1.6. First, we cite some important results of [15]. Define d(x) :≡ dist(x, ∂Ω), and consider thefollowing problem:

wt = ∆w + ϕ(t), (x, t) ∈ Ω × (0, T ),

w(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ),

w(x, 0) = w0(x), x ∈ Ω ,

(4.8)

where ϕ(t) is a nonnegative function. Define Φ(t) =∫ t

0 ϕ(s)ds.

Definition 4.1. We say a nonnegative function ϕ(t) is standard if ϕ(t) satisfies ϕ(t)Φ(t) w (T − t)−1.

J. Zhou, C. Mu / Nonlinear Analysis 69 (2008) 24–34 33

Lemma 4.2 (Proposition 4.7 of [15]). Let w be a solution of Problem (4.8). If w blows up in finite time T , then thereexists some constant c > 0 and some t0 ∈ (0, T ), such that

w(x, t) ≤ cd(x)√

Φ(t) sup0≤s≤t

ϕ(s), ∀(x, t) ∈ Ω × (t0, T ).

Lemma 4.3 (Proposition 4.8 of [15]). Let w be a solution of problem (4.8) which blows up in finite time T . Ifϕ(t) ≥ C(T − t)−1 holds for some constant C > 0, then for every K > 0, there exist some constant m(K ) > 0and tK = t (K ) ∈ (0, T ), such that

w(x, t) ≥ m(K )d(x)

√T − t

,

holds for all (x, t) ∈ Ω × [tK , T ) and d(x) satisfying d(x) ≤ K√

T − t .

By virtue of Lemmas 4.2 and 4.3, we can verify Theorem 1.5.

Proof of Theorem 1.5. From the conclusion (1) of Lemma 4.1 it follows that there exist positive constants k1 ≤ k2and some t1 ∈ (0, T ), such that

k1(T − t)−1≤ gi (t) = G ′

i (t) ≤ k2(T − t)−1, ∀t ∈ (t1, T ), i = 1, 2. (4.9)

By virtue (2) of Theorem 1.3 and the result (2) of Lemma 2.1, we have

G1(t) ∼ ‖u(·, t)‖∞ ∼1p1

| ln(T − t)|, G2(t) ∼ ‖v(·, t)‖∞ ∼1q2

| ln(T − t)|, (4.10)

and hence, there exist positive constants k3 ≤ k4 and some t2 ∈ (0, T ), such that

k3| ln(T − t)| ≤ Gi (t) ≤ k4| ln(T − t)|, ∀t ∈ (t2, T ), i = 1, 2. (4.11)

Set t∗ = maxt0, t1, t2; then by using Lemma 4.2 and (4.9)–(4.11), there exists a constant C > 0, such that

u(x, t) ≤ cd(x)√

G1(t) sup0≤s≤t

g1(s) ≤ Cd(x)‖u(·, t)‖∞

√(T − t)| ln(T − t)|

, (x, t) ∈ (t∗, T ), (4.12)

v(x, t) ≤ cd(x)√

G2(t) sup0≤s≤t

g2(s) ≤ Cd(x)‖v(·, t)‖∞

√(T − t)| ln(T − t)|

, (x, t) ∈ (t∗, T ), (4.13)

which implies the conclusion (i) of Theorem 1.5. The conclusion (ii) of Theorem 1.5 is a direct consequence (3.9) andLemma 4.3. The proof of Theorem 1.5 is complete.

To deduce Theorems 1.4 and 1.6, we need the following lemmas.

Lemma 4.4 (Proposition 4.5 of [15]). Let w be a solution of problem (4.8) which blows up in finite time T . If ϕ(t)is standard, then for any given constant K > 0, there exist some constants C2 ≥ C1 > 0 and some t0 ∈ (0, T ), suchthat

C1d(x)

√T − t

‖w(·, t)‖∞ ≤ w(x, t) ≤ C2d(x)

√T − t

‖w(·, t)‖∞,

holds for all (x, t) ∈ Ω × [t0, T ) and d(x) satisfying d(x) ≤ K√

T − t .

Lemma 4.5 (Proposition 4.6 of [15]). Assume that ϕ(t) is continuous on [0, T ), and Holder continuous in (0, T ). Letw0(x) ∈ C0(Ω), and w be the blow-up solution to problem (4.8) in Ω × (0, T ). If Φ(t) is standard, then there existssome constant C3 > 0 and some t0 ∈ (0, T ), such that(

1 − C3T − t

d2(x)

)‖w(·, t)‖∞ ≤ w(x, t), (x, t) ∈ Ω × [t0, T ).

34 J. Zhou, C. Mu / Nonlinear Analysis 69 (2008) 24–34

Proof of Theorem 1.4. By virtue of the conclusion of Lemma 4.1, we have g1(t) = G ′

1(t) w (T − t)−q1β ,g1(t) = G ′

2(t) w (T − t)−p2α , and hence, g1(t) and g2(t) are standard. Using Lemma 4.4, we draw the conclusion ofTheorem 1.4 immediately.

Proof of Theorem 1.6. By the conclusion (2) of Lemma 2.1 and the conclusion (1) of Theorem 1.3, we know that∫ t

0G1(s)ds = O((T − t)1−α),

∫ t

0G2(s)ds = O((T − t)1−β), as t → T .

For α > 1 and β > 1 it follows G1(t) and G2(t) are standard. Moreover, g1(t) and g2(t) are continuous on [0, T ),and Holder continuous in (0, T ). Theorem 1.6 follows from Lemma 4.5.

Acknowledgements

This work was supported in part by NNSF of China (10571126) and in part by the Program for New CenturyExcellent Talents in University.

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