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

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

Uniform blow-up profiles and boundary layer for a parabolic system with localized sources Jun Zhou a,∗ , Chunlai Mu b a School of Mathematics and Statistics, Southwest University, Chongqing 400715, PR China b 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 sufficient conditions 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 the boundary 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 localized nonlinear sources: u t = ∆u + v q1 (x(t), t)e p1 u(x(t),t) ,

(x, t) ∈ Ω × (0, T ),

q2 v(x(t),t)

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

u(x, 0) = u 0 (x),

v(x, 0) = v0 (x),

(1.1)

x ∈ Ω,

where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω and x(t) : R+ → Ω is H¨older continuous. The initial data u 0 (x), v0 (x) ∈ C0 (Ω ) are nonnegative nontrivial functions. The parameters pi , qi (i = 1, 2) are nonnegative numbers with p2 q1 > 0. The system (1.1) describes chemical reaction–diffusion process in which the nonlinear reaction in a dynamical system 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 [email protected] (J. Zhou). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter 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 frequently observed in biological systems, for instance on chemically active membranes (see [4] and references therein). The additional 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 the comparison principle is true. In recently years, a lot of efforts have been devoted to localized (nonlocal) problems, the blow-up properties of solution to the following problem with a single equation: u t = ∆u + f (u(x(t), t)), u(x, 0) = u 0 (x) ≥ 0,

(x, t) ∈ Ω × (0, T ),

(1.2)

x ∈ Ω,

with homogeneous Dirichlet conditions have been discussed by many authors (see [5,6,14–16] and the references therein). In particular, Souplet [15] proved that if f (u) = u p with p > 1, then lim (T − t)1/( p−1) u(x, t) = lim (T − t)1/( p−1) ku(·, t)k∞ = ( p − 1)−1/( p−1) ,

t→T

t→T

(1.3)

uniformly on the compact subset of Ω , and if f (u) = eu , then lim | ln(T − t)|−1 u(x, t) = lim | ln(T − t)|−1 ku(·, t)k∞ = 1,

t→T

t→T

(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 u t = ∆u + ev(x0 ,t) ,

vt = ∆v + eu(x0 ,t) ,

u(x, t) = v(x, t) = 0,

(x, t) ∈ ∂Ω × (0, T ),

u(x, 0) = u 0 (x) ≥ 0,

(x, t) ∈ Ω × (0, T ),

v(x, 0) = v0 (x) ≥ 0,

(1.5) x ∈ Ω.

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) − u 0 (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: u t = ∆u + e p1 u(x0 ,t)+q1 v(x0 ,t) ,

(x, t) ∈ Ω × (0, T ),

p2 u(x0 ,t)+q2 v(x0 ,t)

vt = ∆v + e u(x, t) = v(x, t) = 0, u(x, 0) = u 0 (x),

, (x, t) ∈ Ω × (0, T ), (x, t) ∈ ∂Ω × (0, T ),

v(x, 0) = v0 (x),

(1.7)

x ∈ Ω,

and of power type: u t = ∆u + u p1 (x0 , t)v q1 (x0 , t),

(x, t) ∈ Ω × (0, T ),

vt = ∆v + u (x0 , t)v (x0 , t),

(x, t) ∈ Ω × (0, T ),

p2

q2

u(x, t) = v(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ), u(x, 0) = u 0 (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 p2 q1 > 0. They obtain the critical exponents and establish the uniform blow-up profiles in the interior.

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

For the Neumann boundary version of the following problem: u t = ∆u + f (v(x0 , t)), u(x, 0) = u 0 (x),

vt = ∆v + g(u(x0 , t)),

v(x, 0) = v0 (x),

(x, t) ∈ Ω × (0, T ),

x ∈ Ω,

(1.9)

we remark on a recent paper [17], in which Xiang et al. establish the blow-up criterion and estimated the blow-up profiles of blow-up solutions for f (v(x0 , t)) = v p (x0 , t), g(u(x0 , t)) = u q (x0 , t) or f (v(x0 , t)) = λe pv(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: u t = ∆u + λe p1 u(x(t),t)+q1 v(x(t),t) ,

(x, t) ∈ R N × (0, T ),

vt = ∆v + µe p2 u(x(t),t)+q2 v(x(t),t) ,

(x, t) ∈ R N × (0, T ),

u(x, 0) = u 0 (x),

v(x, 0) = v0 (x),

(1.10)

x ∈R , N

and u t = ∆u + u p1 (x(t), t)v q1 (x(t), t),

(x, t) ∈ R N × (0, T ),

vt = ∆v + u p2 (x(t), t)v q2 (x(t), t),

(x, t) ∈ R N × (0, T ),

u(x, 0) = u 0 (x),

v(x, 0) = v0 (x),

(1.11)

x ∈ RN .

They first give the blow-up criterion, and then deal with the possibilities of simultaneous blow-up or nonsimultaneous blow-up under some suitable assumptions. Moreover, when simultaneous blow-up occurs, they also establish the precise blow-up rate estimates. Finally, we remark that in connection with the local parabolic systems of semilinear type u t = ∆u + v q1 e p1 u ,

vt = ∆v + u p2 eq2 v ,

(x, t) ∈ Ω × (0, T ),

(1.12)

and of degenerate type u t = ∆u m + v q1 e p1 u ,

vt = ∆v n + u p2 eq2 v ,

(x, t) ∈ Ω × (0, T ),

(1.13)

with initial–boundary values, where the parameters pi , qi (i = 1, 2) are nonnegative numbers with p2 q2 > 0 and m, 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 the references 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 or subsolution. For system (1.1), we use some ideas of Souplet [15] and define a pair of functions (u, v), which are the integrals 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 p2 q1 ≤ 1, then (u, v) exists globally; (2) if p1 = q2 = 0 and p2 q1 > 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 suggest T < ∞ and limt→T kuk∞ + kvk∞ = +∞. However, there is no reason for both components of the system to blow up simultaneously. Since we consider the uniform blow-up profiles and boundary layer for problem (1.1), it needs that u and v blow up simultaneously. To this end, in the following theorem, we give a sufficient condition and a necessary condition 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 p2 q1 > 1, then u and v blow up simultaneously;

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

(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 time T . Then following statements hold uniformly on any compact subset of Ω : q1

1

(1) if p1 = q2 = 0 and p2 q1 > 1, then limt→T u(x, t)(T − t)α = limt→T ku(·, t)k∞ (T − t)α = α p2 q1 −1 β p2 q1 −1 , 1

p2

limt→T v(x, t)(T − t)β = limt→T kv(·, t)k∞ (T − t)β = β p2 q1 −1 α p2 q1 −1 , where α = (q1 + 1)/( p2 q1 − 1), β = ( p2 + 1)/( p2 q1 − 1); (2) if p1 > 0 and q2 > 0, then limt→T u(x, t)| ln(T − t)|−1 = limt→T ku(·, t)k∞ | ln(T − t)|−1 = 1/ p1 ; limt→T v(x, t)| ln(T − t)|−1 = limt→T kv(·, t)k∞ | 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 constant C2 ≥ C1 > 0 and some 0 < t0 < T such that: ku(·, t)k∞ ≤ u(x, t) ≤ C2 √d(x) ku(·, t)k∞ , (i) C1 √d(x) T −t T −t (ii) C1 √d(x) kv(·, t)k∞ ≤ v(x, t) ≤ C2 √d(x) kv(·, t)k∞ hold for all (x, t) ∈ Ω ×[t K , T ) satisfying d(x) ≤ K T −t T −t

√ 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) v(x,t) ku(·,t)k∞ , kv(·,t)k∞

≤

d(x) C √(T −t) , ∀(x, t) ∈ Ω × [t1 , T ). ln(T −t) (ii) For any given constant K > 0, there exist some constant m(K ) > 0 and some 0 < t K = t (K √ ) < T , such that u(x, t), v(x, t) ≥ m(K ) √d(x) T − t, where holds for all (x, t) ∈ Ω × [t , T ) satisfying d(x) ≤ K K T −t 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 and some t0 ∈ (0, T ) such that   ku(·, t)k∞ , ∀(x, t) ∈ Ω × [t0 , T ), (1) u(x, t) ≥ 1 − C dT2−t (x)   kv(·, t)k∞ , ∀(x, t) ∈ Ω × [t0 , T ). (2) v(x, t) ≥ 1 − C dT2−t (x) This paper is organized as follows. In the next section, we give some preliminaries, which are important to our proofs. In Section 3, we consider the blow-up properties of problem (1.1) and prove Theorems 1.1 and 1.2. In Section 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 Z t q1 p1 u(x(t),t) g1 (t) = v (x(t), t)e , G i (t) = gi (s)ds, i = 1, 2, 0 Z t g2 (t) = u p2 (x(t), t)eq2 v(x(t),t) , Hi (t) = G i (s)ds, i = 1, 2.

(2.1)

0

The following lemma comes from Lemma 4.4, Theorem 4.1, and Lemma 4.5 of [15], and plays an important role in our 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 + G 1 (t), v(x, t) ≤ C1 + G 2 (t) in Ω × [T /2, T ); (2) the following arguments hold uniformly on any compact subset of Ω : limt→T u(x,t) G 1 (t) = limt→T 1, limt→T v(x,t) G 2 (t)

=

∞ limt→T kv(·,t)k G 2 (t)

= 1;

ku(·,t)k∞ G 1 (t)

=

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

(3) define K ρ :≡ {y ∈ Ω : dist(y, ∂Ω ) ≥ ρ > 0}; then there exists a constant C2 > 0 such that for 2 2 (1 + H1 (t)), v(x, t) ≥ G 2 (t) − ρ CN +1 (1 + H2 (t)). (x, t) ∈ K ρ × [T /2, T ), u(x, t) ≥ G 1 (t) − ρ CN +1 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 and some 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 , and written by f (t) w g(t). Definition 2.2. Let f (t) and g(t) be two functions defined in [0, T ); if limt→T equivalent to g(t), and written by f (t) ∼ g(t).

f (t) g(t)

= 1, we say we say f (t) is

Obviously, the equipotent relation has the following properties: if f (t) w g(t), then f µ (t) w g µ (t) for all µ ∈ R; if f (t) w g(t) and g(t) w h(t), then f (t) w h(t); if f (t) w g(t) and r (t) w s(t), then r (t) f (t) w s(t)g(t); f (t) if limt→T g(t) = C > 0, then f (t) w g(t); Rt Rt (5) if f (t) w g(t), then a f (s)ds w 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). (1) (2) (3) (4)

Lemma 2.2. Suppose that u and v blow up simultaneously in finite time T . Then we have: (1) if p1 > 0, q p then g1 (t) = G 01 (t) w G 21 e p1 G 1 ; (2) if q2 > 0, then g2 (t) = G 02 (t) w G 1 2 eq2 G 2 ; (3) if p1 = 0, then q1 p2 0 0 g1 (t) = G 1 (t) w G 2 ; (4) if q2 = 0, then g2 (t) = G 2 (t) w G 1 . 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 + G 2 (t))q1 e p1 (C1 +G 1 (t)) ,

lim G i (t) = ∞,

t→T

i = 1, 2.

Thus, (C1 + G 1 (t)) ∼ G 1 (t) and (C1 + G 2 (t)) ∼ G 2 (t). Consequently, there exist some constant k1 > 0 and t1 ∈ (0, T ), such that 0 ≤ g1 (t) ≤ k1 G 21 (t)e p1 G 1 (t) , q

∀t ∈ [t1 , T ).

(2.2)

On the other hand, by virtue of (3) of Lemma 2.1, we know that for all (x, t) ∈ K ρ × [T /2, T ),  q1   C2 p1 C 2 g1 (t) ≥ G 2 (t) − N +1 (1 + H2 (t)) exp p1 G 1 (t) − N +1 (1 + H1 (t)) . ρ ρ Since G i (t) (i = 1, 2) is nondecreasing, it follows that for any ε > 0, R T −ε Rt Rt R T −ε G i (s)ds + T −ε G i (s)ds G i (s)ds Hi (t) 0 0 G i (s)ds = = ≤ 0 + ε, G i (t) G i (t) G i (t) G i (t)

(2.3)

i = 1, 2.

Taking into account that limt→T G i (t) = ∞, we deduce that Hi (t) → 0, G i (t)

as t → T, i = 1, 2.

Then it follows from (2.3) and (2.4) that there exist some constant C > 0 and some t2 ∈ (0, T ), such that   p1 C 2 q1 g1 (t) ≥ C G 2 (t) exp p1 G 1 (t) − N +1 (1 + H1 (t)) , ∀(x, t) ∈ K ρ × [t2 , T ). ρ

(2.4)

(2.5)

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

In view of (2.4) and the fact limt→T G 2 (t) = ∞, we see that there exists t3 ∈ (t2 , T ), such that G 2 (t) ≥ 1,

p1 G 1 (t) −

p1 p1 C 2 (1 + H1 (t)) ≥ G 1 (t), 2 ρ N +1

∀t ∈ [t3 , T ),

and hence, G 01 (t) = g1 (t) ≥ Ce p1 G 1 (t)/2 , ∀t ∈ [t3 , T ). Therefore, G 1 (t) ≤ [t3 , T ). Subsequently, Z t G 1 (s)ds ≤ C, t ∈ [0, T ). H1 (t) =

2 p1 {| ln(T

− t)| + ln( p1 C) − ln 2}, ∀t ∈

(2.6)

0

Combining (2.5) and (2.6) asserts the existence of a constant k2 satisfying g1 (t) ≥ k2 G 21 (t)e p1 G 1 (t) , q

t ∈ [t3 , T ).

(2.7) q k2 G 21 (t)e p1 G 1 (t)

Setting t0 = max{t1 , t3 }, then by virtue of (2.2) and (2.7), we get q g1 (t) = G 01 (t) w G 21 (t)e p1 G 1 (t) . The proof of Lemma 2.2 is complete.

≤ g1 (t) ≤

q k1 G 21 (t)e p1 G 1 (t) ,

i.e.



Lemma 2.3. Suppose that u and v blow up simultaneously in finite time T . If p1 > 0 and q2 > 0, then we have p q G 1 2 e− p1 G 1 (t) w G 21 e−q2 G 2 (t) . Proof. From Lemma 2.2, we have G 01 (t) w G 21 e p1 G 1 (t) and G 02 (t) w G 1 2 eq2 G 2 (t) , and hence q

p

G 01 (t) G 21 e p1 G 1 (t) . w p G 02 (t) G 1 2 eq2 G 2 (t) q

Integrating (2.8) yields Z T Z p e− p1 G 1 (s) G 1 2 (s)G 01 (s)ds w t

(2.8)

T t

e−q2 G 2 (s) G 21 G 02 (s)ds. q

(2.9)

RT p Since p1 > 0 and limt→T G 1 (t) = ∞, the integral t e− p1 G 1 (s) G 1 2 (s)G 01 (s)ds is convergent and R T − p G (s) p2 limt→T t e 1 1 G 1 (s)G 01 (s)ds = 0. Notice that R ∞ p −p s R T − p G (s) p2 1 1 s 2 e 1 ds G 1 (s)G 01 (s)ds −z p2 e− p1 z 1 t e = lim z p − p z = lim = lim , p 2 p −1 − p z p − p z − p G (t) z→∞ z→∞ p2 z 2 e 1 − p1 z 2 e 1 t→T z 2e 1 p1 e 1 1 G 1 (t) namely, T

Z p1 t

e− p1 G 1 (s) G 1 2 (s)G 01 (s)ds ∼ e− p1 G 1 (t) G 1 2 (t). p

p

Analogously, we have Z T q q q2 e−q2 G 2 (s) G 21 (s)G 02 (s)ds ∼ e−q2 G 2 (t) G 21 (t).

(2.10)

(2.11)

t

Therefore, by joining (2.9) with (2.10) and (2.11), we reach the desired conclusion. The proof or Lemma 2.3 is complete.  3. Blow-up properties In this section, we consider the global existence and blow-up properties for problem (1.1). We define Z t u(x, t) = u(t) = v q1 (x(s), s)e p1 u(x(s),s) ds, u = u + C0 ,

(3.1)

0

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

t

Z 0

u p2 (x(s), s)eq2 v(x(s),s) ds,

v = v + C0 ,

(3.2)

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

where C0 = ku 0 k∞ + kv0 k∞ . It is clear that u t − ∆u = u t − ∆u = v q1 (x(t), t)e p1 u(x(t),t) = u t − ∆u, v t − ∆v = v t − ∆v = u p2 (x(t), t)eq2 v(x(t),t) = vt − ∆v, 0 = u|∂ Ω = u|∂ Ω ≤ u|∂ Ω ,

0 = v|∂ Ω = v|∂ Ω ≤ v|∂ Ω .

Since u(x, 0) = 0 ≤ u 0 (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 the maximal 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 u t = v q1 (x(t), t)e p1 u(x(t),t) ≤ v q1 (t)e p1 u(t) ,

(3.3)

v t = u p2 (x(t), t)eq2 v(x(t),t) ≤ u p2 (t)eq2 v(t) , and u t = v q1 (x(t), t)e p1 u(x(t),t) ≥ v q1 (t)e p1 u(t) ,

(3.4)

v t = u p2 (x(t), t)eq2 v(x(t),t) ≥ u p2 (t)eq2 v(t) .

Since the nonlinearities of (3.3) and (3.4) are not necessarily locally Lipschitz, it is not clear whether the comparison principle 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 p2 q1 < 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 − p2 q1 ), q = 2( p2 + 1)/(1 − p2 q1 ). It follows that p, q > 0, p − qq1 = 2 and q − pp2 = 2. After a simple computation, we get w t = p(C + t) p−1 > (C + t) p−2 = z q1 , q−1

z t = q(C + t)

> (C + t)

q−2

=w , p2

w|∂ Ω ≥ u|∂ Ω , z|∂ Ω ≥ v|∂ Ω ,

for C sufficiently large. If we further take C > 0 such that C p ≥ u(0) = C0 and C q ≥ 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 p2 q1 = 1. Given large C > C0 , we take w(x, t) = Ce pt , z = Ceqt , where C is to be chosen later and q = q1−1 p = p2 p. Then w t = C pe pt > C q1 eqq1 t = z q1 ,

w|∂ Ω ≥ u|∂ Ω ,

>C e

z|∂ Ω ≥ v|∂ Ω ,

qt

z t = Cqe

p2 pp2 t

=w , p2

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 p2 q1 > 1. By the claim in the proof of Theorem 1.2, we have u t ≥ v q1 ≥ cu ( p2 q1 +q1 )/(q1 +1) ,

t ∈ (T0 , T ),

namely, −

q1 + 1  (1− p2 q1 )/(q1 +1)  u ≥ c, t p2 q 1 − 1

t ∈ (T0 , T ).

Integrating this inequality from T0 to t, we obtain u (1− p2 q1 )/(q1 +1) (T0 ) − u (1− p2 q1 )/(q1 +1) (t) ≥ c(t − T0 ). Since p2 q1 > 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 noticing that v(t) is nondecreasing in t, we have u(t) ≥ v q1 (1)e p1 u(t) , t > 1. Combining this inequality with u(1) > 0, we see that (u, v) blows up in finite time. Therefore, the solution (u, v) of problem (1.1) also blows up in finite time. The proof of Theorem 1.1 is complete. 

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

Proof of Theorem 1.2. (1) First we claim the following facts. If p1 = q2 = 0 and p2 q1 > 1, then there exist constants C > c > 0 such that cu p2 +1 ≤ v q1 +1 ≤ Cu p2 +1 ,

cu p2 +1 ≤ v q1 +1 ≤ Cu p2 +1 ,

t ∈ (T0 , T ).

We prove that v q1 +1 ≤ Cu p2 +1 holds for all t ∈ (T0 , T ). Let J (t) = C

u p2 +1 v q1 +1 − , p2 + 2 q 1 + 1

where C is a constant to be determined. Notice that v t = v t ≤ u p2 = (u + C0 ) p2 . By (3.3) and (3.4), we easily obtain J (t) = Cu u t − v v t ≥ Cu v p2

0

q1

p2 q 1

− v (u + C0 ) q1

p2

= Cu v

p2 q 1

 1−

1 C



u + C0 u

 p2 

.

On the other hand, note that u and v are nondecreasing in t. Taking C with      p2 + 1 v q1 +1 (T0 ) u(T0 ) + C0 p2 , C ≥ max , u(T0 ) q1 + 1 u p2 +1 (T0 ) we see that J 0 (t) > 0 for all t ∈ (T0 , T ) and J (T0 ) > 0, which implies J (t) > 0 for all t ∈ (T0 , T ). Therefore, there exists some constant C > 0 such that v q1 +1 ≤ Cu p2 +1 holds for all t ∈ (T0 , T ). The other inequalities can be proved by 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 in finite 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, we obtain Z T Z T q − p1 G 1 (s) p2 0 e G 1 G 1 (s)ds w G 21 G 02 (s)ds. t

t

RT p Since limt→T G i (t) = ∞ (i = 1, 2), the integral t e− p1 G 1 (s) G 1 2 G 01 (s)ds is convergent, and the integral R T q1 0 t G 2 G 2 (s)ds is divergent. This is impossible. Therefore, q2 > 0. In a similar way, we can also prove that p1 > 0 when 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 prove Theorems 1.3–1.6. Proof of Theorem 1.3. (1) When p1 = q2 = 0 and p2 q1 > 1. By the conclusions (3)–(4) of Lemma 2.2, we get p

q

G 1 2 G 01 (t) ∼ G 21 G 02 (t).

(4.1)

Integrating (4.1), we have p +1

(q1 + 1)G 1 2

q +1

(t) ∼ ( p2 + 1)G 21

(t).

Then the conclusion of (3) of Lemma 2.2 together with (4.2) shows that   q ( p2 +1) q1 + 1 q1 /(q1 +1) 1q1 +1 q G 01 (t) ∼ G 21 (t) ∼ (t). G1 p2 + 1 Notice that p2 q1 > 1 and limt→T G 1 (t) = ∞; from (4.3) it follows that   q1 +1   q1 q +1 p2 q1 −1 p2 + 1 p2 q1 −1 q1 + 1 − 1 (T − t) p2 q1 −1 , G 1 (t) ∼ p2 q 1 − 1 q1 + 1

(4.2)

(4.3)

32

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

i.e. G 1 (t) ∼ α 1/( p2 q1 −1) β q1 /( p2 q1 −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) ∼ α 1/( p2 q1 −1) β q1 /( p2 q1 −1) (T − t)−α . Similarly, we can get the estimate of v(x, t), that is v(x, t) ∼ β 1/( p2 q1 −1) α p2 /( p2 q1 −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) of Lemmas 2.2 and 2.3 show that q

q

p

p

G 01 w G 21 e p1 G 1 = e p1 G 1 (G 21 e−q2 G 2 )eq2 G 2 w e p1 G 1 (G 1 2 e− p1 G 1 )eq2 G 2 = G 1 2 eq2 G 2 w G 02 . Integrating the above yields G 1 (t) w G 2 (t). Hence, q

G 01 (t) w G 11 (t)e p1 G(t) .

(4.4)

Integrating (4.4), we find that T

Z t

Note that

G 1 1 (s)e− p1 G 1 (s) G 01 (s)ds w T − t. −q

RT t

(4.5)

G 1 1 (s)e− p1 G 1 (s) G 01 (s)ds w G 1 1 (t)e− p1 G 1 (t) ; (4.5) shows that −q

−q

G 1 1 (t)e− p1 G 1 (t) w T − t, −q

that is − p1 G 1 (t) − q1 ln G 1 (t) ∼ ln(T − t).

(4.6)

Recalling the result (2) of Lemma 2.1, we find that − p1 u(x, t) − q1 ln u(x, t) ∼ ln(T − t)

(4.7)

holds on any compact subset of Ω . Since − p1 u(x, t) − q1 ln u(x, t) ∼ − p1 u(x, t) hold uniformly on any compact subset of Ω , we have − p1 u(x, t) ∼ ln(T − t). Like with above process, we get the estimate for v, that is −q2 v(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 01 (t) w

1 , T −t

G 02 (t) w

1 ; T −t

(2) if p1 = q2 = 0 and p2 q1 > 1, then p2 q 1

q1

G 01 (t) ∼ α p2 q1 −1 β p2 q1 −1 (T − t)−q1 β ,

p2 q 1

p2

G 02 (t) ∼ β p2 q1 −1 α p2 q1 −1 (T − t)− p2 α .

In the rest of this paper, we investigate the boundary layer profiles of solutions to problem (1.1) and prove Theorems 1.4–1.6. First, we cite some important results of [15]. Define d(x) :≡ dist(x, ∂Ω ), and consider the following problem: wt = ∆w + ϕ(t), w(x, t) = 0,

(x, t) ∈ Ω × (0, T ),

(x, t) ∈ ∂Ω × (0, T ),

w(x, 0) = w0 (x),

(4.8)

x ∈ Ω,

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

Rt 0

ϕ(s)ds.

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

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 there exists some constant c > 0 and some t0 ∈ (0, T ), such that r w(x, t) ≤ cd(x) Φ(t) sup ϕ(s), ∀(x, t) ∈ Ω × (t0 , T ). 0≤s≤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 . If ϕ(t) ≥ C(T − t)−1 holds for some constant C > 0, then for every K > 0, there exist some constant m(K ) > 0 and t K = t (K ) ∈ (0, T ), such that d(x) w(x, t) ≥ m(K ) √ , T −t holds for all (x, t) ∈ Ω × [t K , 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 ≤ k2 and some t1 ∈ (0, T ), such that k1 (T − t)−1 ≤ gi (t) = G i0 (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 G 1 (t) ∼ ku(·, t)k∞ ∼

1 | ln(T − t)|, p1

G 2 (t) ∼ kv(·, t)k∞ ∼

1 | ln(T − t)|, q2

(4.10)

and hence, there exist positive constants k3 ≤ k4 and some t2 ∈ (0, T ), such that k3 | ln(T − t)| ≤ G i (t) ≤ k4 | ln(T − t)|, Set

t∗

∀t ∈ (t2 , T ), i = 1, 2.

(4.11)

= max{t0 , t1 , t2 }; then by using Lemma 4.2 and (4.9)–(4.11), there exists a constant C > 0, such that ku(·, t)k∞ , (T − t)| ln(T − t)|

(x, t) ∈ (t ∗ , T ),

(4.12)

kv(·, t)k∞ , (T − t)| ln(T − t)|

(x, t) ∈ (t ∗ , T ),

(4.13)

r u(x, t) ≤ cd(x) G 1 (t) sup g1 (s) ≤ Cd(x) √ 0≤s≤t

r v(x, t) ≤ cd(x) G 2 (t) sup g2 (s) ≤ Cd(x) √ 0≤s≤t

which implies the conclusion (i) of Theorem 1.5. The conclusion (ii) of Theorem 1.5 is a direct consequence (3.9) and Lemma 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 ), such that d(x) d(x) kw(·, t)k∞ ≤ w(x, t) ≤ C2 √ kw(·, t)k∞ , C1 √ T −t T −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 H¨older continuous in (0, T ). Let w0 (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 exists some constant C3 > 0 and some t0 ∈ (0, T ), such that   T −t 1 − C3 2 kw(·, t)k∞ ≤ w(x, t), (x, t) ∈ Ω × [t0 , T ). d (x)

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 01 (t) w (T − t)−q1 β , g1 (t) = G 02 (t) w (T − t)− p2 α , and hence, g1 (t) and g2 (t) are standard. Using Lemma 4.4, we draw the conclusion of Theorem 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 Z t Z t G 2 (s)ds = O((T − t)1−β ), as t → T. G 1 (s)ds = O((T − t)1−α ), 0

0

For α > 1 and β > 1 it follows G 1 (t) and G 2 (t) are standard. Moreover, g1 (t) and g2 (t) are continuous on [0, T ), and H¨older 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 Century Excellent Talents in University. References [1] K. Bimpong-Bota, P. Ortoleva, J. Ross, Far-from-equilibrum phenomena at local sites of reactions, J. Chem. Phys. 60 (1974) 3124. [2] J.R. Cannon, H.M. Yin, A class of non-linear non-classical parabolic equations, J. Differential Equations 79 (1989) 266C288. [3] J.M. Chadam, H.M. Yin, An iteration procedure for a class of integrodifferential equations of parabolic type, J. Integral Equations Appl. 2 (1989) 31–47. [4] J.M. Chadam, H.M. Yin, A diffusion equation with localized chemical reactions, Proc. Edinb. Math. Soc. 37 (1993) 101–118. [5] J.R. Cannon, H.M. Yin, A class of non-linear non-classical parabolic equations, J. Differential Equations 79 (1989) 266–288. [6] J.M. Chadam, A. Pierce, H.M. Yin, The blow-up properties of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169 (1992) 313–328. [7] M. Escobedo, M.A. Herrero, A semilinear parabolic system in a bounded domain, Ann. Mat. Pura Appl. CLXV (IV) (1993) 315–336. [8] H.L. Li, M.X. Wang, Blow-up properties for parabolic systems with localized nonlinear sources, Appl. Math. Lett. 17 (2004) 771–778. [9] H.L. Li, M.X. Wang, Uniform blowup profiles and boundary layer for a parabolic system with localized nonlinear reaction terms, Sci. China Ser. A Math. 48 (2005) 185–197. [10] Z.G. Lin, C.H. Xie, M.X. Wang, The blow-up properties of solutions to a parabolic system with localized nonlinear reactions, Acta Math. Sci. 18 (4) (1997) 413–420. [11] C.L. Mu, Y. Su, Global existence and blow-up for a quasilinear degenerate parabolic system in a cylinder, Appl. Math. Lett. 14 (2001) 715–723. [12] P. Ortaleva, J. Ross, Local structures in chemical reactions with heterogeneous catalysis, J. Chem. Phys. 56 (1972) 4397. [13] J.D. Rossi, N. Wolanski, Blowup vs. global existence for a semilinear reaction–diffusion system in a bounded domain, Comm. Partial Differential Equations 20 (1995) 1991–2004. [14] P. Souplet, Blow-up in nonlocal reaction–diffusion equations, SIAM J. Math. Anal. 29 (6) (1998) 1301–1334. [15] P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations 153 (1999) 374–406. [16] L.W. Wang, Q.Y. Chen, The asymptotic behavior of blow-up solution of localized nonlinear equation, J. Math. Anal. Appl. 200 (1996) 315–321. [17] Z.Y. Xiang, X.G. Hu, C.L. Mu, Neumann problem for reaction–diffusion systems with nonlocal nonlinear sources, Nonlinear Anal. 61 (2005) 1209–1224. [18] Z.Y. Xiang, Q. Chen, C.L. Mu, Blowup properties for several diffusion systems with localized sources, ANZIAM J. 48 (2006) 37–56.