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The profile near blowup time for solutions of diffusion systems coupled with localized nonlinear reactions
Nonlinear Analysis 50 (2002) 1013 – 1024
www.elsevier.com/locate/na
The pro le near blowup time for solutions of di$usion systems coupled with localized nonlinear reactions Michael Pedersena;∗ , Lin Zhiguib a Mathematical
Institute, The Technical University of Denmark, DK 2800, Lyngby, Denmark of Mathematics, Yangzhou University, Yangzhou 225002, China
b Department
Received 11 May 2000; received in revised form 30 October 2000; accepted 11 May 2001
MSC: 35K50; 35K60 Keywords: Di$usion equations; Localized source; Blowup behavior; Boundary layer
1. Introduction In this paper, we study positive solutions to the system of di$usion equations coupled with localized source: vt = 7v + 2 eu(x0 ; t) ; x; x0 ∈ ; t ¿ 0; ut = 7u + 1 ev(x0 ; t) ; (1) u = 0; v = 0; x ∈ @; t ¿ 0; u(x; 0) = u0 (x); v(x; 0) = v0 (x); x ∈ ; where i ¿ 0; i = 1; 2. ⊂ RN is a bounded domain with smooth boundary @. u0 (x) and v0 (x) are nonnegative nontrivial continuous functions in 9 vanishing on the boundary. The equations in (1) describe a physical phenomena where the reaction in a dynamical system takes places only at a single point (see [3,17]). For the single more general equation ut = 7u + f(u(x0 (t); t));
x ∈ ; t ¿ 0
the blowup of solutions has been studied by a number of authors. Cannon and Yin [4] study the local solvability, Chadam et al. [5] investigated the above equation with ∗
Corresponding author. E-mail addresses: [email protected] (M. Pedersen), [email protected] (L. Zhigui).
c 2002 Elsevier Science Ltd. All rights reserved. 0362-546X/02/$ - see front matter PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 7 9 8 - 2
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superlinear convex nonlinearities f, in the case when x0 (t) is a constant point x0 and proved that u(x; t) blows up in nite time and the blowup set is the whole domain, the growth rate of solutions near the blowup time is also derived. Wang and Chen [21] studied the solution in the one-dimensional case (with f(u) = up ; p ¿ 1, and x0 (t) ≡ 0), and for a restricted class of initial data, they obtained the growth rate of the blowup solution, the asymptotic behavior and observe that the boundary layer phenomena occurs, that is, when t suKciently near T , u(x; t)(T − t)1=(p−1) approximately equals (p − 1)−1=(p−1) in (−l + ; l − ), where tends 0 as t tends T . The region near the boundary where the solution follows a fast transition between the blowup regime and the assigned zero boundary condition is called the boundary layer. Recently Souplet [20] studied large classes of equations with nonlocal nonlinear reaction terms, and gave the uniform blowup behavior of solutions (in the interior). Moreover, the author derived sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solutions in the boundary layer. We remark that a lot of work have been done in the past few years on the blowup of solutions for local semilinear parabolic equations and systems. In the case of a single equation from the solid fuel ignition model ut − 7u = eu ; it is known that the solution to the Dirichlet problem may blow up in nite time, see [2]. As for the precise nature of the blowup, Friedman and Mcleod [8] established a single point blowup under some assumptions on u0 . Bebernes et al., [1] and Liu [16] gave the blowup rate estimate and characterized the asymptotic behavior of the solution, that is u(x; t) 6 − ln(T − t) + C in BR × (0; T ); 1=2 as t → T; (T − t)eu((T −t) y;t) → 1 uniformly for y 6 C. For the systems ut − 7u = ev ; (x; t) ∈ BR × (0; T ); vt − 7v = eu ; (x; t) ∈ BR × (0; T ); Pao [18] proved that the solution to the Dirichlet problem blow up and Friedman and Giga [7] described the single blow up point for the parabolic system under certain conditions. As we know, there are no further results to describe the asymptotic behavior of blowup solutions. The purpose of this paper is to present blowup results of (1). Namely, we rst prove how the solutions blowup in nite time T (Theorem 2.1), then give the estimates of the blowup rates (Theorem 3.1) and asymptotic behavior (Theorem 3.2). Finally we estimate the size of the boundary layer (Theorems 4.1 and 4.2). 2. Blowup in nite time We assume that is a bounded domain in RN with C 2 boundary and let 1 () and 1 (x) denote the rst eigenvalue and the rst eigenfunction of the following eigenvalue
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problem, 7 + = 0; x ∈ ; = 0; x ∈ @: and
1 (x)
satis es
1 (x) ¿ 0;
x ∈ ;
1 (x) d x = 1:
In this section, we consider the more general problem ui+1 (x0 ; t) ; i = 1; : : : ; k; uk+1 :=u1 ; x ∈ ; t ¿ 0; uit = 7ui + i e ui = 0; x ∈ @; t ¿ 0; ui (x; 0) = ui; 0 (x); x ∈ :
(2)
We rst present the maximum principle that will be used in the sequel. Lemma 2.1. Let (u1 ; : : : ; uk ) be the classical solution of the problem uit − 7ui ¿ ci (x; t)ui+1 (x0 ; t); x ∈ ; 0 ¡ t 6 T; ui = 0; x ∈ @; 0 ¡ t 6 T; ui (x; 0) ¿ 0; x ∈ :
(3)
If 0 6 ci (x; t) 6 Ci (i = 1; : : : ; k); then ui (x; t) ¿ 0; i = 1; : : : ; k for all (x; t) ∈ × [0; T ]: k Proof. Let K = i=1 Ci and wi = e−Kt ui , we conclude that wi ¿ 0 on 9 × [0; T ]. In fact if min(w1 ; w2 ; : : : ; wk )(x; t) ¡ 0 for some (x; t) ∈ 9 ×[0; T ], min(w1 ; w2 ; : : : ; wk )(x; t) takes negative minimum in × (0; T ] (it cannot take negative value at the boundary or the initial time). Without loss of generality, we assume that min(w1 ; w2 ; : : : ; wk )(x; t) takes negative minimum at (x1 ; t1 ) and w1 (x1 ; t1 ) 6 wi (x1 ; t1 ); i = 2; : : : ; k. Using the inequalities in (3) we arrive at w1t − 7w1 ¿ − Kw1 (x; t) + c1 (x; t)u2 (x0 ; t)e−Kt ; (x; t) ∈ × (0; T ] and then w1t (x1 ; t1 ) − 7w1 (x1 ; t1 ) ¿ −Kw1 (x1 ; t1 ) + c1 (x1 ; t1 )u2 (x0 ; t1 )e−Kt1 = −Kw1 (x1 ; t1 ) + c1 (x1 ; t1 )w2 (x0 ; t1 ) ¿ −Kw1 (x1 ; t1 ) + c1 (x1 ; t1 )w1 (x1 ; t1 ) ¿ −Kw1 (x1 ; t1 ) + C1 w1 (x1 ; t1 ) ¿ 0 since 0 6 c1 (x1 ; t1 ) 6 C1 and w2 (x0 ; t1 ) ¿ min(w1 ; : : : ; wk )(x0 ; t1 ) ¿ min(w1 ; : : : ; wk )(x1 ; t1 ) = w1 (x1 ; t1 ): On the other hand, w1 (x; t) attains negative minimum at (x1 ; t1 ), so w1t (x1 ; t1 ) − 7w1 (x1 ; t1 ) 6 0;
(4)
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which leads to a contradiction to the inequality (4). Thus min(w1 ; : : : ; wk ) ¿ 0 on 9 × [0; T ] and ui (x; t) ¿ 0 (i = 1; : : : ; k) on 9 × [0; T ]. Constructing the subsolution, we obtain the blowup result. Theorem 2.1. The solution of (2) blows up in 6nite time if the initial data is su7ciently large in a neighborhood of x0 . Proof. Local existence is clear, there exist T ¿ 0 and a maximal in time solution 9 T ) if ui; 0 is continuous on 9 and vanish (u1 (x; t); : : : ; uk (x; t)) of (2), classical on ×[0; on the boundary. It follows from a standard Schauder’s xed point theorem (see [19]; a detailed account of local existence for nonlocal parabolic equations is given in its k appendix). If T ¡ + ∞, then limt→T i=1 max9 ui (·; t) = + ∞. Moreover, the system is completely coupled, so if T ¡ + ∞, then limt→T max9 ui (·; t) = + ∞(i = 1; : : : ; k). Let B(x0 ; d) be the ball centered at x0 with the radius d ¿ 0 such that B(x0 ; d) ⊂ and let (w1 (x; t); : : : ; wk (x; t))T be the solution of the auxiliary problem wit − 7wi = i ewi+1 (x0 ; t) ; wi = 0;
x ∈ B(x0 ; d); t ¿ 0;
x ∈ @B(x0 ; d); t ¿ 0;
9 0 ; d); wi (x; 0) = wi; 0 (x; 0); x ∈ B(x where wi; 0 (x; 0) are nonnegative smooth symmetric, radially nonincreasing functions which are less that ui (x; 0) on B(x0 ; d). In fact, since ui (x0 ; 0) ¿ 0, there exist such that 0 ¡ ¡ d and ui (x; 0) ¿ ui (x0 ; 0)=2 on B(x0 ; ). Take wi; 0 (x; 0) = "ui (x; 0) on B(x0 ; d) with " a smooth symmetric, radially nonincreasing function satisfying 0 6 " 6 1 in B(x0 ; ) and " = 0 in B(x0 ; d)=B(x0 ; ), and we get that wi; 0 (x; 0) 6 ui (x; 0) on B(x0 ; d). By Lemma 2.1, we may then apply the comparison principle to deduce that ui ¿ wi for t ¿ 0 as long as ui exist. On the other hand, since wi; 0 are symmetric radially nonincreasing, it follows that wi (x; t) are symmetric nonincreasing for all t, so that (w1 ; w2 ; : : : ; wk ) satis es wit − 7wi ¿ i ewi+1 (x; t) ; wi = 0;
x ∈ B(x0 ; d); t ¿ 0;
x ∈ @B(x0 ; d); t ¿ 0;
9 0 ; d); wi (x; 0) = wi; 0 (x; 0); x ∈ B(x since wi (x0 ; t) ¿ wi (x; t) for x ∈ B(x0 ; d); t ¿ 0. From the following lemma, (w1 ; : : : ; wk ) blows up in nite time if wi; 0 are suKciently large, and so does (u1 ; : : : ; uk ) if ui; 0 are suKciently large in a neighborhood of x0 . Lemma 2.2. The solution of the problem uit − 7ui = i eui+1 ; x ∈ ; 0 ¡ t 6 T; x ∈ @; 0 ¡ t 6 T; ui = 0; ui (x; 0) ¿ 0; x∈ blows up in 6nite time if the initial data is large enough.
(5)
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Proof. We introduce the following notation: Ai (t):= ui (x; t) 1 (x) d x;
B(t):=
Ai (t);
:=min i ¿ 0: i
i
Multiplying the equations in (5) by 7
1
+ 1
we obtain uit
1
1,
integrating by parts and using the fact that
= 0;
1 d x = −1
ui
1 dx +
i eui+1
1
d x;
2 ui+1 1 d x: 2 From Jensen’s inequality and the fact that 1 (x) d x = 1 it follows that ¿ −1
ui
Ai ¿ − 1 Ai + A2i+1 ; 2
1
dx +
i = 1; : : : ; k:
Adding these inequalities yields 2 B (t) ¿ −1 B(t) + A (t) 2 i i+1
¿ −1 B(t) + (maxAi+1 (t))2 2
2 1 ¿ −1 B(t) + B(t) 2 k
= B(t) B(t) − 1 2k 2
(6)
for t ¿ 0. Then (6) implies that if 21 k 2 B(0) = ui; 0 1 d x ¿ ; i then B(t) blows up in nite time, hence so does the solution of (5). Remark 2.1. We remark here that for the Cauchy problem with localized reactions we have the following blowup result. Theorem 2.2. All nonnegative solutions of uit = 7ui + i eui+1 (x0 ; t) ; i = 1; : : : ; k; uk+1 :=u1 ; x ∈ RN ; t ¿ 0; x ∈ RN ; ui (x; 0) = ui; 0 (x); blow up in 6nite time.
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This result follows from a comparison principle with (u 1 (x; t); : : : ; u k (x; t)) where t i eui+1 (x0 ; s) ds: u i (x; t):=u i (t):= 0
3. Blowup estimate Throughout this paper, we shall use c and C to denote various generic constants whenever there is no chance of confusion. We denote t t fi (t) = i eui+1 (x0 ; s) ds and Fi (t) = fi (s) ds: (7) 0
0
We rst show some properties of the solutions of (2). Lemma 3.1. Let (u1 ; : : : ; uk ) be the classical solution of (2) in × (0; T ); which blows up in 6nite time T . Then we have (i) −C1 6 − 7ui (x; t) 6 max(sup06s6t i eui+1 (x0 ; s) ; C1 ); i = 1; : : : ; k in × [T=2; T ) for some C1 . (ii) 0 6 ui (x; t) 6 C2 + fi (t); i = 1; : : : ; k in × [T=2; T ) for some C2 . (iii) supx∈K& [fi (t) − ui (x; t)] 6 (C3 =&N +1 )(1 + Fi (t)); where K& = {y ∈ ; dist(y; @) ¿ &}. (iv) limt→T ui (x; t)=fi (t) = limt→T |ui (t)|∞ =fi (t) = 1 uniformly on compact subsets of . (v) limt→T uit (x; t)=i eui+1 (x0 ; t) = 1 and limt→T 7ui (x; t)=i eui+1 (x0 ; t) = 0 uniformly on compact subsets of . Here & is a positive number, in the following chosen according to a x0 and then denoted by &(x0 ). The results are from Lemmas 4:3–4:5; Theorem 4.1 and Proposition 4:2 in [20]. The rst two follows from a maximum principle and the last three follows from the study of linear problems with blowing-up source, and holds here since we assume that a solution exists. Next we establish a relationship among these ui , and then give the estimates of the ui ’s near the blowup time T . Lemma 3.2. Let (u1 ; : : : ; uk ) be the classical solution of (2) in × (0; T ). Then there exist T1 ¡ T and C4 such that 0 6 ui (x; t) 6 ui (x0 ; t) + C4 ;
i = 1; : : : ; k; x ∈ ; t ∈ (T1 ; T ):
(8)
Proof. It follows from (v) of Lemma 3.1 that there exists T1 ¡ T such that uit (x0 ; t) ¿ 12 i eui+1 (x0 ; t) ;
i = 1; : : : ; k; T1 6 t ¡ T;
(9)
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hence (u1 + u2 + · · · + uk )t (x0 ; t) ¿ 12 1 eu2 (x0 ; t) + · · · + 12 k eu1 (x0 ; t) ; ¿ c(eu2 (x0 ; t) + · · · + euk (x0 ; t) + eu1 (x0 ; t) );
(u1 + u2 + · · · + uk )(x0 ; t) ¿ ck exp k for T1 6 t ¡ T . By an integration from t to T , we have e
(u1 +u2 +···+uk )(x0 ; t) k
6
1 ; c(T − t)
T1 6 t ¡ T;
(u1 + u2 + · · · + uk )(x0 ; t) 6 kC + k|log(T − t)|;
(10) T1 6 t ¡ T:
(11)
Now we show that eui (x0 ; t) 6 C(T − t)−3=2 ;
i = 1; : : : ; k; T1 6 t ¡ T
(12)
for some C. If (12) was false then there would exist sequences {tn }; {Cn } with T1 6 tn → T − and Cn → ∞ as n → ∞ and m ∈ {1; : : : ; k} such that eum (x0 ; tn ) ¿ Cn (T − tn )−3=2 :
(13)
Without loss of generality we may assume that m = 1. Integrating the inequality (9) for i = k from tn to t, making use of (13), we have t u1 (x0 ; s) 1 uk (x0 ; t) ¿ uk (x0 ; tn ) + ds 2 k e tn
¿ 12 k eu1 (x0 ; tn ) (t − tn ) ¿ 12 k Cn (T − tn )−3=2 (t − tn );
t ¿ tn ;
Taking t = t9 = (T + tn )=2 in the above inequality, we obtain that uk (x0 ; t9) ¿ 12 k Cn (T − tn )−3=2 (t9 − tn ) = 12 k Cn [2(T − t9)]−3=2 (T − t9) = 2−5=2 k Cn (T − t9)−1=2 :
(14)
On the other hand, (11) implies that uk (x0 ; t) 6 kC + k|log(T − t)|;
T1 6 t ¡ T:
(15)
Combining (14) and (15) gives that kC + k|log(T − t9)| ¿ uk (x0 ; t9) ¿ 2−5=2 k Cn (T − t9)−1=2 : Taking n large enough in the above inequality, we achieve a contradiction. This concludes (12).
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Therefore,
fi (t) = Fi (t) =
t
0 t
0
i eui+1 (x0 ; s) ds 6 C (T − t)−1=2 ; fi (s) ds 6 C ;
i = 1; : : : ; k;
T1 6 t ¡ T; T1 6 t ¡ T:
(16) (17)
From (iii) of Lemma 3.1 and (17); we have fi (t) − ui (x0 ; t) 6
C3 (1 + Fi (t)) 6 C; &N +1 (x0 )
T1 6 t ¡ T;
which together with (ii) of Lemma 3.1 gives the desired result. For the case k = 2, it is easy to establish a relationship between the solutions. Lemma 3.3. Let (u; v) be the classical solution of (1) in × (0; T ); which blows up in 6nite time T . Then there exist T1 ¡ T and C5 such that u(x0 ; t) 6 v(x0 ; t) + C5 ;
v(x0 ; t) 6 u(x0 ; t) + C5 ; t ∈ (T1 ; T ):
(18)
Proof. We only prove the left inequality, since the right can be shown in similar way. It follows from (v) of Lemma 3.1 that there exists T1 ¡ T such that v(x0 ; t) 1 2 1 e
6 ut (x0 ; t) 6 21 ev(x0 ; t) ;
T1 6 t ¡ T;
(19)
u(x0 ; t) 1 2 2 e
6 vt (x0 ; t) 6 22 eu(x0 ; t) ;
T1 6 t ¡ T:
(20)
Let w(x; t):=u(x; t) − v(x0 ; t) − C5 , where C5 = C4 + |u(x; T1 )|∞ + |v(x; T1 )|∞ + |log(21 =2 )| + |log(22 =1 )| and C4 is from (8). Using the equation of u in (1) and the left inequality of (20) yields wt − 7w 6 1 ev(x0 ; t) − 12 2 eu(x0 ; t)
21 v(x0 ; t)−u(x0 ; t) u(x0 ; t) 1 e −1 = 2 2 e 2
21 v(x0 ; t)−u(x; t)+C4 6 12 2 eu(x0 ; t) e − 1 by (8) 2 6 12 2 eu(x0 ; t) (ev(x0 ; t)+C5 −u(x; t) − 1) = 12 2 eu(x0 ; t) (e−w(x; t) − 1);
x ∈ ; T1 ¡ t ¡ T:
Since w 6 0 for t = T1 and on the boundary @, the maximum principle implies that w = u(x; t) − v(x0 ; t) − C5 6 0 on 9 × [T1 ; T ). So we have u(x0 ; t) 6 v(x0 ; t) + C5 on [T1 ; T ). Now we can easily derive the estimate of the blowup rate, which will be useful when studying the blowup behavior of the solution.
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Theorem 3.1. Let (u; v) be the classical solution of (1) in × (0; T ); which blow up in 6nite time T . Then there exist a constant C such that |log(T − t)| − C 6 max u(x; t) 6 |log(T − t)| + C;
0 ¡ t ¡ T;
(21)
|log(T − t)| − C 6 max v(x; t) 6 |log(T − t)| + C;
0 ¡ t ¡ T:
(22)
9 ×[0; t] 9 ×[0; t]
Proof. We only need to prove it for t near T . From Lemma 3.2, we only need to prove that |log(T − t)| − C 6 u(x0 ; t) 6 |log(T − t)| + C;
0 ¡ t ¡ T;
|log(T − t)| − C 6 v(x0 ; t) 6 |log(T − t)| + C;
0¡t¡T
for t near T . It follows from (ii) and (iii) of Lemma 3.1 and Lemma 3.3 that there exists T2 ¡ T such that f1 (t) −
C3 (1 + F1 (t)) 6 u(x0 ; t) 6 f1 (t) + C2 ; &N +1 (x0 )
T2 6 t ¡ T;
(23)
f2 (t) −
C3 N +1 & (x
T2 6 t ¡ T;
(24)
0)
(1 + F2 (t)) 6 v(x0 ; t) 6 f2 (t) + C2 ;
v(x0 ; t) − C5 6 u(x0 ; t) 6 v(x0 ; t) + C5 ; where
f1 (t) =
0
f2 (t) =
t
0
t
1 ev(x0 ; t) ds;
F1 (t) =
1 eu(x0 ; t) ds;
F2 (t) =
t
0
0
t
T2 6 t ¡ T;
(25)
f1 (s) ds; f2 (s) ds:
Noticing that Fi (t) 6 C ; i = 1; 2 from (17); we have f1 (t) = 1 ev(x0 ; t) 6 1 eu(x0 ; t)+C5 6 1 ef1 (t)+C ; f1 (t) = 1 ev(x0 ; t) ¿ 1 eu(x0 ; t)−C5 ¿ 1 ef1 (t)−C ; f2 (t) = 2 eu(x0 ; t) 6 2 ev(x0 ; t)+C5 6 2 ef2 (t)+C ; f2 (t) = 2 eu(x0 ; t) ¿ 2 ev(x0 ; t)−C5 ¿ 2 ef2 (t)−C for T2 6 t ¡ T . Integrating these inequalities gives that |log(T − t)| − C 6 fi (t) 6 |log(T − t)| + C;
i = 1; 2; T2 ¡ t ¡ T;
which together with (23) and (24) gives the desired result.
(26)
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Combining (21), (22) and (iv) of Lemma 3.1 gives the behavior of solution near the blowup time. Theorem 3.2. Let (u; v) be the classical solution of (1) in × (0; T ); which blow up in 6nite time T . Then we have lim |log(T − t)|−1 u(x; t) = lim |log(T − t)|−1 |u(t)|∞ = 1;
(27)
lim |log(T − t)|−1 v(x; t) = lim |log(T − t)|−1 |u(t)|∞ = 1;
(28)
t→T t→T
t→T
t→T
uniformly on compact subsets of . The above result shows that the solution has global blowup and that the rate of the blowup is uniform in all compact subsets of the domain . 4. Boundary layer In this section, we estimate the size of the boundary layer. The following results are due to Souplet, his method in [20] for the case k = 1 is suitable to handle the general case. We present it here for completeness and signi cance. We denote d(x) = dist(x; @); and consider the problem ut = 7u + g(t); x ∈ ; t ¿ 0; u = 0; x ∈ @; t ¿ 0; u(x; 0) = u0 (x); x ∈ :
(29)
Lemma 4.1. Assume that g is nonnegative; and let u be the classical solution of (29) in × (0; T ); which blows up in 6nite time T . Then there exist a constant C6 and some t0 ∈ (0; T ); such that u satis6es the estimate 9 (30) u(x; t) 6 C6 d(x) g(t)G(t) t for all (x; t) in × [t0 ; T ); where g(t) 9 = sup06s6t g(s) and G(t) = 0 g(s) ds. Lemma 4.2. Let u be the classical solution of (29) in × (0; T ); which blows up in 6nite time T . Assume that g(t) ¿ C(T − t)−1 for some C. Then; for all K ¿ 0; there exist constants C7 = C7 (K) ¿ 0 and t0 = t0 (K) ∈ (0; T ) such that d(x) C7 √ 6 u(x; t) (31) T −t √ for all (x; t) in × [t0 ; T ) such that d(x) 6 K T − t. These two lemmas follow from Propositions 4:7 and 4:8 in [20]. The lower estimates in Lemma 4.2 is obtained from comparison under the assumption that g(t) ¿ C(T −t)−1
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for some C. The upper estimate in Lemma 4.1 is obtained by interpolation. Under the assumptions of Theorem 3.1, it is easy to check that 2 ev(x0 ; t) 6 C(T − t)−1 ; C(T − t)−1 6 1 eu(x0 ; t) ; t t |log(T − t)| − C 6 1 eu(x0 ; s) ds; 2 ev(x0 ; s) ds 6 |log(T − t)| + C 0
0
for some C, so we have Theorem 4.1. Assume that (u; v) be the classical solution of (1) in × (0; T ); which blow up in 6nite time T . Then; for all K ¿ 0; there exist C = C(K) and some t0 = t0 (K) ∈ (0; T ) such that d(x) u(x; t) C√ 6 (32) v(x; t) T −t √ for all (x; t) in × [t0 ; T ) such that d(x) 6 K T − t. Theorem 4.2. Assume that (u; v) be the classical solution of (1) in × (0; T ); which blow up in 6nite time T . Then there exist C and t1 ∈ (0; T ) such that
|log(T − t)| u(x; t) C d(x) ¿ (33) v(x; t) T −t for all (x; t) in × [t1 ; T ). From Theorems 3.2 and 4.2, we have d(x) |u(t)|∞ ; u(x; t) 6 C (T − t)|log(T − t)| v(x; t) 6 C
d(x) (T − t)|log(T − t)|
which means that u(x; t) → 0; |u(x; t)|∞
|v(t)|∞ ;
as t → T and
d(x) (T − t)|log(T − t)|
→0
and v(x; t) → 0; |v(x; t)|∞
d(x)
as t → T and
(T − t)|log(T − t)|
→ 0:
In other words, the size of the boundary layer is at least of order
(T − t)|log(T − t)|.
References [1] J. Bebernes, A. Bressan, D. Eberly, A description of blowup for the solid fuel ignition model, Indiana Univ. Math. J. 36 (1987) 295–305.
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[2] J. Bebernes, D. Kassoy, A mathematics analysis for blowup for theronal reactions, SIAM J. Appl. Math. 40 (1981) 476–484. [3] K. Bimpong-Bota, P. Ortoleva, J. Ross, Far-from-equilibrium phenomena at local sites of reaction, J. Chem. Phys. 60 (1974) 3124. [4] J.R. Cannon, H.M. Yin, A class of non-linear non-classical parabolic equations, J. Di$erential Equations 79 (1989) 266–288. [5] J.M. Chadam, A. Peirce, H.M. Yin, The blow-up property of solutions to some di$usion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169 (1992) 313–328. [7] A. Friedman, Y. Giga, A single point blow-up for solutions of semilinear parabolic systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987) 65–79. [8] A. Friedman, J.B. Mcleod, Blow-up of positive solutions of semilinear heat equation, Indiana Univ. Math. J. 34 (1985) 425–447. [16] W.X. Liu, The blowup rate of solutions of semilinear heat equations, J. Di$erential Equations 77 (1989) 104–122. [17] P. Ortoleva, J. Ross, Local structures in chemical reaction with heterogeneous catalysis, J. Chem. Phys. 56 (1972) 4397. [18] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, London, 1992. [19] P. Souplet, Blow up in nonlocal reaction-di$usion equation, SIAM J. Math. Anal. 29 (6) (1998) 1301– 1334. [20] P. Souplet, Uniform blow-up pro les and boundary behavior for di$usion equations with nonlocal nonlinear source, J. Di$erential Equations 153 (1999) 374–406. [21] L. Wang, Q. Chen, The asymptotic behavior of blow-up solution of localized nonlinear equation, J. Math. Anal. Appl. 200 (1996) 315–321.
www.elsevier.com/locate/na
The pro le near blowup time for solutions of di$usion systems coupled with localized nonlinear reactions Michael Pedersena;∗ , Lin Zhiguib a Mathematical
Institute, The Technical University of Denmark, DK 2800, Lyngby, Denmark of Mathematics, Yangzhou University, Yangzhou 225002, China
b Department
Received 11 May 2000; received in revised form 30 October 2000; accepted 11 May 2001
MSC: 35K50; 35K60 Keywords: Di$usion equations; Localized source; Blowup behavior; Boundary layer
1. Introduction In this paper, we study positive solutions to the system of di$usion equations coupled with localized source: vt = 7v + 2 eu(x0 ; t) ; x; x0 ∈ ; t ¿ 0; ut = 7u + 1 ev(x0 ; t) ; (1) u = 0; v = 0; x ∈ @; t ¿ 0; u(x; 0) = u0 (x); v(x; 0) = v0 (x); x ∈ ; where i ¿ 0; i = 1; 2. ⊂ RN is a bounded domain with smooth boundary @. u0 (x) and v0 (x) are nonnegative nontrivial continuous functions in 9 vanishing on the boundary. The equations in (1) describe a physical phenomena where the reaction in a dynamical system takes places only at a single point (see [3,17]). For the single more general equation ut = 7u + f(u(x0 (t); t));
x ∈ ; t ¿ 0
the blowup of solutions has been studied by a number of authors. Cannon and Yin [4] study the local solvability, Chadam et al. [5] investigated the above equation with ∗
Corresponding author. E-mail addresses: [email protected] (M. Pedersen), [email protected] (L. Zhigui).
c 2002 Elsevier Science Ltd. All rights reserved. 0362-546X/02/$ - see front matter PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 7 9 8 - 2
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superlinear convex nonlinearities f, in the case when x0 (t) is a constant point x0 and proved that u(x; t) blows up in nite time and the blowup set is the whole domain, the growth rate of solutions near the blowup time is also derived. Wang and Chen [21] studied the solution in the one-dimensional case (with f(u) = up ; p ¿ 1, and x0 (t) ≡ 0), and for a restricted class of initial data, they obtained the growth rate of the blowup solution, the asymptotic behavior and observe that the boundary layer phenomena occurs, that is, when t suKciently near T , u(x; t)(T − t)1=(p−1) approximately equals (p − 1)−1=(p−1) in (−l + ; l − ), where tends 0 as t tends T . The region near the boundary where the solution follows a fast transition between the blowup regime and the assigned zero boundary condition is called the boundary layer. Recently Souplet [20] studied large classes of equations with nonlocal nonlinear reaction terms, and gave the uniform blowup behavior of solutions (in the interior). Moreover, the author derived sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solutions in the boundary layer. We remark that a lot of work have been done in the past few years on the blowup of solutions for local semilinear parabolic equations and systems. In the case of a single equation from the solid fuel ignition model ut − 7u = eu ; it is known that the solution to the Dirichlet problem may blow up in nite time, see [2]. As for the precise nature of the blowup, Friedman and Mcleod [8] established a single point blowup under some assumptions on u0 . Bebernes et al., [1] and Liu [16] gave the blowup rate estimate and characterized the asymptotic behavior of the solution, that is u(x; t) 6 − ln(T − t) + C in BR × (0; T ); 1=2 as t → T; (T − t)eu((T −t) y;t) → 1 uniformly for y 6 C. For the systems ut − 7u = ev ; (x; t) ∈ BR × (0; T ); vt − 7v = eu ; (x; t) ∈ BR × (0; T ); Pao [18] proved that the solution to the Dirichlet problem blow up and Friedman and Giga [7] described the single blow up point for the parabolic system under certain conditions. As we know, there are no further results to describe the asymptotic behavior of blowup solutions. The purpose of this paper is to present blowup results of (1). Namely, we rst prove how the solutions blowup in nite time T (Theorem 2.1), then give the estimates of the blowup rates (Theorem 3.1) and asymptotic behavior (Theorem 3.2). Finally we estimate the size of the boundary layer (Theorems 4.1 and 4.2). 2. Blowup in nite time We assume that is a bounded domain in RN with C 2 boundary and let 1 () and 1 (x) denote the rst eigenvalue and the rst eigenfunction of the following eigenvalue
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1015
problem, 7 + = 0; x ∈ ; = 0; x ∈ @: and
1 (x)
satis es
1 (x) ¿ 0;
x ∈ ;
1 (x) d x = 1:
In this section, we consider the more general problem ui+1 (x0 ; t) ; i = 1; : : : ; k; uk+1 :=u1 ; x ∈ ; t ¿ 0; uit = 7ui + i e ui = 0; x ∈ @; t ¿ 0; ui (x; 0) = ui; 0 (x); x ∈ :
(2)
We rst present the maximum principle that will be used in the sequel. Lemma 2.1. Let (u1 ; : : : ; uk ) be the classical solution of the problem uit − 7ui ¿ ci (x; t)ui+1 (x0 ; t); x ∈ ; 0 ¡ t 6 T; ui = 0; x ∈ @; 0 ¡ t 6 T; ui (x; 0) ¿ 0; x ∈ :
(3)
If 0 6 ci (x; t) 6 Ci (i = 1; : : : ; k); then ui (x; t) ¿ 0; i = 1; : : : ; k for all (x; t) ∈ × [0; T ]: k Proof. Let K = i=1 Ci and wi = e−Kt ui , we conclude that wi ¿ 0 on 9 × [0; T ]. In fact if min(w1 ; w2 ; : : : ; wk )(x; t) ¡ 0 for some (x; t) ∈ 9 ×[0; T ], min(w1 ; w2 ; : : : ; wk )(x; t) takes negative minimum in × (0; T ] (it cannot take negative value at the boundary or the initial time). Without loss of generality, we assume that min(w1 ; w2 ; : : : ; wk )(x; t) takes negative minimum at (x1 ; t1 ) and w1 (x1 ; t1 ) 6 wi (x1 ; t1 ); i = 2; : : : ; k. Using the inequalities in (3) we arrive at w1t − 7w1 ¿ − Kw1 (x; t) + c1 (x; t)u2 (x0 ; t)e−Kt ; (x; t) ∈ × (0; T ] and then w1t (x1 ; t1 ) − 7w1 (x1 ; t1 ) ¿ −Kw1 (x1 ; t1 ) + c1 (x1 ; t1 )u2 (x0 ; t1 )e−Kt1 = −Kw1 (x1 ; t1 ) + c1 (x1 ; t1 )w2 (x0 ; t1 ) ¿ −Kw1 (x1 ; t1 ) + c1 (x1 ; t1 )w1 (x1 ; t1 ) ¿ −Kw1 (x1 ; t1 ) + C1 w1 (x1 ; t1 ) ¿ 0 since 0 6 c1 (x1 ; t1 ) 6 C1 and w2 (x0 ; t1 ) ¿ min(w1 ; : : : ; wk )(x0 ; t1 ) ¿ min(w1 ; : : : ; wk )(x1 ; t1 ) = w1 (x1 ; t1 ): On the other hand, w1 (x; t) attains negative minimum at (x1 ; t1 ), so w1t (x1 ; t1 ) − 7w1 (x1 ; t1 ) 6 0;
(4)
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which leads to a contradiction to the inequality (4). Thus min(w1 ; : : : ; wk ) ¿ 0 on 9 × [0; T ] and ui (x; t) ¿ 0 (i = 1; : : : ; k) on 9 × [0; T ]. Constructing the subsolution, we obtain the blowup result. Theorem 2.1. The solution of (2) blows up in 6nite time if the initial data is su7ciently large in a neighborhood of x0 . Proof. Local existence is clear, there exist T ¿ 0 and a maximal in time solution 9 T ) if ui; 0 is continuous on 9 and vanish (u1 (x; t); : : : ; uk (x; t)) of (2), classical on ×[0; on the boundary. It follows from a standard Schauder’s xed point theorem (see [19]; a detailed account of local existence for nonlocal parabolic equations is given in its k appendix). If T ¡ + ∞, then limt→T i=1 max9 ui (·; t) = + ∞. Moreover, the system is completely coupled, so if T ¡ + ∞, then limt→T max9 ui (·; t) = + ∞(i = 1; : : : ; k). Let B(x0 ; d) be the ball centered at x0 with the radius d ¿ 0 such that B(x0 ; d) ⊂ and let (w1 (x; t); : : : ; wk (x; t))T be the solution of the auxiliary problem wit − 7wi = i ewi+1 (x0 ; t) ; wi = 0;
x ∈ B(x0 ; d); t ¿ 0;
x ∈ @B(x0 ; d); t ¿ 0;
9 0 ; d); wi (x; 0) = wi; 0 (x; 0); x ∈ B(x where wi; 0 (x; 0) are nonnegative smooth symmetric, radially nonincreasing functions which are less that ui (x; 0) on B(x0 ; d). In fact, since ui (x0 ; 0) ¿ 0, there exist such that 0 ¡ ¡ d and ui (x; 0) ¿ ui (x0 ; 0)=2 on B(x0 ; ). Take wi; 0 (x; 0) = "ui (x; 0) on B(x0 ; d) with " a smooth symmetric, radially nonincreasing function satisfying 0 6 " 6 1 in B(x0 ; ) and " = 0 in B(x0 ; d)=B(x0 ; ), and we get that wi; 0 (x; 0) 6 ui (x; 0) on B(x0 ; d). By Lemma 2.1, we may then apply the comparison principle to deduce that ui ¿ wi for t ¿ 0 as long as ui exist. On the other hand, since wi; 0 are symmetric radially nonincreasing, it follows that wi (x; t) are symmetric nonincreasing for all t, so that (w1 ; w2 ; : : : ; wk ) satis es wit − 7wi ¿ i ewi+1 (x; t) ; wi = 0;
x ∈ B(x0 ; d); t ¿ 0;
x ∈ @B(x0 ; d); t ¿ 0;
9 0 ; d); wi (x; 0) = wi; 0 (x; 0); x ∈ B(x since wi (x0 ; t) ¿ wi (x; t) for x ∈ B(x0 ; d); t ¿ 0. From the following lemma, (w1 ; : : : ; wk ) blows up in nite time if wi; 0 are suKciently large, and so does (u1 ; : : : ; uk ) if ui; 0 are suKciently large in a neighborhood of x0 . Lemma 2.2. The solution of the problem uit − 7ui = i eui+1 ; x ∈ ; 0 ¡ t 6 T; x ∈ @; 0 ¡ t 6 T; ui = 0; ui (x; 0) ¿ 0; x∈ blows up in 6nite time if the initial data is large enough.
(5)
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Proof. We introduce the following notation: Ai (t):= ui (x; t) 1 (x) d x;
B(t):=
Ai (t);
:=min i ¿ 0: i
i
Multiplying the equations in (5) by 7
1
+ 1
we obtain uit
1
1,
integrating by parts and using the fact that
= 0;
1 d x = −1
ui
1 dx +
i eui+1
1
d x;
2 ui+1 1 d x: 2 From Jensen’s inequality and the fact that 1 (x) d x = 1 it follows that ¿ −1
ui
Ai ¿ − 1 Ai + A2i+1 ; 2
1
dx +
i = 1; : : : ; k:
Adding these inequalities yields 2 B (t) ¿ −1 B(t) + A (t) 2 i i+1
¿ −1 B(t) + (maxAi+1 (t))2 2
2 1 ¿ −1 B(t) + B(t) 2 k
= B(t) B(t) − 1 2k 2
(6)
for t ¿ 0. Then (6) implies that if 21 k 2 B(0) = ui; 0 1 d x ¿ ; i then B(t) blows up in nite time, hence so does the solution of (5). Remark 2.1. We remark here that for the Cauchy problem with localized reactions we have the following blowup result. Theorem 2.2. All nonnegative solutions of uit = 7ui + i eui+1 (x0 ; t) ; i = 1; : : : ; k; uk+1 :=u1 ; x ∈ RN ; t ¿ 0; x ∈ RN ; ui (x; 0) = ui; 0 (x); blow up in 6nite time.
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This result follows from a comparison principle with (u 1 (x; t); : : : ; u k (x; t)) where t i eui+1 (x0 ; s) ds: u i (x; t):=u i (t):= 0
3. Blowup estimate Throughout this paper, we shall use c and C to denote various generic constants whenever there is no chance of confusion. We denote t t fi (t) = i eui+1 (x0 ; s) ds and Fi (t) = fi (s) ds: (7) 0
0
We rst show some properties of the solutions of (2). Lemma 3.1. Let (u1 ; : : : ; uk ) be the classical solution of (2) in × (0; T ); which blows up in 6nite time T . Then we have (i) −C1 6 − 7ui (x; t) 6 max(sup06s6t i eui+1 (x0 ; s) ; C1 ); i = 1; : : : ; k in × [T=2; T ) for some C1 . (ii) 0 6 ui (x; t) 6 C2 + fi (t); i = 1; : : : ; k in × [T=2; T ) for some C2 . (iii) supx∈K& [fi (t) − ui (x; t)] 6 (C3 =&N +1 )(1 + Fi (t)); where K& = {y ∈ ; dist(y; @) ¿ &}. (iv) limt→T ui (x; t)=fi (t) = limt→T |ui (t)|∞ =fi (t) = 1 uniformly on compact subsets of . (v) limt→T uit (x; t)=i eui+1 (x0 ; t) = 1 and limt→T 7ui (x; t)=i eui+1 (x0 ; t) = 0 uniformly on compact subsets of . Here & is a positive number, in the following chosen according to a x0 and then denoted by &(x0 ). The results are from Lemmas 4:3–4:5; Theorem 4.1 and Proposition 4:2 in [20]. The rst two follows from a maximum principle and the last three follows from the study of linear problems with blowing-up source, and holds here since we assume that a solution exists. Next we establish a relationship among these ui , and then give the estimates of the ui ’s near the blowup time T . Lemma 3.2. Let (u1 ; : : : ; uk ) be the classical solution of (2) in × (0; T ). Then there exist T1 ¡ T and C4 such that 0 6 ui (x; t) 6 ui (x0 ; t) + C4 ;
i = 1; : : : ; k; x ∈ ; t ∈ (T1 ; T ):
(8)
Proof. It follows from (v) of Lemma 3.1 that there exists T1 ¡ T such that uit (x0 ; t) ¿ 12 i eui+1 (x0 ; t) ;
i = 1; : : : ; k; T1 6 t ¡ T;
(9)
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1019
hence (u1 + u2 + · · · + uk )t (x0 ; t) ¿ 12 1 eu2 (x0 ; t) + · · · + 12 k eu1 (x0 ; t) ; ¿ c(eu2 (x0 ; t) + · · · + euk (x0 ; t) + eu1 (x0 ; t) );
(u1 + u2 + · · · + uk )(x0 ; t) ¿ ck exp k for T1 6 t ¡ T . By an integration from t to T , we have e
(u1 +u2 +···+uk )(x0 ; t) k
6
1 ; c(T − t)
T1 6 t ¡ T;
(u1 + u2 + · · · + uk )(x0 ; t) 6 kC + k|log(T − t)|;
(10) T1 6 t ¡ T:
(11)
Now we show that eui (x0 ; t) 6 C(T − t)−3=2 ;
i = 1; : : : ; k; T1 6 t ¡ T
(12)
for some C. If (12) was false then there would exist sequences {tn }; {Cn } with T1 6 tn → T − and Cn → ∞ as n → ∞ and m ∈ {1; : : : ; k} such that eum (x0 ; tn ) ¿ Cn (T − tn )−3=2 :
(13)
Without loss of generality we may assume that m = 1. Integrating the inequality (9) for i = k from tn to t, making use of (13), we have t u1 (x0 ; s) 1 uk (x0 ; t) ¿ uk (x0 ; tn ) + ds 2 k e tn
¿ 12 k eu1 (x0 ; tn ) (t − tn ) ¿ 12 k Cn (T − tn )−3=2 (t − tn );
t ¿ tn ;
Taking t = t9 = (T + tn )=2 in the above inequality, we obtain that uk (x0 ; t9) ¿ 12 k Cn (T − tn )−3=2 (t9 − tn ) = 12 k Cn [2(T − t9)]−3=2 (T − t9) = 2−5=2 k Cn (T − t9)−1=2 :
(14)
On the other hand, (11) implies that uk (x0 ; t) 6 kC + k|log(T − t)|;
T1 6 t ¡ T:
(15)
Combining (14) and (15) gives that kC + k|log(T − t9)| ¿ uk (x0 ; t9) ¿ 2−5=2 k Cn (T − t9)−1=2 : Taking n large enough in the above inequality, we achieve a contradiction. This concludes (12).
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Therefore,
fi (t) = Fi (t) =
t
0 t
0
i eui+1 (x0 ; s) ds 6 C (T − t)−1=2 ; fi (s) ds 6 C ;
i = 1; : : : ; k;
T1 6 t ¡ T; T1 6 t ¡ T:
(16) (17)
From (iii) of Lemma 3.1 and (17); we have fi (t) − ui (x0 ; t) 6
C3 (1 + Fi (t)) 6 C; &N +1 (x0 )
T1 6 t ¡ T;
which together with (ii) of Lemma 3.1 gives the desired result. For the case k = 2, it is easy to establish a relationship between the solutions. Lemma 3.3. Let (u; v) be the classical solution of (1) in × (0; T ); which blows up in 6nite time T . Then there exist T1 ¡ T and C5 such that u(x0 ; t) 6 v(x0 ; t) + C5 ;
v(x0 ; t) 6 u(x0 ; t) + C5 ; t ∈ (T1 ; T ):
(18)
Proof. We only prove the left inequality, since the right can be shown in similar way. It follows from (v) of Lemma 3.1 that there exists T1 ¡ T such that v(x0 ; t) 1 2 1 e
6 ut (x0 ; t) 6 21 ev(x0 ; t) ;
T1 6 t ¡ T;
(19)
u(x0 ; t) 1 2 2 e
6 vt (x0 ; t) 6 22 eu(x0 ; t) ;
T1 6 t ¡ T:
(20)
Let w(x; t):=u(x; t) − v(x0 ; t) − C5 , where C5 = C4 + |u(x; T1 )|∞ + |v(x; T1 )|∞ + |log(21 =2 )| + |log(22 =1 )| and C4 is from (8). Using the equation of u in (1) and the left inequality of (20) yields wt − 7w 6 1 ev(x0 ; t) − 12 2 eu(x0 ; t)
21 v(x0 ; t)−u(x0 ; t) u(x0 ; t) 1 e −1 = 2 2 e 2
21 v(x0 ; t)−u(x; t)+C4 6 12 2 eu(x0 ; t) e − 1 by (8) 2 6 12 2 eu(x0 ; t) (ev(x0 ; t)+C5 −u(x; t) − 1) = 12 2 eu(x0 ; t) (e−w(x; t) − 1);
x ∈ ; T1 ¡ t ¡ T:
Since w 6 0 for t = T1 and on the boundary @, the maximum principle implies that w = u(x; t) − v(x0 ; t) − C5 6 0 on 9 × [T1 ; T ). So we have u(x0 ; t) 6 v(x0 ; t) + C5 on [T1 ; T ). Now we can easily derive the estimate of the blowup rate, which will be useful when studying the blowup behavior of the solution.
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Theorem 3.1. Let (u; v) be the classical solution of (1) in × (0; T ); which blow up in 6nite time T . Then there exist a constant C such that |log(T − t)| − C 6 max u(x; t) 6 |log(T − t)| + C;
0 ¡ t ¡ T;
(21)
|log(T − t)| − C 6 max v(x; t) 6 |log(T − t)| + C;
0 ¡ t ¡ T:
(22)
9 ×[0; t] 9 ×[0; t]
Proof. We only need to prove it for t near T . From Lemma 3.2, we only need to prove that |log(T − t)| − C 6 u(x0 ; t) 6 |log(T − t)| + C;
0 ¡ t ¡ T;
|log(T − t)| − C 6 v(x0 ; t) 6 |log(T − t)| + C;
0¡t¡T
for t near T . It follows from (ii) and (iii) of Lemma 3.1 and Lemma 3.3 that there exists T2 ¡ T such that f1 (t) −
C3 (1 + F1 (t)) 6 u(x0 ; t) 6 f1 (t) + C2 ; &N +1 (x0 )
T2 6 t ¡ T;
(23)
f2 (t) −
C3 N +1 & (x
T2 6 t ¡ T;
(24)
0)
(1 + F2 (t)) 6 v(x0 ; t) 6 f2 (t) + C2 ;
v(x0 ; t) − C5 6 u(x0 ; t) 6 v(x0 ; t) + C5 ; where
f1 (t) =
0
f2 (t) =
t
0
t
1 ev(x0 ; t) ds;
F1 (t) =
1 eu(x0 ; t) ds;
F2 (t) =
t
0
0
t
T2 6 t ¡ T;
(25)
f1 (s) ds; f2 (s) ds:
Noticing that Fi (t) 6 C ; i = 1; 2 from (17); we have f1 (t) = 1 ev(x0 ; t) 6 1 eu(x0 ; t)+C5 6 1 ef1 (t)+C ; f1 (t) = 1 ev(x0 ; t) ¿ 1 eu(x0 ; t)−C5 ¿ 1 ef1 (t)−C ; f2 (t) = 2 eu(x0 ; t) 6 2 ev(x0 ; t)+C5 6 2 ef2 (t)+C ; f2 (t) = 2 eu(x0 ; t) ¿ 2 ev(x0 ; t)−C5 ¿ 2 ef2 (t)−C for T2 6 t ¡ T . Integrating these inequalities gives that |log(T − t)| − C 6 fi (t) 6 |log(T − t)| + C;
i = 1; 2; T2 ¡ t ¡ T;
which together with (23) and (24) gives the desired result.
(26)
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Combining (21), (22) and (iv) of Lemma 3.1 gives the behavior of solution near the blowup time. Theorem 3.2. Let (u; v) be the classical solution of (1) in × (0; T ); which blow up in 6nite time T . Then we have lim |log(T − t)|−1 u(x; t) = lim |log(T − t)|−1 |u(t)|∞ = 1;
(27)
lim |log(T − t)|−1 v(x; t) = lim |log(T − t)|−1 |u(t)|∞ = 1;
(28)
t→T t→T
t→T
t→T
uniformly on compact subsets of . The above result shows that the solution has global blowup and that the rate of the blowup is uniform in all compact subsets of the domain . 4. Boundary layer In this section, we estimate the size of the boundary layer. The following results are due to Souplet, his method in [20] for the case k = 1 is suitable to handle the general case. We present it here for completeness and signi cance. We denote d(x) = dist(x; @); and consider the problem ut = 7u + g(t); x ∈ ; t ¿ 0; u = 0; x ∈ @; t ¿ 0; u(x; 0) = u0 (x); x ∈ :
(29)
Lemma 4.1. Assume that g is nonnegative; and let u be the classical solution of (29) in × (0; T ); which blows up in 6nite time T . Then there exist a constant C6 and some t0 ∈ (0; T ); such that u satis6es the estimate 9 (30) u(x; t) 6 C6 d(x) g(t)G(t) t for all (x; t) in × [t0 ; T ); where g(t) 9 = sup06s6t g(s) and G(t) = 0 g(s) ds. Lemma 4.2. Let u be the classical solution of (29) in × (0; T ); which blows up in 6nite time T . Assume that g(t) ¿ C(T − t)−1 for some C. Then; for all K ¿ 0; there exist constants C7 = C7 (K) ¿ 0 and t0 = t0 (K) ∈ (0; T ) such that d(x) C7 √ 6 u(x; t) (31) T −t √ for all (x; t) in × [t0 ; T ) such that d(x) 6 K T − t. These two lemmas follow from Propositions 4:7 and 4:8 in [20]. The lower estimates in Lemma 4.2 is obtained from comparison under the assumption that g(t) ¿ C(T −t)−1
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1023
for some C. The upper estimate in Lemma 4.1 is obtained by interpolation. Under the assumptions of Theorem 3.1, it is easy to check that 2 ev(x0 ; t) 6 C(T − t)−1 ; C(T − t)−1 6 1 eu(x0 ; t) ; t t |log(T − t)| − C 6 1 eu(x0 ; s) ds; 2 ev(x0 ; s) ds 6 |log(T − t)| + C 0
0
for some C, so we have Theorem 4.1. Assume that (u; v) be the classical solution of (1) in × (0; T ); which blow up in 6nite time T . Then; for all K ¿ 0; there exist C = C(K) and some t0 = t0 (K) ∈ (0; T ) such that d(x) u(x; t) C√ 6 (32) v(x; t) T −t √ for all (x; t) in × [t0 ; T ) such that d(x) 6 K T − t. Theorem 4.2. Assume that (u; v) be the classical solution of (1) in × (0; T ); which blow up in 6nite time T . Then there exist C and t1 ∈ (0; T ) such that
|log(T − t)| u(x; t) C d(x) ¿ (33) v(x; t) T −t for all (x; t) in × [t1 ; T ). From Theorems 3.2 and 4.2, we have d(x) |u(t)|∞ ; u(x; t) 6 C (T − t)|log(T − t)| v(x; t) 6 C
d(x) (T − t)|log(T − t)|
which means that u(x; t) → 0; |u(x; t)|∞
|v(t)|∞ ;
as t → T and
d(x) (T − t)|log(T − t)|
→0
and v(x; t) → 0; |v(x; t)|∞
d(x)
as t → T and
(T − t)|log(T − t)|
→ 0:
In other words, the size of the boundary layer is at least of order
(T − t)|log(T − t)|.
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