BLOWUP FOR DEGENERATE AND SINGULAR PARABOLIC SYSTEM WITH NONLOCAL SOURCE JUN ZHOU, CHUNLAI MU, AND ZHONGPING LI Received 23 January 2006; Revised 3 April 2006; Accepted 7 April 2006 We deal with the blowup properties of the solution to the degenerate and singular par- abolic system with nonlocal source and homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained. Furthermore, under certain conditions it is proved that the blowup set of the solution is the whole domain. Copyright © 2006 Jun Zhou et al. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this paper, we consider the following degenerate and singular nonlinear reaction- diffusion equations with nonlocal source: x q 1 u t −  x r 1 u x  x =  a 0 v p 1 dx,(x, t) ∈(0,a) ×(0,T), x q 2 v t −  x r 2 v x  x =  a 0 u p 2 dx,(x, t) ∈(0,a) ×(0,T), u(0,t) = u(a,t) =v(0,t) =v(a,t) =0, t ∈ (0,T), u(x,0) = u 0 (x), v(x,0) = v 0 (x), x ∈ [0,a], (1.1) where u 0 (x), v 0 (x) ∈ C 2+α (D)forsomeα ∈ (0,1) are nonnegative nontrivial functions. u 0 (0) = u 0 (a) = v 0 (0) = v 0 (a) = 0, u 0 (x) ≥ 0, v 0 (x) ≥ 0, u 0 , v 0 satisfy the compatibility condition, T>0, a>0, r 1 ,r 2 ∈ [0,1), |q 1 |+ r 1 = 0, |q 2 |+ r 2 = 0, and p 1 > 1, p 2 > 1. Let D = (0,a)andΩ t = D ×(0,t], D and Ω t are their closures, respectively. Since |q 1 |+ r 1 = 0, |q 2 |+ r 2 = 0, the coefficients of u t , u x , u xx and v t , v x , v xx may tend to 0 or ∞ as x tends to 0, we can regard the equations as degenerate and singular. Hindawi Publishing Cor poration Boundary Value Problems Volume 2006, Article ID 21830, Pages 1–19 DOI 10.1155/BVP/2006/21830 2 Blowup for degenerate and singular parabolic system Floater [9] and Chan and Liu [4] investigated the blowup properties of the following degenerate parabolic problem: x q u t −u xx = u p ,(x,t) ∈(0,a) ×(0,T), u(0,t) = u(a,t) =0, t ∈(0,T), u(x,0) = u 0 (x), x ∈ [0,a], (1.2) where q>0andp>1. Under certain conditions on the initial datum u 0 (x), Floater [9] proved that the solution u(x,t)of(1.2) blows up at the boundary x = 0forthecase1< p ≤ q + 1. This contrasts with one of the results in [10], which showed that for the case q = 0, the blowup set of solution u(x,t)of(1.2)isapropercompactsubsetofD. The motivation for studying problem (1.2) comes from Ockendon’s model (see [14]) for the flow in a channel of a fluid whose viscosity depends on temperature xu t = u xx + e u , (1.3) where u represents the temperature of the fluid. In [9] Floater approximated e u by u p and considered (1.2). Budd et al. [2] generalized the results in [9] to the following degenerate quasilinear parabolic equation: x q u t =  u m  xx + u p , (1.4) with homogeneous Dirichlet conditions in the cr itical exponent q = (p −1)/m,whereq> 0, m ≥ 1, and p>1. They pointed out that the general classification of blowup solution for the degenerate equation (1.4) stays the same for the quasilinear equation (see [2, 17]) u t =  u m  xx + u p . (1.5) For the case p>q+1,in[4] Chan and Liu continued to study problem (1.2). Under certain conditions, they proved that x = 0 is not a blowup point and the blowup set is a propercompactsubsetofD. In [7], Chen and Xie discussed the following degenerate and singular semilinear para- bolic equation: u t −  x α u x  x =  a 0 f  u(x,t)  dx,(x, t) ∈(0,a) ×(0,T), u(0,t) = u(a,t) =0, t ∈(0,T), u(x,0) = u 0 (x), x ∈ [0,a], (1.6) they established the local existence and uniqueness of a classical solution. Under appro- priate hypotheses, they obtained some sufficient conditions for the global existence and blowup of a positive solution. Jun Zhou et al. 3 In [6], Chen et al. consider the following degenerate nonlinear reaction-diffusion equation w ith nonlocal source: x q u t −  x γ u x  x =  a 0 u p dx,(x, t) ∈(0,a) ×(0,T), u(0,t) = u(a,t) =0, t ∈(0,T), u(x,0) = u 0 (x), x ∈ [0,a], (1.7) they established the local existence and uniqueness of a classical solution. Under appro- priate hypotheses, they also got some sufficient conditions for the global existence and blowup of a positive solution. Furthermore, under certain conditions, it is proved that the blowup set of the solution is the whole domain. In this paper, we generalize the results of [6] to parabolic system and investigate the effect of the singularity, degeneracy, and nonlocal reaction on the behavior of the solution of (1.1). The difficulties are the establishment of the corresponding comparison principle and the construction of a supersolution of (1.1). It is different from [4, 9]thatunder certain conditions the blowup set of the solution of (1.1) is the whole domain. But this is consistent with the conclusions in [1, 18, 19]. This paper is organized as follows: in the next section, we show the existence of a unique classical solution. In Section 3, we give some criteria for the solution (u(x,t),v(x, t)) to exist globally or blow up in finite time and in the last section, we discuss the blowup set. 2. Local existence In order to prove the existence of a unique positive solution to (1.1), we start with the following comparison principle. Lemma 2.1. Let b 1 (x, t) and b 2 (x, t) be continuous nonnegative functions defined on [0,a] × [0,r] for any r ∈ (0,T),andlet(u(x,t),v(x,t)) ∈ (C(Ω r ) ∩C 2,1 (Ω r )) 2 satisfy x q 1 u t −  x r 1 u x  x ≥  a 0 b 1 (x, t)v(x,t)dx,(x, t) ∈(0,a) ×(0,r], x q 2 v t −  x r 2 v x  x ≥  a 0 b 2 (x, t)u(x,t)dx,(x,t) ∈ (0,a) ×(0,r], u(0,t) ≥ 0, u(a,t) ≥0, v(0,t) ≥0, v(a,t) ≥ 0, t ∈(0,r], u(x,0) ≥ 0, v(x,0) ≥ 0, x ∈[0,a]. (2.1) Then, u(x,t) ≥ 0, v(x,t) ≥0 on [0,a] ×[0,T). Proof. At first, similar to the proof of Lemma 2.1 in [20], by using [15, Lemma 2.2.1], we can easily obtain the following conclusion. 4 Blowup for degenerate and singular parabolic system If W(x,t)andZ(x, t) ∈ C(Ω r ) ∩C 2,1 (Ω r ) satisfy x q 1 W t −  x r 1 W x  x ≥  a 0 b 1 (x, t)Z(x,t)dx,(x,t) ∈(0,a) ×(0,r], x q 2 Z t −  x r 2 Z x  x ≥  a 0 b 2 (x, t)W(x,t)dx,(x,t) ∈(0,a) ×(0,r], W(0,t) > 0, W(a,t) ≥ 0, Z(0, t) > 0, Z(a,t) ≥0, t ∈(0, r], W(x,0) ≥ 0, Z(x,0) ≥ 0, x ∈[0,a], (2.2) then, W(x,t) > 0, Z(x, t) > 0, (x,t) ∈ (0,a) ×(0,r]. Next let r  1 ∈ (r 1 ,1), r  2 ∈ (r 2 ,1) be positive constants and W(x,t) = u(x,t)+η  1+x r  1 −r 1  e ct , Z(x,t) =v(x,t)+η  1+x r  2 −r 2  e ct , (2.3) where η>0issufficiently small and c is a positive constant to be determined. Then W(x,t) > 0, Z(x,t) > 0 on the parabolic boundar y of Ω r , and in (0,a) ×(0, r], we have x q 1 W t −  x r 1 W x  x −  a 0 b 1 (x, t)Z(x,t)dx ≥ x q 1 η  1+x r  1 −r 1  ce ct +  r  1 −r 1  1 −r  1  ηe ct x 2−r  1 −  a 0 b 1 (x, t)η  1+x r  2 −r 2  e ct dx ≥ ηe ct  cx q 1 +  r  1 −r 1  1 −r  1  x 2−r  1 −a  1+a r  2 −r 2  max (x,t)∈[0,a]×[0,r] b 1 (x, t)  , x q 2 Z t −  x r 2 Z x  x −  a 0 b 2 (x, t)W(x,t)dx ≥ ηe ct  cx q 2 +  r  2 −r 2  1 −r  2  x 2−r  2 −a  1+a r  1 −r 1  max (x,t)∈[0,a]×[0,r] b 2 (x, t)  . (2.4) We w ill prove that the above inequalities are nonnegative in three cases. Case 1. When max (x,t)∈[0,a]×[0,r] b 1 (x, t) ≤  r  1 −r 1  1 −r  1  a 3−r  1  1+a r  2 −r 2  , max (x,t)∈[0,a]×[0,r] b 2 (x, t) ≤  r  2 −r 2  1 −r  2  a 3−r  2  1+a r  1 −r 1  . (2.5) It is obvious that x q 1 W t −  x r 1 W x  x −  a 0 b 1 (x, t)Z(x,t)dx ≥0, x q 2 Z t −  x r 2 Z x  x −  a 0 b 2 (x, t)W(x,t)dx ≥0. (2.6) Jun Zhou et al. 5 Case 2. If max (x,t)∈[0,a]×[0,r] b 1 (x, t) >  r  1 −r 1  1 −r  1  a 3−r  1  1+a r  2 −r 2  , max (x,t)∈[0,a]×[0,r] b 2 (x, t) >  r  2 −r 2  1 −r  2  a 3−r  2  1+a r  1 −r 1  . (2.7) Let x 0 and y 0 be the root of the algebraic equations a  1+a r  2 −r 2  max (x,t)∈[0,a]×[0,r] b 1 (x, t) =  r  1 −r 1  1 −r  1  x 2−r  1 , a  1+a r  1 −r 1  max (x,t)∈[0,a]×[0,r] b 2 (x, t) =  r  2 −r 2  1 −r  2  y 2−r  2 , (2.8) and C 1 ,C 2 > 0besufficient large such that C 1 > ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩  max (x,t)∈[0,a]×[0,r] b 1 (x, t)  a  1+a r  2 −r 2  x q 1 0 for q 1 ≥ 0,  max (x,t)∈[0,a]×[0,r] b 1 (x, t)  a  1+a r  2 −r 2  a q 1 for q 1 < 0, C 2 > ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩  max (x,t)∈[0,a]×[0,r] b 2 (x, t)  a  1+a r  1 −r 1  y q 2 0 for q 2 ≥ 0,  max (x,t)∈[0,a]×[0,r] b 2 (x, t)  a  1+a r  1 −r 1  a q 2 for q 2 < 0. (2.9) Set c = max{C 1 ,C 2 },thenwehave x q 1 W t −  x r 1 W x  x −  a 0 b 1 (x, t)Z(x,t)dx ≥ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ηe ct   r  1 −r 1  1 −r  1  x 2−r  1 −a  1+a r  2 −r 2  max (x,t)∈[0,a]×[0,r] b 1 (x, t)  for x ≤ x 0 , ηe ct  cx q 1 −a  1+a r  2 −r 2  max (x,t)∈[0,a]×[0,r] b 1 (x, t)  for x>x 0 , ≥ 0, x q 2 Z t −  x r 2 Z x  x −  a 0 b 2 (x, t)W(x,t)dx ≥ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ηe ct   r  2 −r 2  1 −r  2  x 2−r  2 −a  1+a r  1 −r 1  max (x,t)∈[0,a]×[0,r] b 2 (x, t)  for x ≤ y 0 , ηe ct  cx q 2 −a  1+a r  1 −r 1  max (x,t)∈[0,a]×[0,r] b 2 (x, t)  for x>y 0 , ≥ 0. (2.10) 6 Blowup for degenerate and singular parabolic system Case 3. When max (x,t)∈[0,a]×[0,r] b 1 (x, t) ≤  r  1 −r 1  1 −r  1  a 3−r  1  1+a r  2 −r 2  , max (x,t)∈[0,a]×[0,r] b 2 (x, t) >  r  2 −r 2  1 −r  2  a 3−r  2  1+a r  1 −r 1  , (2.11) or max (x,t)∈[0,a]×[0,r] b 2 (x, t) ≤  r  2 −r 2  1 −r  2  a 3−r  2  1+a r  1 −r 1  , max (x,t)∈[0,a]×[0,r] b 1 (x, t) >  r  1 −r 1  1 −r  1  a 3−r  1  1+a r  2 −r 2  . (2.12) Combining Cases 1 with 2,itiseasytoprove x q 1 W t −  x r 1 W x  x −  a 0 b 1 (x, t)Z(x,t)dx ≥0, x q 2 Z t −  x r 2 Z x  x −  a 0 b 2 (x, t)W(x,t)dx ≥0, (2.13) so we omit the proof here. From the above three cases, we know that W(x,t) > 0, Z(x,t) > 0on[0,a] ×[0,r]. Letting η → 0 + ,wehaveu(x,t) ≥ 0, v(x,t) ≥ 0on[0,a] ×[0,r]. By the arbitrariness of r ∈ (0,T), we complete the proof of Lemma 2.1.  Obviously, (u,v) =(0,0) is a subsolution of (1.1), we need to construct a supersolu- tion. Lemma 2.2. There exists a positive constant t 0 (t 0 <T) such that the problem (1.1)hasa supersolution (h 1 (x, t),h 2 (x, t)) ∈(C(Ω t 0 ) ∩C 2,1 (Ω t 0 )) 2 . Proof. Let ψ(x) =  x a  1−r 1  1 − x a  +  x a  (1−r 1 )/2  1 − x a  1/2 , ϕ(x) =  x a  1−r 2  1 − x a  +  x a  (1−r 2 )/2  1 − x a  1/2 , (2.14) and let K 0 be a p ositive constant such that K 0 ψ(x) ≥u 0 (x), K 0 ϕ(x) ≥ v 0 (x). Denote the positive constant  1 0 [s 1−r 1 (1 −s)+s (1−r 1 )/2 (1 −s) 1/2 ] p 2 ds by b 20 and  1 0 [s 1−r 2 (1 − s)+s (1−r 2 )/2 (1 − s) 1/2 ] p 1 ds by b 10 .LetK 10 ∈ (0, (1 − r 1 )/(2 − r 1 )), K 20 ∈ (0,(1 −r 2 )/(2 −r 2 )) be positive constants such that K 10 ≤  2 p 1 +1 a 3−r 1 b 10 K p 1 −1 0  −2/(1−r 1 ) , K 20 ≤  2 p 2 +1 a 3−r 2 b 20 K p 2 −1 0  −2/(1−r 2 ) . (2.15) Jun Zhou et al. 7 Let (K 1 (t), K 2 (t)) be the positive solution of the following initial value problem: K  1 (t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ b 10 K p 1 2 (t) a q 1 −1 K q 1 10  K 10  1 −K 10  1−r 1 + K 1/2 10  1 −K 10  (1−r 1 )/2  , q 1 ≥ 0, b 10 K p 1 2 (t) a q 1 −1  1 −K 10  q 1  K 10  1 −K 10  1−r 1 + K 1/2 10  1 −K 10  (1−r 1 )/2  , q 1 < 0, K 1 (0) =K 0 , K  2 (t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ b 20 K p 2 1 (t) a q 2 −1 K q 2 20  K 20  1 −K 20  1−r 2 + K 1/2 20  1 −K 20  (1−r 2 )/2  , q 2 ≥ 0, b 20 K p 2 1 (t) a q 2 −1  1 −K 20  q 2  K 20  1 −K 20  1−r 2 + K 1/2 20  1 −K 20  (1−r 2 )/2  , q 2 < 0, K 2 (0) =K 0 . (2.16) Since K 1 (t), K 2 (t) are increasing functions, we can choose t 0 > 0suchthatK 1 (t) ≤ 2K 0 , K 2 (t) ≤2K 0 for all t ∈[0,t 0 ]. Set h 1 (x, t) =K 1 (t)ψ(x), h 2 (x, t) =K 2 (t)ϕ(x), then h 1 (x, t) ≥ 0, h 2 (x, t) ≥ 0onΩ t 0 . We would like to show that (h 1 (x, t),h 2 (x, t)) is a supersolution of (1.1)inΩ t 0 . To do this, let us construct two functions J 1 , J 2 by J 1 = x q 1 h 1t −  x r 1 h 1x  x −  a 0 h p 1 2 dx,(x, t) ∈Ω t 0 , J 2 = x q 2 h 2t −  x r 2 h 2x  x −  a 0 h p 2 1 dx,(x, t) ∈Ω t 0 . (2.17) Then, J 1 = x q 1 h 1t −  x r 1 h 1x  x −  a 0 h p 1 2 dx = x q 1 K  1 ψ(x)+  2 −r 1 a 2−r 1 +   1 −r 1  2 4 x (r 1 −3)/2 (a −x) 1/2 + 1 2 x (r 1 −1)/2 (a −x) −1/2 + 1 4 x (1+r 1 )/2 (a −x) −3/2  × 1 a 1−r 1 /2  K 1 (t) −ab 10 K p 1 2 (t) ≥ x q 1 K  1 (t)ψ(x)+x (r 1 −1)/2 (a −x) −1/2 K 1 (t) 2a 1−r 1 /2 −ab 10 K p 1 2 (t), J 2 ≥ x q 2 K  2 (t)ϕ(x)+x (r 2 −1)/2 (a −x) −1/2 K 2 (t) 2a 1−r 2 /2 −ab 20 K p 2 1 (t). (2.18) 8 Blowup for degenerate and singular parabolic system For (x,t) ∈ (0,aK 10 ) ×(0,t 0 ] ∪(a(1 −K 10 ),a) ×(0,t 0 ], by (2.15), we have J 1 ≥ x (r 1 −1)/2 (a −x) −1/2 K 1 (t) 2a 1−r 1 /2 −ab 10 K p 1 2 (t) ≥  K (r 1 −1)/2 10 2a 2−r 1  K 1 (t) −ab 10 K p 1 2  t 0  ≥  K (r 1 −1)/2 10 2a 2−r 1  K 0 −ab 10  2K 0  p 1 ≥ 0. (2.19) For (x,t) ∈ (0,aK 20 ) ×(0,t 0 ] ∪(a(1 −K 20 ),a) ×(0,t 0 ], by (2.15), we have J 2 ≥  K (r 2 −1)/2 20 2a 2−r 2  K 0 −ab 20  2K 0  p 2 ≥ 0. (2.20) For (x,t) ∈ [aK 10 ,a(1 −K 10 )] ×(0,t 0 ]by(2.16), we have J 1 ≥ x q 1 K  1 (t)ψ(x) −ab 10 K p 1 2 (t) ≥ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a q 1 K q 1 10 K  1 (t)  K 10  1 −K 10  1−r 1 + K 1/2 10  1 −K 10  (1−r 1 )/2  − ab 10 K p 1 2 (t), q 1 ≥0, a q 1  1 −K 10  q 1 K  1 (t)  K 10  1 −K 10  1−r 1 + K 1/2 10  1 −K 10  (1−r 1 )/2  − ab 10 K p 1 2 (t), q 1 <0, ≥ 0, (2.21) For (x,t) ∈ [aK 20 ,a(1 −K 20 )] ×(0,t 0 ]by(2.16), we have J 2 ≥ x q 2 K  2 (t)ϕ(x) −ab 20 K p 2 1 (t) ≥ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a q 2 K q 2 20 K  2 (t)  K 20  1 −K 20  1−r 2 + K 1/2 20  1 −K 20  (1−r 2 )/2  − ab 20 K p 2 1 (t), q 2 ≥0, a q 2  1−K 20  q 2 K  2 (t)  K 20  1−K 20  1−r 2 + K 1/2 20  1−K 20  (1−r 1 )/2  − ab 20 K p 2 1 (t), q 2 <0, ≥ 0. (2.22) Thus, J 1 (x, t)≥0, J 2 (x, t)≥0inΩ t 0 .Itfollowsfromh 1 (0,t)=h 1 (a,t)=h 2 (0,t)=h 2 (a,t)=0 and h 1 (x,0)= K 0 ψ(x) ≥u 0 (x), h 2 (x,0)= K 0 ϕ(x) ≥ v 0 (x)that(h 1 (x, t),h 2 (x, t)) is a super- solution of (1.1)inΩ t 0 .TheproofofLemma 2.2 is complete.  To show the existence of the classical solution (u(x,t),v(x,t)) of (1.1), let us intro- duce a cutoff function ρ(x). By Dunford and Schwartz [8, page 1640], there exists a Jun Zhou et al. 9 nondecreasing ρ(x) ∈ C 3 (R)suchthatρ(x) = 0ifx ≤ 0andρ(x) = 1ifx ≥ 1. Let 0 < δ<min {(1 −r 1 )/(2 −r 1 )a,(1−r 2 )/(2 −r 2 )a}, ρ δ (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, x ≤ δ, ρ  x δ −1  , δ<x<2δ, 1, x ≥ 2δ, (2.23) and u 0δ (x) = ρ δ (x) u 0 (x), v 0δ (x) = ρ δ (x) v 0 (x). We note that ∂u 0δ (x) ∂δ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, x ≤ δ, − x δ 2 ρ   x δ −1  u 0 (x), δ<x<2δ, 0, x ≥ 2δ, ∂v 0δ (x) ∂δ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, x ≤ δ, − x δ 2 ρ   x δ −1  v 0 (x), δ<x<2δ, 0, x ≥ 2δ. (2.24) Since ρ is nondecreasing, we have ∂u 0δ (x) /∂δ ≤0, ∂v 0δ (x) /∂δ ≤0. From 0 ≤ρ(x) ≤1, we have u 0 (x) ≥ u 0δ (x), v 0 (x) ≥ v 0δ (x)andlim δ→0 u 0δ (x) = u 0 (x), lim δ→0 v 0δ (x) = v 0 (x). Let D δ = (δ,a), let w δ = D δ ×(0,t 0 ], let D δ and w δ be their respective closures, and let S δ ={δ,a}×(0,t 0 ]. We consider the following regular ized problem: x q 1 u δt −  x r 1 u δx  x =  a δ v p 1 δ dx,(x,t) ∈w δ , x q 2 v δt −  x r 2 v δx  x =  a δ u p 2 δ dx,(x,t) ∈w δ , u δ (δ,t) =u δ (a,t) =v δ (δ,t) =v δ (a,t) =0, t ∈  0,t 0  , u δ (x,0)= u 0δ (x), v δ (x,0)= v 0δ (x), x ∈ D δ . (2.25) By using Schauder’s fixed point theorem, we have t he following. Theorem 2.3. The problem (2.25) admits a unique nonnegative solution (u δ ,v δ ) ∈ (C 2+α,1+α/2 (w δ )) 2 .Moreover,0 ≤u δ ≤ h 1 (x, t), 0 ≤v δ ≤ h 2 (x, t), (x,t) ∈w δ ,whereh 1 (x, t), h 2 (x, t) are given by Lemma 2.2. Proof. By the proof of Lemma 2.1, we know that there exists at most one nonnegative solution (u δ ,v δ ). To prove existence, we use Schauder’s fixed point theorem. Let X 1 =  v 1 ∈ C α,α/2  w δ  :0≤ v 1 (x, t) ≤h 2 (x, t), (x,t) ∈w δ  , X 2 =  u 1 ∈ C α,α/2  w δ ):0≤ u 1 (x, t) ≤h 1 (x, t), (x,t) ∈w δ  . (2.26) 10 Blowup for degenerate and singular parabolic system Obviously, X 1 , X 2 are closed convex subsets of Banach space C α,α/2 (w δ ). In order to get the conclusion, we have to define another set: X = X 1 ×X 2 .Obviously(C α,α/2 (w δ )) 2 is a Banach space with the norm    v 1 ,u 1    α,α/2 =   v 1   α,α/2 +   u 1   α,α/2 ,forany  v 1 ,u 1  ∈  C α,α/2  w δ  2 , (2.27) and X is a closed convex subset of Banach space (C α,α/2 (w δ )) 2 .Foranyv 1 ∈ X 1 , u 1 ∈ X 2 , let us consider the following linearized uniformly parabolic problem: x q 1 W δt −  x r 1 W δx  x =  a δ v p 1 1 dx,(x, t) ∈w δ , x q 2 Z δt −  x r 2 Z δx  x =  a δ u p 2 1 dx,(x,t) ∈w δ , W δ (δ,t) =W δ (a,t) =Z δ (δ,t) =Z δ (a,t) =0, t ∈  0,t 0  , W δ (x,0)= u 0δ (x), Z δ (x,0)= v 0δ (x), x ∈ [δ,a]. (2.28) It is easy to see that (W (x, t),Z(x,t)) = (0,0) and (W(x,t),Z(x,t)) = (h 1 (x, t),h 2 (x, t)) are subsolution and supersolution of problem (2.28). We also note that x −q 1 +r 1 , x −q 1 −1+r 1 , x −q 1 , x −q 2 +r 2 , x −q 2 −1+r 2 , x −q 2 ∈ C α,α/2 (w δ ), and x −q 1  a δ v p 1 1 dx, x −q 2  a δ u p 2 1 dx ∈ C α,α/2 (w δ ), u 0δ (x), v 0δ (x) ∈ C 2+α (D δ ).ItfollowsfromTheorem4.2.2ofLaddleetal.[11, page 143] that the problem (2.28) has a unique solution (W δ (x, t;v 1 ,u 1 ),Z δ (x, t;v 1 ,u 1 )) ∈ (C 2+α,1+α/2 (w δ )) 2 , which satisfies 0 ≤ W δ (x, t;v 1 ,u 1 ) ≤h 1 (x, t), 0≤Z δ (x, t;v 1 ,u 1 )≤h 2 (x, t). Thus, we can define a mapping Y from X into (C 2+α,1+α/2 (w δ ) 2 ,suchthat Y  v 1 (x, t),u 1 (x, t)  =  W δ  x, t;v 1 ,u 1  ,Z δ  x, t;v 1 ,u 1  , (2.29) where (W δ (x, t;v 1 ,u 1 ),Z δ (x, t;v 1 ,u 1 )) denotes the unique solution of (2.28) correspond- ing to (v 1 (x, t),u 1 (x, t)) ∈X. To use Schauder’s fixed point theorem, we need to verify the fact that Y maps X into itself is continuous and compact. In fact, YX ⊂ X and the embedding operator form Banach space (C 2+α,1+α/2 (w δ )) 2 to the Banach space (C α,α/2 (w δ )) 2 is compact. Therefore Y is compact. To show Y is contin- uous in X 1 let us consider a sequence {v 1n (x, t)} which converges to v 1 (x, t)uniformlyin the norm · α,α/2 .Weknowthatv 1 (x, t) ∈X 1 .Analogously,inX 2 we consider a sequence {u 1n (x, t)} which converges to u 1 (x, t) uniformly in the norm · α,α/2 and u 1 (x, t) ∈X 2 . So we get a sequence {(v 1n (x, t),u 1n (x, t))}⊂X, which converges to (v 1 (x, t),u 1 (x, t)) uni- formly in the norm (·,·) α,α/2 and (v 1 (x, t),u 1 (x, t)) ∈ X.Let(W δ n(x,t),Z δ n(x,t)) and (W δ (x, t),Z δ (x, t)) be the solution of problem (2.28) corresponding to (v 1n (x, t),u 1n (x, t)) and (v 1 (x, t),u 1 (x, t)), respectively. Without loss of generality, let us assume that   v 1n (x, t)   α,α/2 ≤   v 1 (x, t)   α,α/2 +1, foranyn ≥1,   u 1n (x, t)   α,α/2 ≤   u 1 (x, t)   α,α/2 +1, foranyn ≥1. (2.30) [...]... and Applied Mathematics 113 (2000), no 1-2, 353–364 [6] Y P Chen, Q Liu, and C H Xie, Blow-up for degenerate parabolic equations with nonlocal source, Proceedings of the American Mathematical Society 132 (2004), no 1, 135–145 [7] Y P Chen and C H Xie, Blow-up for degenerate, singular, semilinear parabolic equations with nonlocal source, Acta Mathematica Sinica 47 (2004), no 1, 41–50 [8] N Dunford and. .. 965–992 [3] C Y Chan and W Y Chan, Existence of classical solutions for degenerate semilinear parabolic problems, Applied Mathematics and Computation 101 (1999), no 2-3, 125–149 [4] C Y Chan and H T Liu, Global existence of solutions for degenerate semilinear parabolic problems, Nonlinear Analysis 34 (1998), no 4, 617–628 [5] C Y Chan and J Yang, Complete blow-up for degenerate semilinear parabolic equations,... (a1 ,a2 ) × (t2 ,t3 ] such that (x1 ,t1 ) ∈ Q ⊂ Q ⊂ Q ⊂ (0,a) × (0,t0 ] with 0 < a1 < a1 < a1 < x1 < a2 < a2 < a2 < a, 0 ≤ t2 ≤ t2 ≤ t2 < t1 < t3 ≤ t3 ≤ t3 ≤ t0 Since 12 Blowup for degenerate and singular parabolic system (uδ (x,t),vδ (x,t)) ≤ (h1 (x,t),h2 (x,t)) in Q and h1 (x,t), h2 (x,t) are finite on Q , for any constant q > 1 and some positive constants K3 , K4 , we have (i) uδ Lq (Q ) a (ii) x−q1... v p1 (y,t − τ)d y dτ t ∈ (0,T) It follows from the above inequality and (3.25) that limsupt→T u(x,t) = +∞ (3.26) 18 Blowup for degenerate and singular parabolic system For any x ∈ {0,a}, we can choose a sequence {(xn ,tn )} such that (xn ,tn ) → (x,T) (n → +∞) and limn→∞ u(xn ,tn ) = +∞ Thus the blowup set is the whole domain [0,a], and we complete the proof of Theorem 3.6 Case 2 q1 = 0, 0 ≤ r1 < 1... Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992 [16] Y.-W Qi and H A Levine, The critical exponent of degenerate parabolic systems, Zeitschrift f¨ r u Angewandte Mathematik und Physik 44 (1993), no 2, 249–265 [17] A A Samarskii, V A Galaktionov, S P Kurdyumov, and A P Mikhailoi, Blow-up in Qusilinear Parabolic Equations, Nauka, Moscow, 1987 [18] P Souplet, Blow-up in nonlocal reaction-diffusion... Floater, Blow-up at the boundary for degenerate semilinear parabolic equations, Archive for Rational Mechanics and Analysis 114 (1991), no 1, 57–77 [10] A Friedman and B McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana University Mathematics Journal 34 (1985), no 2, 425–447 [11] G S Laddle, V Lakshmikantham, and A S Vatsala, Monotone Iterative Techniques for Nonlinear Differential... V A Solonikiv, and N N Ural’ceva, Linear and Quasilinear Equations of z Parabolic Type, Translations of Mathematical Monographs, vol 23, American Mathematical Society, Rhode Island, 1967 Jun Zhou et al 19 [13] N W Mclachlan, Bessel Functions for Engineers, 2nd ed., Clarendon Press, Oxford University Press, London, 1955 [14] H Ockendon, Channel flow with temperature-dependent viscosity and internal viscous... associated with the operator L = xq1 (∂/∂t) − ∂2 /∂x2 with the first boundary condition, and obtained the following lemmas Lemma 3.4 (a) For t > τ, G(x,ξ,t − τ) is continuous for (x,t,ξ,τ) ∈ ([0,a] × (0,T]) × ((0,a] × [0,T)) (b) For each fixed (ξ,τ) ∈ (0,a] × [0,T), Gt (x,ξ,t − τ) ∈ C([0,a] × (τ,T]) (c) In {(x,t,ξ,τ) : x and ξ are in (0,a), T ≥ t > τ ≥ 0}, G(x,ξ,t − τ) is positive Lemma 3.5 For fixed x0... supported in part by NNSF of China (10571126) and in part by Program for New Century Excellent Talents in University References [1] C Budd, B Dold, and A Stuart, Blowup in a partial differential equation with conserved first integral, SIAM Journal on Applied Mathematics 53 (1993), no 3, 718–742 [2] C Budd, V A Galaktionov, and J Chen, Focusing blow-up for quasilinear parabolic equations, Proceedings of the... theorem, we know that u H 2+α ,1+α /2 (Q ) ≤ K8 , v H 2+α ,1+α /2 (Q ) ≤ K8 , (2.37) for some α ∈ (0,α) and some positive constant K8 independent of δ, and that the derivatives of u and v are uniform limits of the corresponding partial derivatives of uδ Jun Zhou et al 13 and vδ , respectively Hence (u(x,t),v(x,t)) satisfies (1.1), and limt→0 (u(x,t),v(x,t)) = limt→0 limδ →0 (uδ (x,t),vδ (x,t)) = limδ →0 (u0δ . BLOWUP FOR DEGENERATE AND SINGULAR PARABOLIC SYSTEM WITH NONLOCAL SOURCE JUN ZHOU, CHUNLAI MU, AND ZHONGPING LI Received 23 January 2006; Revised 3 April 2006; Accepted 7 April 2006 We deal with. inequality and (3.25) that limsup t→T u(x,t) =+∞. 18 Blowup for degenerate and singular parabolic system For any x ∈{0,a}, we can choose a sequence {(x n ,t n )} such that (x n ,t n ) →(x,T)(n → +∞)andlim n→∞ u(x n ,t n ). Blow-up for degenerate parabolic equations with nonlocal source, Proceedings of the American Mathematical Society 132 (2004), no. 1, 135–145. [7] Y.P.ChenandC.H.Xie,Blow-up for degenerate, singular,