Multiple blow-up rates in a coupled heat system with mixed-type nonlinearities

Nonlinear Analysis 70 (2009) 2176–2197 www.elsevier.com/locate/na

Multiple blow-up rates in a coupled heat system with mixed-type nonlinearitiesI Sining Zheng ∗ , Lan Qiao 1 Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China Received 18 December 2007; accepted 28 February 2008

Abstract This paper studies heat equations with inner absorptions and coupled boundary fluxes of mixed-type nonlinearities. At first, the critical exponent is obtained, and simply described via a characteristic algebraic system introduced by us. Then, as the main results of the paper, three blow-up rates are established under different dominations of nonlinearities for the one-dimensional case, and represented in another characteristic algebraic system. In particular, it is observed that unlike those in previous literature on parabolic models with absorptions, two of the multiple blow-up rates obtained here do depend on the absorption exponents. In the known works, the absorptions affect the blow-up criteria, the blow-up time, as well as the initial data required for the blow-up of solutions, all without changing the blow-up rates. To our knowledge, this is the first example of absorption-dependent blowup rates, exploiting the significant interactions among diffusions, inner absorptions and nonlinear boundary fluxes in the coupled system. It is also proved that the blow-up of solutions in the model occurs on the boundary only. c 2008 Elsevier Ltd. All rights reserved.

MSC: primary 35K57; 35B40; 35K33 Keywords: Coupled heat equations; Critical exponents; Nonlinear absorption; Nonlinear boundary flux; Multiple blow-up rates; Blow-up set; Characteristic algebraic system

1. Introduction In this paper we consider the following parabolic system with inner absorptions and coupled nonlinear boundary fluxes of mixed-type nonlinearities:  u t = 1u − a1 u m , vt = 1v − a2 env , (x, t) ∈ Ω × (0, T ),     ∂u ∂v = e pv , = u q , (x, t) ∈ ∂Ω × (0, T ), (1.1)  ∂η ∂η    u(x, 0) = u 0 (x), v(x, 0) = v0 (x), x ∈ Ω¯ , I Supported by the National Natural Science Foundation of China (10771024). ∗ Corresponding author.

E-mail address: [email protected] (S. Zheng). 1 Current address: College of Science, Tianjin University of Science and Technology, Tianjin 300457, PR China. c 2008 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2008.02.118

2177

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

where m, n ≥ 0, p, q > 0, a1 , a2 > 0; Ω ⊂ Rn is a bounded domain with smooth boundary ∂Ω , and η is the exterior normal vector on ∂Ω ; u 0 (x), v0 (x) are nontrivial and nonnegative functions satisfying compatibility conditions. Parabolic models like (1.1) can be used to describe heat propagation of two mixed media with nonlinear absorptions and nonlinear fluxes [1–4,11,18,25]. The nonlinear Neumann boundary conditions in (1.1) represent cross-boundary fluxes. Parabolic systems with absorptions and coupled boundary fluxes of one type nonlinearities  u t = 1u − a1 u m , vt = 1v − a2 v n , (x, t) ∈ Ω × (0, T ),     ∂u ∂v = v p, = u q , (x, t) ∈ ∂Ω × (0, T ), (1.2)  ∂η ∂η    u(x, 0) = u 0 (x), v(x, 0) = v0 (x), x ∈ Ω¯ and  u t = 1u − a1 emu , vt = 1v − a2 env , (x, t) ∈ Ω × (0, T ),     ∂u ∂v = e pv , = equ , (x, t) ∈ ∂Ω × (0, T ),  ∂η ∂η    u(x, 0) = u 0 (x), v(x, 0) = v0 (x), x ∈ Ω¯ ,

(1.3)

were studied in [30,27] respectively. The following characteristic algebraic systems ! ! ! −µ p τ11 1 = q −γ τ12 1 with µ=1+



m−1 2

 +



 m+1 = max ,1 , 2

γ =1+



n−1 2





+

 n+1 = max ,1 , 2

and  1 − 2 m   q



! p  τ ! 1 21  = 1  τ22 1 − n 2 were introduced to describe the critical blow-up exponents for (1.2) and (1.3) correspondingly. In this paper, the critical blow-up exponent for (1.1) will be represented via such a characteristic algebraic system of the form:   ! ! −µ p τ1 1  = , (1.4) 1  τ2 1 q − n 2 namely, τ1 =

2p + n , 2 pq − µn

τ2 =

2(q + µ) 2 pq − µn

(1.5)

with µ=1+



m−1 2

 +



 m+1 = max ,1 . 2

(1.6)

Indeed, we will show that the solutions of (1.1) are global if 1/τ1 , 1/τ2 < 0, and nonglobal for large initial data with either 1/τ1 > 0 or 1/τ2 > 0; if 1/τ1 = 1/τ2 = 0, then blow-up or not of solutions will depend on a1 , a2 , and m. Here (1/τ1 , 1/τ2 ) = (0, 0) is defined with the limit of (1/τ1 , 1/τ2 ) (determined by (1.5)) as pq → µn.

2178

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

The main goal of the paper is to study multiple blow-up rates for one-dimensional case (1.1), i.e.,  u t = u x x − a1 u m , vt = vx x − a2 env , (x, t) ∈ (0, 1) × (0, T ),    u (1, t) = e pv(1,t) , vx (1, t) = u q (1, t), t ∈ (0, T ), x  u x (0, t) = 0, vx (0, t) = 0, t ∈ (0, T ),    u(x, 0) = u 0 (x), v(x, 0) = v0 (x), x ∈ [0, 1]

(1.7)

00 nv0 ≥ 0. We will determine three different blow-up with initial data u 0 , v0 ≥ δ > 0, u 00 , v00 ≥ 0, u 000 − a1 u m 0 , v0 − a2 e rates, represented by another characteristic algebraic system ! ! ! −1 θ1 α 1 = (1.8) θ2 0 β 1

with   if ( p, q) ,        2p 1 p, q( − ) , if (θ1 , θ2 ) =  n q        2 p(q + 1)   , q , if m+1

m ≤ 2q + 1, n <

2 pq , q +1

m ≤ 2q + 1, n ≥

2 pq , q +1

m > 2q + 1, n <

2 pq , q +1

(1.9)

namely,    1 q +1  (α , β ) = ,  1 1   q pq       n 2q (α, β) = (α2 , β2 ) = ,  2 pq − n 2 pq − n        1 m+1   (α3 , β3 ) = , q 2 pq

for

m ≤ 2q + 1, n <

2 pq , q +1

for

m ≤ 2q + 1, n ≥

2 pq , q +1

for

m > 2q + 1, n <

2 pq . q +1

(1.10)

By means of the above (1.8)–(1.10), the multiple blow-up rates will be briefly described as c ≤ max u(·, t)(T − t)αi /2 ≤ C, [0,1]

c ≤ exp{max v(·, t)}(T − t)βi /2 ≤ C [0,1]

with i = 1, 2, 3, as t tends to the blow-up time T . Currently, a parabolic system with coupled mixed-type boundary fluxes without absorptions  ut = u x x , vt = vx x , (x, t) ∈ (0, 1) × (0, T ),    u (1, t) = e pv(1,t) , vx (1, t) = u q (1, t), t ∈ (0, T ), x  u x (0, t) = 0, vx (0, t) = 0, t ∈ (0, T ),    u(x, 0) = u 0 (x), v(x, 0) = v0 (x), x ∈ [0, 1]

(1.11)

was studied by Song [22]. The blow-up rate for (1.11) was determined as 1

c ≤ max u(·, t)(T − t) 2q ≤ C, [0,1]

q+1

c ≤ exp{max v(·, t)}(T − t) 2 pq ≤ C [0,1]

(1.12)

2 pq as t → T , which coincides with the situation of (α1 , β1 ) for (1.7) under weak absorptions that m ≤ 2q + 1, n ≤ q+1 . So, the inner absorptions included in the model yield much more complicated multi-nonlinearity interactions. It is easy to understand for parabolic equations that either nonlinear diffusions (subject null Dirichlet boundary data) or inner absorptions make negative contributions to the blowing up of solutions. However, in general, they affect

2179

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

the blow-up criteria, the blow-up time as well as the initial data required for blow-up of solutions, all without changing the blow-up rates. For example, in the Cauchy or Dirichlet problems to nonlinear diffusion equation u t = 1u m + u p ,

(1.13)
the occurrence of blow-up requires p > m ≥ 1, while the blow-up rate was known as O (T − t)−1/( p−1) , independent of the nonlinear diffusion exponent m [7,9,10]. The similar ones happen with inner absorptions as shown in the following examples: Souplet [24] studied Z u t − 1u = u p (y, t)dy − u q (1.14) Ω

with p > q ≥ 1 and blow-up time T , and proved that 1

lim (T − t) p−1 u(x, t) = [( p − 1)|Ω |]

1 − p−1

t→T

(1.15)

uniformly on compact subsets of Ω . Rossi [21] obtained that the scalar problem  q in (0, 1) × (0, T ),  u t = u x x − λu p u x (0, t) = 0, u x (1, t) = u (1, t) in (0, T ), (1.16)   u(x, 0) = u 0 (x) on [0, 1]   − 1 possesses the blow-up rate O (T − t) 2( p−1) for either 1 < q < 2 p − 1 or 1 < q = 2 p − 1 with 0 < λ < p. Currently, for the nonlinear diffusion scalar problem  m (u )t = 1u − λu q in Ω × (0, T ),    ∂u = up on ∂Ω × (0, T ), (1.17)  ∂η   u(x, 0) = u 0 (x) on Ω¯ with m ≥ 1, the blow-up rate was determined as O((T − t)−1/[2 p−(m+1)] ) [13,17]. Notice that the blow-up rates for the three examples (1.14), (1.16) and (1.17) are all independent of the absorption exponent q. Differently, two of the multiple simultaneous blow-up rates established in this paper for (1.7) (described via (αi , βi ) (i = 2, 3)) do depend on the absorption exponents m and n. To our knowledge, this is the first example with absorption-dependent blow-up rates for nonlinear parabolic problems, which exploits the significant interactions among diffusions, inner absorptions and nonlinear boundary fluxes in the coupled system, and shows a substantial difference on blow-up rates of solutions between scalar and coupled parabolic equations with absorptions. The rest of the paper is organized as follows. The critical exponents for (1.1) will be established in Section 2. Section 3 deals with the multiple blow-up rates for the one-dimensional model (1.7), and Section 4 is arranged for the blow-up sets. Some remarks with a figure are given in the last section to illustrate the main results of the paper by summarizing the complete classification for the nonlinear parameters of the model. Throughout the paper c and C denote positive constants independent of t, and may change from line to line. 2. Critical exponent This section deals with critical blow-up exponents for (1.1), stated in the following three theorems: Theorem 2.1. If 1/τ1 > 0 or 1/τ2 > 0, then the solutions of (1.1) blow up in finite time with large initial data. Theorem 2.2. If 1/τi < 0, i = 1, 2, then the solutions of (1.1) are global. Theorem 2.3. Assume 1/τ1 = 1/τ2 = 0. (i) If either µ = 1, or µ > 1 with a1 , a2 large enough, the solutions of (1.1) are global. (ii) If µ > 1 with a1 , a2 properly small, the solutions of (1.1) blow up in finite time with large initial data. To prove these theorems, we begin with some lemmas.

2180

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

Lemma 2.1. When n = 0, the solutions of (1.1) blow up in finite time for large initial data. Proof. For u, v ≥ 0 with p, q > 0, there exists K ∈ N such that K q > µ and e pv ≥ problem  v t = 1v − a2 , (x, t) ∈ Ω × (0, T ), u t = 1u − a1 u m ,      K K p v ∂v ∂u = , = u q , (x, t) ∈ ∂Ω × (0, T ),  ∂η K ! ∂η     u(x, 0) = u 0 (x), v(x, 0) = v0 (x), x ∈ Ω¯ .

pK vK K!

. Consider the auxiliary

Clearly, (u, v) is a pair of subsolution of (1.1). By the result for (1.2) in [30], (u, v) blows up in finite time for large initial data (u 0 , v0 ).  Let ϕ0 be the first eigenfunction of 1ϕ + λϕ = 0

in Ω ,

ϕ=0

on ∂Ω

(2.1)

with the first eigenvalue λ0 , normalized by kϕ0 k∞ = 1, ϕ0 > 0 in Ω . Then   ∂ϕ0 = − ∂ϕ0 ≤ c2 = max |∇ϕ0 | c1 ≤ ∂η ∂ Ω ∂η ∂ Ω Ω¯ for some constants c1 , c2 > 0. In addition, there exist positive constants ε0 and c3 such that |∇ϕ0 | ≥ ϕ0 ≥ c 3

1 c1 for x ∈ Ω1 = {x ∈ Ω : dist(x, ∂Ω ) ≤ ε0 }, 2 for x ∈ Ω2 = {x ∈ Ω : dist(x, ∂Ω ) ≥ ε0 }.

It is well known that λ0 , ε0 and ci (i = 1, 2, 3) depend on the size and shape of Ω . Lemma 2.2. If 2 pq > µn with m > 1 and n > 0, then the solutions of (1.1) blow up in finite time for large initial data. Proof. Construct u(x, t) =

A , [ϕ Aµ−1 + (1 − ct) K ]1/(µ−1)

v(x, t) = log

B , [ϕ B n/2 + (1 − ct) L ]2/n

where ϕ = Mϕ0 with positive constants M, A, B, K , L, c to be determined. Simple computations show   Aµ ∂ϕ Aµ ∂u = − ≤ Mc2 , Kµ Kµ ∂η ∂η (µ − 1)(1 − ct) µ−1 (µ − 1)(1 − ct) µ−1   ∂v 2B n/2 ∂ϕ 2B n/2 = − ≤ Mc2 , L ∂η n(1 − ct) ∂η n(1 − ct) L Bp Aq q e pv = , u = Kq 2L p (1 − ct) n (1 − ct) µ−1   on ∂Ω × (0, T ). Because of 2 pq > µn and µ = 1 + m−1 > 1 for m > 1, the inequalities 2 +

µ−1 2 p(µ − 1) L


Mc2 µ−1

1

p

µ

Ap < B <



n 2Mc2

2 n

2q

An

are obviously true for some large A, B, K and L, and thus  1  2 n µK 2L 2L 2q Mc2 p µp n − (µ−1) − 2q K n p A (1 − ct)
(2.2)

2181

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

which yields ∂u ≤ e pv , ∂η

∂v ≤ uq ∂η

on ∂Ω × (0, T ).

(2.3)

In Ω × (0, T ), we have ut =

AK c(1 − ct) K −1 µ µ−1

,

vt =

2Lc(1 − ct) L−1 , n[ϕ B n/2 + (1 − ct) L ]

(µ − 1)[ϕ Aµ−1 + (1 − ct) K ] A µ λ0 ϕ µAm |∇ϕ|2 , 1u = µ + 2µ−1 (µ − 1)[ϕ Aµ−1 + (1 − ct) K ] µ−1 (µ − 1)2 [ϕ Aµ−1 + (1 − ct) K ] µ−1 2B n/2 λ0 ϕ 2B n |∇ϕ|2 1v = + . n[ϕ B n/2 + (1 − ct) L ] n[ϕ B n/2 + (1 − ct) L ]2 Furthermore,     µM 2 c12 a1 (µ−1)A2(µ−1)  AK c − − 1   K (M Aµ−1 +1) 4(µ−1)2 a1    µ   (µ − 1)[ϕ Aµ−1 + (1 − ct) K ] µ−1 m u t − 1u + a1 u ≤     M 2 c32 λ0  a1 (µ−1)Aµ−1  AK c − − 1  K Mc3 (µ−1)a1     µ (µ − 1)[ϕ Aµ−1 + (1 − ct) K ] µ−1

for x ∈ Ω1 ,

for x ∈ Ω 2 = Ω \ Ω1 ,

and similarly,

v t − 1v + a2 e

nv

    n M 2 c12 a n B 2  2L c − 2L(M B n/2 +1) 2na2 − 1       n[ϕ B n/2 + (1 − ct) L ] ≤    2 2   a2 n B n/2 2M c3 λ0  2L c − − 1  2L Mc3 na2    n[ϕ B n/2 + (1 − ct) L ]

for x ∈ Ω1 ,

for x ∈ Ω 2 .

Letting 8a1 (µ − 1)2 2a1 (µ − 1) 4a2 n a2 n M = max , , , µc12 λ0 c32 c12 λ0 c32

!

2

,

a2 n B n a1 A2(µ−1) (µ − 1) a1 Aµ−1 (µ − 1) a2 n B n/2 , c = min , , K Mc3 K (M Aµ−1 + 1) 2L(M B n/2 + 1) 2M Lc3

! ,

we derive u t ≤ 1u − a1 u m ,

v t ≤ 1v − a2 env

in Ω × (0, T ).

(2.4)

With the initial data u 0 (x) ≥ u(x, 0), v0 (x) ≥ v(x, 0) on Ω¯ , we obtain from (2.3) and (2.4) that (u, v) is a blowing up subsolution of (1.1).  Lemma 2.3. If 2 pq > µn with m = 1 and n > 0, the solutions for (1.1) blow up in finite time for large initial data. Proof. Let u(x, t) =

A ϕ

(1 − ct) K e 1−ct

,

v(x, t) = log

B , [ϕ B n/2 + (1 − ct) L ]2/n

(2.5)

where ϕ = Mϕ0 , and M, A, B, K , L, c are positive constants It is known from the proof of   to be determined.   4a2 n a2 n a2 n B n a2 n B n/2 nv 2 Lemma 2.2 that v t ≤ 1v − a2 e with M = max , , c = min , . 2 2 2L(M B n/2 +1) 2M Lc3 c1

λ0 c3

2182

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

We have on the boundary that ∂u ∂v A 2B n/2 Mc , Mc2 , ≤ ≤ 2 ∂η ∂η n(1 − ct) L (1 − ct) K +1 Bp Aq q e pv = = , u . (1 − ct) K q (1 − ct)2L p/n Since 2 pq > µn = n, there exist large A, B, K and L such that   2p 2Mc2 1/q n/2q 1 1 B p, L < K < K +1< L, B


Lemma 2.4. If 2 pq > µn with m < 1 and n > 0, the solutions for (1.1) blow up in finite time for large initial data. Proof. Define u(x, t) =

A , (ϕ + 1 − ct) K

v(x, t) = log

[ϕ B n/2

B + (1 − ct) L ]2/n

with ϕ = Mϕ0 . As 2 pq > µn = n and n > 0, there exist large K , L satisfying 1 2p L < K < K +1< L, q n and large A, B such that   2Mc2 1/q n/2q 1 B


2 K +2 a1 K λ0 c3

2

4a2 n a2 n M = max , , c12 λ0 c32  K +1  2 a1 a2 n B n a2 n B n/2 c = min , , . K 2L(M B n/2 + 1) 2M Lc3 2

! ,

(2.6)

2183

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

Then, we can show via similar arguments as those for Lemmas 2.2 and 2.3 that (u, v) is a subsolution of (1.1) with large initial data u 0 (x) ≥ u(x, 0), v0 (x) ≥ v(x, 0) on Ω¯ .  Observing 1/τ1 > 0 (or 1/τ2 > 0) is equivalent to 2 pq > µn, we derive Theorem 2.1 from Lemmas 2.1–2.4 directly. Similarly, Theorem 2.2 will come from the Lemmas 2.5 and 2.6. Lemma 2.5. If 2 pq < µn with m > 1, the solutions of (1.1) are globally bounded. Proof. Define time-independent functions u¯ =

A 2 − (1 − ϕ)

Aµ−1

,

v¯ = log

B 2 − (1 − ϕ) B

n/2

with ϕ = εϕ0 , and A, B > 1 to be determined. Since 2 pq < µn, there exist A and B large such that (εc1 )−1/µ B p/µ < A < (εc1 )−1/q B n/2q , and hence ∂ u¯ ≥ e pv¯ , ∂η

∂ v¯ ≥ u¯ q ∂η

on ∂Ω × (0, T )

by a simple computation. Moreover, 1u¯ =

Aµ λ0 ϕ(1 − ϕ) A (2 − (1 − ϕ) A

µ−1 −1

µ−1

)2

+

Aµ (Aµ−1 − 1)(1 − ϕ) A (2 − (1 − ϕ) A

µ−1 −2

µ−1

|∇ϕ0 |2

)2

+

2A2µ−1 (1 − ϕ)2(A

µ−1 −1)

(2 − (1 − ϕ) A

µ−1

|∇ϕ0 |2

)3

≤ Aµ λ0 ε + Am ε 2 c22 + 2Am ε 2 c22 , 1v¯ ≤ B n/2 λ0 ε + 2B n ε 2 c22 and a1 A m

m

a1 u¯ =

(2 − (1 − ϕ) A

µ−1

)m

 m A ≥ a1 , 2

n v¯

a2 e

 n B ≥ a2 = n/2 2 B 2 (2 − (1 − ϕ) ) a2 B n

in Ω × (0, T ). Setting ! a1 a2 ε = min , , 1 , 2m (λ0 + 3c22 ) 2n (λ0 + 2c22 ) we have u¯ t ≥ 1u¯ − a1 u¯ m ,

v¯t ≥ 1v¯ − a2 en v¯

in Ω × (0, T ).

(2.7)

Consequently, we can choose A and B even large such that u(x, ¯ 0) ≥ u 0 (x), v(x, ¯ 0) ≥ v0 (x) on Ω¯ to ensure (u, ¯ v) ¯ is a time-independent supersolution of (1.1).  Lemma 2.6. If 2 pq ≤ µn with m ≤ 1, the solutions of (1.1) are global. Proof. Construct   ϕ0  A +M , u¯ = e K t 1− 2

  n Lt 2 v¯ = log Be Lt e−(Be ) ϕ0 + M

with  M = max ku 0 k∞ , ekv0 k∞ , 1,  2(B(M + 1)) p A = max , 2 , c1

c22 a2

!

1 n+1

 ,



 B=

(M + 1)q+1 c1

 n2

,

2184

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

K = pL ,

 L = max

 1 2λ0 (M + 1) 2 A−1 (Aλ0 + A2 c22 ), . p 2 M(2e(M + 1) − n)

A simple computation shows u¯ t ≥

K eK t , 2A

n L(−e−1 ) , 2(M + 1)

v¯t ≥ L +

B n en Lt c22 1 Kt 1 λ0 e Aλ0 ϕ0 + e K t A2 |∇ϕ0 |2 , 1v¯ ≤ + 2 4 eM M in Ω × (0, T ), and thus (2.7) is true. Moreover, we know on the boundary that  p 1 ∂ u¯ ≥ Ae K t c1 , e pv¯ ≤ Be Lt (1 + M) , ∂η 2  q ∂ v¯ B n/2 n Lt u¯ q ≤ e K t (1 + M) , ≥ e 2 c1 , ∂η M +1 1u¯ ≤

and hence ∂ u¯ ≥ e pv¯ , ∂η

∂ v¯ ≥ u¯ q ∂η

on ∂Ω × (0, T )

(2.8)

due to 2 pq ≤ µn = n with m ≤ 1 and the above A, B, K , L. We have shown that (u, ¯ v) ¯ is a global supersolution of (1.1) with u(x, ¯ 0) ≥ u 0 (x), v(x, ¯ 0) ≥ v0 (x) on Ω¯ .  Now consider the critical case 1/τ1 = 1/τ2 = 0 to prove Theorem 2.3. Notice that the case 1/τ1 = 1/τ2 = 0 with µ = 1 is covered by Lemma 2.6. Next treat the other case µ > 1 for 1/τ1 = 1/τ2 = 0 via two lemmas. Lemma 2.7. Suppose 1/τ1 = 1/τ2 = 0 with µ > 1. If   m  2 ! n c22 λ0 1 1 c2 , a2 ≥ a1 ≥ +1 + +2 +1 c1 c1 c1 c1 c1

c2 λ0 + 2 + c1 c1



c2 c1

2 !

,

the solutions of (1.1) are globally bounded. Proof. Observe that 1/τ1 = 1/τ2 = 0 is equivalent to 2 pq = µn, and define time-independent functions u¯ =

A 1 c1

+1−

1 Aµ−1 c1 (1 − ϕ0 )

,

v¯ = log

B 1 c1

+1−

1 B n/2 c1 (1 − ϕ0 )

with A, B > 0 to be fixed later. Noticing ∂ u¯ ≥ Aµ , ∂η

∂ v¯ ≥ B n/2 , ∂η

e pv¯ = B p ,

u¯ q = Aq ,

(x, t) ∈ ∂Ω × (0, T ),

n

we know that (2.8) holds with A = B 2q . For (x, t) ∈ Ω × (0, T ), we have by simple computations,  2 1 µ 1 m 2 m c2 1u¯ ≤ A λ0 + A c2 + 2A , a1 u¯ m ≥ a1 c1 c1 c1  2 1 n/2 1 n 2 n c2 1v¯ ≤ B λ0 + B c2 + B , a2 en v¯ ≥ a2 c1 c1 c1

A 1 c1

,

+1 B

1 c1

!m

+1

!n .

Let  a1 ≥

1 +1 c1

m

 2 ! c22 λ0 c2 + +2 , c1 c1 c1

 a2 ≥

n 1 +1 c1

c2 λ0 + 2 + c1 c1



c2 c1

2 !

,

2185

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197





4q

B = max 22/n , 2 (m−1)n ,

 2q

    n 1 1 . + 1 ekv0 k∞ , + 1 ku 0 k∞ c1 c1

Then the time-dependent (u, ¯ v) ¯ is just a supersolution of (1.1).



Lemma 2.8. Assume 1/τ1 = 1/τ2 = 0 with µ > 1. If ! ! (µ − 1)λ0 c32 µc12 nλ0 c32 nc12 a1 ≤ min , , a2 ≤ min , , 2c22 2c22 4c22 4c22 then the solutions of (1.1) blow up in finite time for large initial data. Proof. Construct u(x, t) =

1 [(µ − 1)ϕ + (1 − ct) K ]

1 µ−1

,

v(x, t) = log

1 [ n2 ϕ

2

+ (1 − ct) L ] n

,

with ϕ = ϕ0 /c2 , and K , L , c > 0 to be determined. For (x, t) ∈ ∂Ω × (0, T ) ∂u − Kµ ≤ (1 − ct) µ−1 , ∂η

∂v ≤ (1 − ct)−L , ∂η

e pv = (1 − ct)−

2 pL n

Because of 1/τ1 = 1/τ2 = 0 (i.e., 2 pq = µn) and µ > 1, we can take L = On the other hand, noticing ut =

K c(1 − ct) K −1  µ ,  (µ − 1) (µ − 1)ϕ + (1 − ct) K µ−1

vt =

2q K

u q = (1 − ct) µ−1 .

,

qK µ−1

such that (2.3) holds on the boundary.

2Lc(1 − ct) L−1 , n[ n2 ϕ + (1 − ct) L ]

λ0 ϕ µ|∇ϕ|2 1u =   µ +  2µ−1 , (µ − 1)ϕ + (1 − ct) K µ−1 (µ − 1)ϕ + (1 − ct) K µ−1 n|∇ϕ|2 λ0 ϕ + , n + (1 − ct) L 2[ 2 ϕ + (1 − ct) L ]2 a1 a2 , a2 env = n a1 u m = 2µ−1 [ 2 ϕ + (1 − ct) L ]2 [(µ − 1)ϕ + (1 − ct) K ] µ−1 1v =

n 2ϕ

for (x, t) ∈ Ω × (0, T ), we have  Kc c2 + m u t − 1u + a1 u ≤  v t − 1v + a2 env ≤

Lc c2

+

Kc µ−1

− λ0 (µ − 1)ϕ 2 − µ|∇ϕ|2 + a1



2µ−1

[(µ − 1)ϕ + (1 − ct) K ] µ−1 2Lc n

− n2 λ0 ϕ 2 − n2 |∇ϕ|2 + a2 n 2ϕ

,

 .

+ (1 − ct) L

Let c = min

a1 ( c12 +

1 2µ )K

,

a2

( c12 + n2 )L ! (µ − 1)λ0 c32 µc12 a1 ≤ min , , 2c22 2c22

! , a2 ≤ min

nλ0 c32 4c22

,

nc12 4c22

! .

Then (2.4) is true, and thus (u, v) is a subsolution of (1.1) if in addition u 0 (x) ≥ u(x, 0), v0 (x) ≥ v(x, 0) on Ω¯ .



2186

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

3. Multiple blow-up rates Now study the multiple blow-up rates for (1.7) (the one-dimensional case of (1.1)). Firstly, the coupled boundary nonlinearities imply that any blow-up must be simultaneous in (1.7). Furthermore, by Theorems 2.1–2.3, the blow-up of solutions in (1.7) requires τ11 ≥ 0 or τ12 ≥ 0, namely, 2 pq ≥ µn if µ > 1, or 2 pq > n if µ = 1. In addition, the comparison principle ensures that the solutions of (1.7) are monotonic with respect to t and x, since u 0 , v0 ≥ δ > 0, 00 nv0 ≥ 0. Thus, u 00 , v00 ≥ 0, u 000 − a1 u m 0 , v0 − a2 e max v(·, t) = v(1, t).

max u(·, t) = u(1, t), [0,1]

[0,1]

We will show that the simultaneous blow-up rates of solutions to (1.7) possess the form c ≤ u(1, t)(T − t)α/2 ≤ C,

c ≤ ev(1,t) (T − t)β/2 ≤ C

(3.1)

with (α, β) = (αi , βi ) defined by (1.10), i = 1, 2, 3, under different nonlinear dominations. Denote N (t) = v(1, t).

M(t) = u(1, t),

To treat the asymptotic analysis on the coupled system, we have to determine at first the comparison relationship between M(t) and N (t). Lemma 3.1. Let (u, v) be a solution for (1.7) with simultaneous blow-up time T . If m ≤ 2q + 1, n ≥ exists ε0 > 0 independent of t such that M 2q (t) ≥ ε0 en N (t) ,

2 pq q+1 ,

then there (3.2)

t → T.

Proof. We exploit the scaling method [12] for the proof. Suppose (3.2) fails for any ε0 > 0. Then there would exist a sequence t j → T ( j → ∞) such that M 2q (t j )e−n N (t j ) → 0.

(3.3)

Set ϕ j (y, s) =

u(1 + ay, a 2 s + t j ) n

e 2q

N (t j )

ev(1+ay,a s+t j ) ψ j (y, s) = e N (t j ) 2

,

for (y, s) ∈ [−1/a, 0] × (−t j /a 2 , 0] with a = e− 2 N (t j ) . Then  (−1+ m−1 2q )n N (t j ) (ϕ )m ,  (y, s) ∈ (−1/a, 0) × (−t j /a 2 , 0], j (ϕ j )s = (ϕ j ) yy − a1 e    ψ (ψ ) = ψ (ψ ) − (ψ )2 − a (ψ )n+2 , (y, s) ∈ (−1/a, 0) × (−t /a 2 , 0],  n

j

j s

j

j yy

j y

2

j

j

2 pq−n(q+1) N (t j ) 2q

   (ϕ j ) y (0, s) = e (ψ j ) p (0, s), (ϕ j ) y (−1/a, s) = 0, s ∈ (−t j /a 2 , 0],     (ψ j ) y (0, s) = (ϕ j )q (ψ j )(0, s), (ψ j ) y (−1/a, s) = 0, s ∈ (−t j /a 2 , 0]. Notice that the solutions of (1.7) are monotonic with respect to t and x for the assumed initial data, and hence (ϕ j )s , (ψ j )s ≥ 0, (ϕ j ) y , (ψ j ) y ≥ 0. Uniformly bounded solutions to porous medium-type equations turn out to be equicontinuous in compact subsets of their common domain, see [6,14,32]. Observe that for every K , S > 0 the domain of ψ j contains the compact set A = [−K , 0] × [−S, 0], if t j → T as j → ∞. Therefore, passing to a subsequence if necessary, we have that (ϕ j , ψ j ) → (ϕ, ψ) uniformly on compact sets of [−K , 0]×[−S, 0], where ϕ, ψ are continuous functions and satisfy ϕ ≡ 0, ψ(0, 0) = 1. Hence, there exists a neighbourhood of (0, 0), U ⊂ A, such that ψ > 12 in U . As we have uniform convergence in U¯ (we can assume that U¯ is compact), for j large enough we obtain 41 ≤ ψ j ≤ 1 in U¯ . Thus the functions ψ j are

2187

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

solutions of uniformly parabolic equations in U¯ . The assumption m ≤ 2q + 1 with n ≥

2 pq q+1

implies −1 +

m−1 2q

≤0

2 pq−n(q+1) 2q

and ≤ 0. Together with ϕ j → 0 as j → ∞, 0 ≤ ψ j ≤ 1, we can use Schauder’s estimates (see e.g. [15]) for the above system to get kϕ j k

C

2+α,1+ α 2

≤ C,

kψ j k

C

2+α,1+ α 2

in U¯

≤C

β

with some α ∈ (0, 1). Therefore, we derive nonnegative functions ϕ and ψ such that (ϕ j , ψ j ) → (ϕ, ψ) in C 2+β,1+ 2 (β < α), satisfying  m  ϕs = ϕ yy − a1 θ1 ϕ , (y, s) ∈ U,  ψψs = ψψ yy − ψ y2 − a2 ψ n+2 , (y, s) ∈ U,    ϕ y (0, s) = θ2 ψ p (0, s), ψ y (0, s) = (ϕ q ψ)(0, s), s ∈ (−S, 0] ∩ U, with ψ(0, 0) = 1, 0 ≤ ψ ≤ 1, ϕs , ψs , ϕ y , ψ y ≥ 0, θi ∈ {0, 1}, i = 1, 2. From the second equation above with ψs ≥ 0, we know ψ yy ≥ 0, which implies that ψ y is monotonic with respect to y. Notice that (3.3) means ϕ ≡ 0, and thus ψ y (0, s) ≡ 0. Furthermore, in virtue of ψ y ≥ 0, we obtain ψ y ≡ 0 for (y, s) ∈ U , and so ψ yy ≡ 0. This concludes ψ ≡ 0, which contradicts ψ > 21 in U .  Lemma 3.2. Let (u, v) be a solution of (1.7) with blow-up time T . If m ≥ 2q + 1 and n < c ≤ M m+1 (t)e−2 pN (t) ≤ C,

2 pq q+1 ,

then (3.4)

t → T.

Proof. Suppose the left inequality in (3.4) is not true. Then there exists a sequence t j → T ( j → ∞) such that M m+1 (t j )e−2 pN (t j ) → 0,

(3.5)

t j → T.

Let ϕ j (y, s) =

u(1 + ay, a 2 s + t j ) 2p

e m+1 N (t j )

ev(1+ay,a s+t j ) ψ j (y, s) = e N (t j ) 2

,

p(m−1)

for (y, s) ∈ [−1/a, 0] × (−t j /a 2 , 0] with a = e− m+1 N (t j ) . Then  (ϕ j )s = (ϕ j ) yy − a1 (ϕ j )m , (y, s) ∈ (−1/a, 0) × (−t j /a 2 , 0],      ψ j (ψ j )s = ψ j (ψ j ) yy − (ψ j )2y − a2 eλ1 N (t j ) (ψ j )n+2 , (y, s) ∈ (−1/a, 0) × (−t j /a 2 , 0],   (ϕ j ) y (0, s) = (ψ j ) p (0, s), (ϕ j ) y (−1/a, s) = 0, s ∈ (−t j /a 2 , 0],     (ψ j ) y (0, s) = eλ2 N (t j ) (ϕ j )q (ψ j )(0, s), (ψ j ) y (−1/a, s) = 0, s ∈ (−t j /a 2 , 0] n(m+1)−2 p(m−1) , m+1

with λ1 =

λ2 =

− p(m−1−2q) . m+1

0 ≤ ϕ j (y, s) ≤ M m+1 (t j )e−2 pN (t j ) ,

In addition, 0 ≤ ψ j (y, s) ≤ 1,

ψ j (0, 0) = 1.

(3.6)

2 pq The blow-up of (u, v) implies 2 pq ≥ µn. Together with the assumption m ≥ 2q +1 and n < q+1 , we have λ1 , λ2 ≤ 0. Similarly to the proof of Lemma 3.1 with Schauder’s estimates, we know that there exists a compact neighbourhood U of (0, 0) such that

kϕ j k

C

2+α,1+ α 2

≤ C,

kψ j k

C

2+α,1+ α 2

in U¯

≤C

with some α ∈ (0, 1). Thus, there exist a subsequence of (ϕ j , ψ j ), still denoted by (ϕ j , ψ j ), and nonnegative functions β

ϕ and ψ such that (ϕ j , ψ j ) → (ϕ, ψ) in C 2+β,1+ 2 (β < α). Moreover, (ϕ, ψ) solves ( ϕs = ϕ yy − a1 ϕ m , (y, s) ∈ U, ϕ y (0, s) = ψ p (0, s),

s ∈ (−S, 0] ∩ U.

2188

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

Recalling (3.5) and (3.6), we know ϕ ≡ 0 and ψ(0, 0) = 1, which contradicts the boundary condition ϕ y (0, s) = ψ p (0, s). Now, suppose the right inequality in (3.4) is not true. Then there exists a sequence t j → T ( j → ∞) such that M −(m+1) (t j )e2 pN (t j ) → 0,

(3.7)

t j → T.

Let ϕ j (y, s) =

u(1 + ay, a 2 s + t j ) , M(t j )

ψ j (y, s) =

ev(1+ay,a M

m+1 2p

2 s+t

j)

(t j )

1−m

for (y, s) ∈ [−1/a, 0] × (−t j /a 2 , 0] with a = M 2 (t j ). Then ( (ϕ j )s = (ϕ j ) yy − a1 (ϕ j )m , (y, s) ∈ (−1/a, 0) × (−t j /a 2 , 0], (ϕ j ) y (0, s) = (ψ j ) p (0, s),

(ϕ j ) y (−1/a, s) = 0,

s ∈ (−t j /a 2 , 0].

Since ψ j → 0 as j → ∞, 0 ≤ ϕ ≤ 1, by Schauder’s estimates, kϕ j k

C

2+α,1+ α 2

≤C

with some α ∈ (0, 1). Thus, there exist a subsequence of ϕ j , still denoted by ϕ j , and a nonnegative function ϕ such β

that ϕ j → ϕ in C 2+β,1+ 2 (β < α). Moreover, ϕ solves ( ϕs = ϕ yy − a1 ϕ m , (y, s) ∈ (−∞, 0) × (−∞, 0], ϕ y (0, s) = 0,

s ∈ (−∞, 0]

with ϕ(0, 0) = 1, 0 ≤ ϕ ≤ 1, ϕs , ϕ y ≥ 0. We have clearly ϕ yy ≥ 0. For y ∈ (−∞, 0], ϕ y ≥ 0 with ϕ y (0, s) = 0 implies ϕ y ≡ 0, and so ϕ yy ≡ 0. This concludes ϕ ≡ 0, a contradiction with ϕ(0, 0) = 1.  Lemma 3.3. Let (u, v) be a solution of (1.7) with blow-up time T . If m ≤ 2q + 1, and n ≤ ε0 > 0 independent of t such that M q+1 (t) ≥ ε0 e pN (t) ,

2 pq q+1 ,

then there exists (3.8)

t → T.

Proof. Assume for contradiction that (3.8) is not true. We have a sequence t j → T ( j → ∞) such that M q+1 (t j )e− pN (t j ) → 0,

(3.9)

t j → T.

Put ϕ j (y, s) =

u(1 + ay, a 2 s + t j ) p

e q+1

N (t j )

ev(1+ay,a s+t j ) ψ j (y, s) = e N (t j ) 2

,

−

pq

N (t )

for (y, s) ∈ [−1/a, 0] × (−t j /a 2 , 0] with a = e q+1 j . Then  (ϕ j )s = (ϕ j ) yy − a1 eλ1 N (t j ) (ϕ j )m , (y, s) ∈ (−1/a, 0) × (−t j /a 2 , 0],      ψ j (ψ j )s = ψ j (ψ j ) yy − (ψ j )2y − a2 eλ2 N (t j ) (ψ j )n+2 , (y, s) ∈ (−1/a, 0) × (−t j /a 2 , 0],   (ϕ j ) y (0, s) = (ψ j ) p (0, s), (ϕ j ) y (−1/a, s) = 0, s ∈ (−t j /a 2 , 0],     (ψ j ) y (0, s) = (ϕ j )q (ψ j )(0, s), (ψ j ) y (−1/a, s) = 0, s ∈ (−t j /a 2 , 0] with λ1 =

p(−2q+m−1) , q+1

λ2 =

−2 pq+n(q+1) . q+1

0 ≤ ϕ j (y, s) ≤ M q+1 (t j )e− pN (t j ) ,

Furthermore, 0 ≤ ϕ j (y, s) ≤ 1,

ψ j (0, 0) = 1.

(3.10)

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

2189

We know λ1 , λ2 ≤ 0 by 2q ≥ m − 1 and 2 pq ≥ n(q + 1). Similarly to the proof of Lemma 3.1 with Schauder’s estimates, we know that there exists a compact neighbourhood U of (0, 0) such that kϕ j k

C

2+α,1+ α 2

≤ C,

kψ j k

C

2+α,1+ α 2

≤C

in U¯

with some α ∈ (0, 1). Passing to a subsequence if necessary (still denoted by itself), we have nonnegative functions β ϕ, ψ such that (ϕ j , ψ j ) → (ϕ, ψ) in C 2+β,1+ 2 (β < α). Moreover, ( ϕs = ϕ yy − a1 θ ϕ m , (y, s) ∈ U, ϕ y (0, s) = ψ p (0, s),

s ∈ (−S, 0] ∩ U

with θ ∈ {0, 1}. Noting (3.9) and (3.10), we get ϕ ≡ 0 with ψ(0, 0) = 1, which contradicts the boundary condition ϕ y (0, s) = ψ p (0, s).  Lemma 3.4. The following estimates are true r √ 1 T − z pN (t) M(t) ≤ M(z) + e + C ∗ T − z M(t), 2 π r √ T −z q M (t) + C ∗ T − z N (t) N (t) ≤ N (z) + π

(3.11) (3.12)

for 0 ≤ z < t < T and some C ∗ > 0. Proof. Let Γ (x, t) be the fundamental solution of the heat equation, namely,  2 −x 1 Γ (x, t) = √ exp . 4t 2 πt It is known that Γ satisfies (see [8]) Z 1 Γ (x − y, t − z)dy ≤ 1, 0 Z t Z t √ 1 1 √ ∗ Γ (1, t − τ ) dτ ≤ C t − z, Γ (0, t − τ )dτ = √ t − z, 2(t − τ ) π z z ∂Γ x−y (x − y, t − τ ) = Γ (x − y, t − τ ), x, y ∈ [0, 1], 0 ≤ z < t. ∂η y 2(t − τ ) By Green’s identity, we have from (1.7) that Z 1 Z tZ 1 u(x, t) = Γ (x − y, t − z)u(y, z)dy − Γ (x − y, t − τ )u m (y, τ )dy dτ 0 z 0 Z t Z t ∂u ∂Γ + (1, τ )Γ (x − 1, t − τ )dτ − (x − 1, t − τ )u(1, τ )dτ z ∂x z ∂η y Z t ∂Γ + (x, t − τ )u(0, τ )dτ, ∂η y z Z 1 Z tZ 1 v(x, t) = Γ (x − y, t − z)v(y, z)dy − Γ (x − y, t − τ )env(y,τ ) dy dτ 0 z 0 Z t Z t ∂v ∂Γ + (1, τ )Γ (x − 1, t − τ )dτ − (x − 1, t − τ )v(1, τ )dτ z ∂x z ∂η y Z t ∂Γ + (x, t − τ )v(0, τ )dτ z ∂η y

(3.13)

(3.14)

2190

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

for 0 ≤ z < t < T , 0 < x < 1. Let x → 1 in (3.13) with the jump relation to obtain r Z t 1 T − z pN (t) 1 M(t) ≤ M(z) + e + Γ (1, t − τ )u(0, τ )dτ 2 π z 2(t − τ ) r √ T − z pN (t) ≤ M(z) + e + C ∗ T − z M(t). π On the other hand, by taking the supremum in (3.14), we know r √ T −z q N (t) ≤ N (z) + M (t) + C ∗ T − z N (t).  π Now we deal with the main results of the paper to establish the multiple blow-up rates for (1.7). Theorem 3.1. If m ≤ 2q + 1, n < by (1.10).

2 pq q+1 ,

then the blow-up rate estimates in (3.1) hold with (α, β) = (α1 , β1 ) defined

Proof. The proof is motivated by that of lemma 3.1 and 3.2 in [22]. By Lemmas 3.3 and 3.4, we know s √ 1 T − z q+1 M(t) ≤ M(z) + M (t) + C ∗ T − z M(t) 2 2 π ε0 √ √ ≤ M(z) + C0 T − z M q+1 (t) + C ∗ T − z M(t) r √ 1 ∗ T − z ≤ 1 , and set M(t) = 4M(z). We derive the lower estimate of the with C0 = 2 . Fix z such that C 8 πε0

blow-up rate for u −1

(3.15)

M(z) ≥ c(T − z) 2q with c = (C0 22(q+1)+1 ) Set

−1 q

.

F(x, t) = u t − εu 2q+1 ,

G(x, t) = vt − ε

2q + 1 2 vx . p

A simple computation with m ≤ 2q + 1 shows Ft − Fx x + a1 mu m−1 F = ε(2q + 1)2qu 2q−1 u 2x + εa1 (2q + 1 − m)u m+2q ≥ 0, 2(2q + 1) 2 2q + 1 G t − G x x + a2 nenv G = ε vx x + ε a2 nenv vx2 ≥ 0, p p Fx (1, t) = pe pN (t) G(1, t),   2(2q + 1) q q 2(2q + 1)2 G x (1, t) = q M q−1 (t)F(1, t) − ε M (t)G(1, t) + εM 2q (t) −ε p 2 m2   q 2q 2(2q + 1) n N (t) +ε M (t) − a2 e M q (t), 2 p Fx (0, t) = 0, G x (0, t) = 0. By Lemma 3.3 with n <

2 pq q+1 ,

we know

M 2q (t)e−n N (t) → ∞ as t → T. Hence, there exists t0 ∈ (0, T ) such that q2 M 2q (t) − 2(2q+1) a2 en N (t) ≥ 0 for t ∈ [t0 , T ). Take ε small enough such p that F(x, t0 ), G(x, t0 ) ≥ 0. Then, the comparison principle yields F(x, t), G(x, t) ≥ 0, or equivalently, u t ≥ εu 2q+1 ,

vt ≥ ε

2q + 1 2 vx p

for (x, t) ∈ [0, 1] × [t0 , T ).

(3.16)

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

2191

The first inequality in (3.16) concludes −1

as t → T.

M(t) ≤ C(T − t) 2q

Combining with the second inequality in (3.16) and Lemma 3.3, we obtain 2q

Nt ≥ ε

2q + 1 2q 2q + 1  pN  q+1 M (t) ≥ ε ε0 e p p

as t → T,

and thus, N (t) ≤ log C(T − t)

−(q+1) 2 pq

as t → T.

(3.17)

√ √ Let z = At − (A − 1)T with A > 1 such that C ∗ T − z = C ∗ A(T − t) ≤ 41 . Then, by Lemma 3.4, √

T − ze pN (t) ≥

√ π (M(t) − 4M(z)), 4

t → T.

From the upper and lower estimates of the blow-up rate for u, it follows that −1

e

pN (t)

c − 4C A 2q ≥ 4

r

−(q+1) π (T − t) 2q A

with T − z = A(T − t). This concludes N (t) ≥ log c(T − t)

−(q+1) 2 pq

as t → T. 

Theorem 3.2. If m < 2q + 1, n ≥ by (1.10).

2 pq q+1 ,

then the blow-up rate estimates in (3.1) hold with (α, β) = (α2 , β2 ) defined

Proof. By Lemmas 3.1 and 3.4, we know that √ √ 2 pq 1 M(t) ≤ M(z) + C0 T − z M n (t) + C ∗ T − z M(t) 2 p √ 1 with C0 = 1/(π 2 ε0n ). Choosing M(t) = 4M(z), and taking T − z small enough such that 4C ∗ T − z ≤ 12 , we can get the lower estimate of the blow-up rate for u:

√ 2 pq 1 M(z) ≤ C0 T − z M n (z), 2 i.e., −n

M(z) ≥ c0 (T − t) 2(2 pq−n) −n

with c0 = (2C0 ) 2 pq−n . Inspired by [12,20], let ϕ M (y, s) =

u(1 + ay, a 2 s + t) , M(t)

ψ N (y, s) =

ev(1+by,b s+t) , e N (t)

y ∈ [−1/a, 0], s ∈ [−t/a 2 , 0],

2

y ∈ [−1/b, 0], s ∈ [−t/b2 , 0],

2192

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197 2 pq

with a = M 1− n (t), b = M −q (t). It is easy to see that a → 0 and b → 0 as t → T , since 2 pq > n for (u, v) blowing up. Moreover, (ϕ M , ψ N ) solves  m+1− 4 npq (ϕ M )m ,  (ϕ M )s = (ϕ M ) yy − a1 M    ψ N (ψ N )s = ψ N (ψ N ) yy − (ψ N )2y − a2 en N M −2q (ψ N )n+2 ,    −2 pq (ψ N ) y (0, s) = (ϕ M )q ψ N (0, s), (ϕ M ) y (0, s) = e pN M n (ψ N ) p (0, s),    (ψ N ) y (−1/a, s) = 0, (ϕ M ) y (−1/a, s) = 0,    1 1  ϕ (y, 0) = u 0 (ay + 1), ψ N (y, 0) = v0 (ay + 1) M M(t) N (t) with 0 ≤ ϕ M , ψ N ≤ 1,

ϕ M (0, 0) = ψ N (0, 0) = 1,

(ϕ M )s , (ψ N )s ≥ 0.

4 pq

By Lemma 3.1, we know that M m+1− n and en N M −2q are both uniformly bounded as t → T , since 2 pq ≥ nµ ≥ 2+α,1+ α2 . We claim n m+1 2 for (u, v) blowing up. By using Schauder’s estimates again, ϕ M is uniformly bounded in C that there exists a positive constant c > 0 such that (ϕ M )s (0, 0) ≥ c. Otherwise, there would be a sequence (ϕ M j , ψ N j ) and nonnegative functions ϕ and ψ such that (ϕ M j , ψ N j ) → (ϕ, ψ) uniformly as t j → T , and ( ϕs = ϕ yy − c2 ϕ m ,

(y, s) ∈ (−∞, 0) × (−∞, 0),

ϕ y (0, s) = c1 ψ (0, s), p

s ∈ (−∞, 0),

(3.18)

with ϕs (0, 0) = 0, ϕs ≥ 0, 0 ≤ ϕ, ψ ≤ 1, ϕ(0, 0) = ψ(0, 0) = 1, c1 , c2 ≥ 0. Set w = ϕs ≥ 0. Then w has a minimum at (0, 0). By Hopf’s Lemma, w ≡ 0, which means ϕ is independent of s. For c2 = 0, (3.18) becomes ( ϕ yy = 0, y ∈ (−∞, 0), (3.19) ϕ y (0) = c1 ψ p (0, s), s ∈ (−∞, 0). The boundary condition in (3.19) implies that ψ(0, s) is independent of s. So, ϕ y (0) = c1 due to ψ(0, 0) = 1. Furthermore, ϕ = 1 + c1 y since ϕ yy = 0 with ϕ(0, 0) = 1. Obviously, c1 > 0 results in a unbounded ϕ for y ∈ (−∞, 0), a contradiction with 0 ≤ ϕ ≤ 1. In the case of c1 = 0, we deduce ϕ ≡ 1 for y ∈ (−∞, 0). Hence, we know lim

t j →T

u(1 + ay, a 2 s + t j ) =1 M(t j )

for every y ∈ (−∞, 0). We obtain lim

t j →T

u(x, t j ) = 1, M(t j )

or equivalently, lim

t j →T

u(x, t j ) − u(1, t j ) = 0. M(t j )

Consequently, u x (1, t j )  M(t j ) = u(1, t j ) as t j → T,

2193

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

which contradicts the blow-up of u. On the other hand, if c2 > 0, i.e., a1 , a2 small enough by Theorem 2.3, where (3.18) becomes ( ϕ yy = c2 ϕ m , (y, s) ∈ (−∞, 0) × (−∞, 0),

1 τ1

=

1 τ2

= 0, the blow-up of (u, v) requires

ϕ y (0, s) = c1 ψ p (0, s), s ∈ (−∞, 0), 1 c2 Hence, we have ϕ y2 ≥ c1 − 2a m+1 for c1 > 0. We get a contradiction with the boundedness of ϕ for y ∈ (−∞, 0) provided a1 < c1 (m+1) 2c2 . For c2 > 0 with c1 = 0, we have ϕ ≡ 0, a contradiction with ϕ(0, 0) = 1. So, the claimed (ϕ M )s (0, 0) ≥ c is true, and hence −n

M(t) ≤ C(T − t) 2(2 pq−n)

as t → T.

By Lemma 3.1 with the blow-up rate estimates of u, similarly to the arguments used for Theorem 3.1, we can get the required blow-up rate estimates for v also: −q

−q

log c(T − t) 2 pq−n ≤ N (t) ≤ log C(T − t) 2 pq−n ,

t → T.



The proof is complete.

2 pq , with in addition (v000 − a2 env0 )00 − a2 nenv0 (v000 − a2 env0 ) ≥ 0. Then Theorem 3.3. Assume m > 2q + 1 and n < q+1 the blow-up rate estimates in (3.1) hold with (α, β) = (α3 , β3 ) defined by (1.10).

Proof. By Lemmas 3.2 and 3.4, we know that √ √ 2 pq N (t) ≤ N (z) + C T − ze m+1 N (t) + C ∗ T − z N (t) √ 2 pq ≤ N (z) + C T − ze m+1 N (t) . Choose N (t) = N (z) + C0 for fixed C0 > 0. We have √ 2 pq C0 ≤ C T − ze m+1 (N (z)+C0 ) , and thus N (z) ≥ log c(T − z)

−(m+1) 4 pq

.

(3.20)

Denote w = vt . Then  wt = wx x − a2 nenv w, (x, t) ∈ (0, 1) × (0, T ),    w (1, t) = qu q−1 u (1, t), t ∈ (0, T ), x t  w (0, t) = 0, t ∈ (0, T ),  x   nv0 00 x ∈ [0, 1]. w(x, 0) = v0 − a2 e , The maximum principle with (v000 − a2 env0 )00 − a2 nenv0 (v000 − a2 env0 ) ≥ 0 yields wt ≥ 0. Consequently, wx x ≥ 0, and hence vt x = wx ≥ 0 for x ∈ [0, 1] due to wx (0, t) = 0. We obtain from (1.7) with vt x ≥ 0 that Z 1 Z 1 1 2q 1 a2 u (1, t) = vx2 (1, t) = vx vx x dx = vx (vt + a2 env )dx ≤ v(1, t)vt (1, t) + env(1,t) , 2 2 n 0 0 and thus, v(1, t)vt (1, t) ≥

4 pq 1 2q a2 c2q 4 pq v(1,t) a2 nv(1,t) u (1, t) − env(1,t) ≥ e m+1 − e ≥ c1 e m+1 v(1,t) 2 n 2 n

by Lemma 3.2 for t close to T , where either 2 pq > µn = N (t) − log N

m+1 4 pq

(t) ≤ log C(T − t)

−(m+1) 4 pq

n(m+1) , 2

,

namely, 1 − N −1 (t) log N

m+1 4 pq

(t) ≤ N −1 (t) log C(T − t)

−(m+1) 4 pq

.

or 2 pq =

n(m+1) 2

with a2 small. We obtain

2194

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

Since N −1 (t) log N

m+1 4 pq

(t) = o(1) when t → T , we get directly that

N (t) ≤ log C(T − t)

−(m+1) 4 pq

,

t → T.

(3.21)

By Lemma 3.2, the desired estimates for u follow from the estimates (3.20) and (3.21) for v: −1

−1

c(T − t) 2q ≤ M(t) ≤ C(T − t) 2q . 

The proof is complete.

00 nv0 ≥ 0, with in Remark 3.1. In Theorem 3.3, it is assumed that u 0 , v0 ≥ δ > 0, u 00 , v00 ≥ 0, u 000 − a1 u m 0 , v0 − a2 e 00 00 nv 00 nv nv 0 0 0 addition (v0 − a2 e ) − a2 ne (v0 − a2 e ) ≥ 0 and the compatibility conditions. Such initial data do exist. For √ √ √ −2 n A nA 2 ( n A x)), with example, we can take u 0 = (B cos π2 x + 1) m−1 , v0 = n1 log( 2a sec A tan 2 2 = 1 and 2

( max

1+

n 2a2

p n

 p ) √ n r  p π2 n( n 2 + 1) 2 2a1 a1 πB n(A + 1) n   , < < . ,2 = π m+1 m−1 2a2 2a2

4. Blow-up sets Now determine the blow-up sets for (1.7). Theorem 4.1. Let (u, v) be a solution of (1.7) with blow-up time T . Then there exist two positive constants A, B such that   u(x, t) ≤ A(1 − x 2 )−α , v(x, t) ≤ log B(1 − x 2 )−β for all (x, t) ∈ [0, 1)×[0, T ), where (α, β) is defined by (1.10). That is to say the blow-up only occurs on the boundary {x = 1}. Proof. Construct w(x, t) = with C1 =

[(1 −

x 2 )2

A , + C1 (T − t)]α/2

z = log

[(1 −

x 2 )2

B + C2 (T − t)]β/2

4 + 16( α2

+ 1), C2 = 20, and  α  α A = max C12 C, (1 + C1 T ) 2 u 0 (1) ,

 β  β v0 (1) 2 2 B = max C2 C, (1 + C2 T ) e ,

where C is the positive constant in the inequalities (3.1). Then, w ≥ u, z ≥ v on the parabolic boundary of (0, 1) × (0, T ) by Theorems 3.1–3.3. In addition, simple computations derive   16( α2 +1)x 2 (1−x 2 )2 Aα C1 − 4(1 − 3x 2 ) − (1−x 2 )2 +C (T −t) 1 wt − wx x + a1 w m ≥ α +1 2 2 2[(1 − x ) + C1 (T − t)] 2 Aα(C1 − 4 − 16( α2 + 1)) ≥ = 0, α 2[(1 − x 2 )2 + C1 (T − t)] 2 +1   16x 2 (1−x 2 )2 β C2 − 4(1 − 3x 2 ) − (1−x 2 )2 +C (T −t) 2 z t − z x x + a2 enz ≥ 2[(1 − x 2 )2 + C2 (T − t)] β (C2 − 4 − 16) ≥ =0 2[(1 − x 2 )2 + C2 (T − t)] for (x, t) ∈ (0, 1) × (0, T ). Thus, by the comparison principle, w(x, t) ≥ u(x, t),

z(x, t) ≥ v(x, t),

2195

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

Fig. 5.1. Multiple blowup rates in model (1.7) G: Global solutions; I: (α, β) = (α1 , β1 ) (Theorem 3.1); II: (α, β) = (α2 , β2 ) (Theorem 3.2); III: (α, β) = (α3 , β3 ) (Theorem 3.3).

and consequently, u(x, t) ≤ A(1 − x 2 )−α ,

  v(x, t) ≤ log B(1 − x 2 )−β

for (x, t) ∈ [0, 1] × (0, T ). So, the solutions of (1.7) must keep bounded up to time T whenever x 6= 1.



5. Discussion Let us illustrate the main results of the paper for the model with absorptions and coupled boundary fluxes of mixedtype nonlinearities. Obviously, the two absorption terms −u m and −env benefit the global existence of solutions. In fact, without them, the solutions of (1.11) are always nonglobal [22]. When the absorptions are strong enough such that both 1/τ1 and 1/τ2 are negative, namely, µn > 2 pq with µ = max( m+1 2 , 1) = m (or n ≥ 2 pq with m ≤ 1), then the solutions of (1.1) are global (Theorem 2.2). While the positivity of 1/τ1 or 1/τ2 results in nonglobal solutions (Theorem 2.1). In the balance situation with 1/τ1 = 1/τ2 = 0, whether the solutions blow up depend on the absorption coefficients a1 and a2 . For example, the global solutions require in addition large a1 and a2 with m > 1 (Theorem 2.3 (i)) for the balance case. It is mentioned that any sublinear absorption for u in the model will work as a linear one, namely, µ = max( m+1 2 , 1) = 1 for m ∈ (0, 1) by (1.6). Now we analyze the influence of absorptions to the blow-up rates in (1.7). Due to the mixed-type multinonlinearities in the model, we obtained a total of three simultaneous blow-up rates as c ≤ u(1, t)(T − t)α/2 ≤ C,

c ≤ ev(1,t) (T − t)β/2 ≤ C

with (α, β) = (αi , βi ) defined by (1.10), i = 1, 2, 3, which can be demonstrated by Fig. 5.1. 2 pq It can be found that in the region I with the two absorption exponents both small that m ≤ 2q + 1 and n < q+1 ,   2 pq 1 q+1 the related (α1 , β1 ) = q , pq is independent of the absorption exponents m, n. In II with n ≥ q+1 , m ≤ 2q + 1,   2q n the related (α2 , β2 ) = 2 pq−n , 2 pq−n will rely on n, one of the two absorption exponents. If m > 2q + 1 with   2 pq n < q+1 , as illustrated via III, then the related (α3 , β3 ) = q1 , m+1 does depend on m. Differently, the blow2 pq up rates for nonlinear parabolic models with absorptions obtained in previous literature are all independent of the absorption parameters [5,13,16,19,21,23,24,26,28,29,31], to our knowledge. Together with the global solution region G, the above classification for the parameters is complete.

2196

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197

It is well known that the scalar problem u t = 1u + u p with homogeneous Dirichlet boundary condition and large 1 − p−1

initial data possesses the blow-up rate O((T − t)

). This is to say that when p (the exponent of the superlinear −

1

source) is decreasing with p > 1, the blow-up time T will be delayed, while the blow-up rate O((T − t) p−1 ) will become larger. Similar phenomena happen in the model (1.7) of the paper. Clearly, the negative contribution of the absorption terms to the blow-up of solutions requires even larger initial data and even larger blow-up time T for the blowing up of solutions. However, as shown via Theorems 3.2 and 3.3, in the stronger absorption cases II and III we have αi ≥ α1 , βi ≥ β1 , i = 2, 3. Naturally, in the system (1.7) with the mixed-type nonlinearities, there exist important differences between the two stronger absorption situations II and III. For example, in II the blow-up rates for both u and v depend on the absorption exponent n, while in III only the blow-up rate of v depends on m. We point out that even though the absorption exponent n is assumed to be nonnegative in this paper, the proofs for Lemmas 2.1–2.3 and Theorems 3.1, 3.3 show that the corresponding conclusions can be extended to n < 0 without difficulties. For example, the regions I and III can be enlarged downward with the additional parts {n < 0, 0 ≤ m ≤ 2q + 1} and {n < 0, m > 2q + 1} respectively. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations 72 (1988) 201–269. J. Bebernes, A. Bressan, D. Eberly, A description of blowup for the solid fuel ignition model, Indiana Univ. Math. J. 36 (1987) 295–305. J. Bebernes, D. Kassy, A mathematical analysis of blow-up for thermal reactions, SIAM J. Appl. Math. 40 (1981) 476–484. G. Caristi, E. Mitidieri, Blow-up estimates of positive solutions of a parabolic system, J. Differential Equations 113 (1994) 265–271. Y.P. Chen, Blow-up for a system of heat equations with nonlocal sources and absorptions, Comput. Math. Appl. 48 (2004) 361–372. E. Dibenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983) 83–118. R. Ferreira, P. Groisman, J. Rossi, Numerical blow-up for the porous medium equation with a source, Numer. Methods Partial Differential Equations 20 (2004) 552–575. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964. V.A. Galaktionov, S.A. Posashkov, On so some investigation method for unbounded solutions of quasilinear parabolic equations, USSR Comput. Math. Phys. 28 (1988) 148–156. V.A. Galaktionov, On asymptotic self-similar behaviour for a quasilinear heat equation: single point blow-up, SIAM J. Math. Anal. 26 (1995) 675–693. Y. Giga, R.V. Kohn, Asymptotic self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985) 297–319. B. Hu, H.M. Yin, The profile near blow-up time for the solution of the heat equation with a nonlinear boundary condition, Trans. Amer. Math. Soc. 346 (1994) 117–135. Z.X. Jiang, S.N. Zheng, Blow-up rate for a nonlinear diffusion equation with absorption and nonlinear boundary flux, Adv. Math. (China) 33 (2004) 615–620. O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’ceva, Linear and quasilinear equations of parabolic type, in: Transl. Math. Mono., vol. 23, AMS, Providence, RI, 1968. G.M. Liberman, Second Order Parabolic Differential Equations, World scientific, River Edge, 1996. H.L. Li, M.X. Wang, Critical exponents and lower bounds of blow-up rate for a reaction-diffusion system, Nonlinear Anal. 63 (2005) 1083–1093. H.L. Li, M.X. Wang, Properties of positive solutions to a nonlinear parabolic system, Sci. China Ser. A, 50, 590–608. Z.G. Lin, C.H. Xie, The blow-up rate for a system of heat equations with nonlinear boundary conditions, Nonlinear Anal. 34 (1998) 767–778. Q.L. Liu, Y.X. Li, C.H. Xie, The blowup property of solutions to degenerate parabolic equation with localized nonlinear reactions, Acta. Math. Sinica (Chin. Ser.) 46 (6) (2003) 1135–1142. F. Quir´os, J.D. Rossi, Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions, Indiana Univ. Math. J. 50 (2001) 629–654. J.D. Rossi, The blow up rate for a semilinear parabolic equation with a nonlinear boundary condition, Acta Math. Univ. Comenian. (N.S.) 67 (1998) 343–350. X.F. Song, Blow-up analysis for a system of heat equations coupled via nonlinear boundary conditions, Math. Methods Appl. Sci. 30 (10) (2007) 1135–1146. P. Souplet, Uniform blow-up profile and boundary behaviour for a non-local reaction-diffusion equation with critical damping, Math. Methods Appl. Sci. 27 (2004) 1819–1829. P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations 153 (1999) 374–406. F.B. Weissler, Single point blow-up of semi-linear initial value problems, J. Differential Equations 55 (1984) 204–224. S.N. Zheng, R.H. Ji, Asymptotic estimates to non-global solutions of a multi-coupled parabolic system, preprint. S.N. Zheng, F.J. Li, Critical exponents for a reaction-diffusion model with absorptions and coupled boundary flux, Proc. Edinb. Math. Soc. 48 (2005) 241–252. S.N. Zheng, F.J. Li, B.C. Liu, Asymptotic behavior for a reaction-diffusion equation with inner absorption and boundary flux, Appl. Math. Lett. 19 (2006) 942–948.

S. Zheng, L. Qiao / Nonlinear Analysis 70 (2009) 2176–2197 [29] [30] [31] [32]

2197

S.N. Zheng, H. Su, A quasilinear reaction-diffusion system coupled via nonlocal sources, Appl. Math. Comput. 180 (2006) 295–308. S.N. Zheng, W. Wang, Critical exponents for a nonlinear parabolic system, Nonlinear Anal. 67 (2007) 1190–1210. S.N. Zheng, W. Wang, Blow-up rate for a nonlinear diffusion equation, Appl. Math. Lett. 19 (2006) 1385–1389. W.P. Ziemer, Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. Amer. Math. Soc. 271 (1982) 733–748.