SYSTEM WITH NONLOCAL SOURCE

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^q1ut− x^r1ux

x= _a

0v^p1dx, (x,t)∈(0,a)×(0,T), x^q2v_t−

x^r2v_xx= _a

0 u^p2dx, (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)=u0(x), v(x, 0)=v0(x), x∈[0,a],

(1.1)

whereu0(x),v0(x)∈C^2+α(D) for someα∈(0, 1) are nonnegative nontrivial functions.

u0(0)=u0(a)=v0(0)=v0(a)=0,u0(x)≥0,v0(x)≥0,u0,v0 satisfy the compatibility condition,T >0,a >0,r1,r2∈[0, 1),|q1|+r1=0,|q2|+r2=0, andp1>1,p2>1.

LetD=(0,a) andΩt=D×(0,t],DandΩtare their closures, respectively. Since|q1|+ r1=0,|q2|+r2=0, the coefficients ofu_t,u_x,u_xxandv_t,v_x,v_xx may tend to 0 or∞asx tends to 0, we can regard the equations as degenerate and singular.

Hindawi Publishing Corporation Boundary Value Problems

Volume 2006, Article ID 21830, Pages1–19 DOI10.1155/BVP/2006/21830

Floater [9] and Chan and Liu [4] investigated the blowup properties of the following degenerate parabolic problem:

x^qut−uxx=u^p, (x,t)∈(0,a)×(0,T), u(0,t)=u(a,t)=0, t∈(0,T),

u(x, 0)=u0(x), x∈[0,a],

(1.2)

whereq >0 andp >1. Under certain conditions on the initial datumu0(x), Floater [9]

proved that the solutionu(x,t) of (1.2) blows up at the boundaryx=0 for the case 1<

p≤q+ 1. This contrasts with one of the results in [10], which showed that for the case q=0, the blowup set of solutionu(x,t) of (1.2) is a proper compact subset ofD.

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)

whereurepresents the temperature of the fluid. In [9] Floater approximatede^ubyu^pand considered (1.2). Budd et al. [2] generalized the results in [9] to the following degenerate quasilinear parabolic equation:

x^qut=

u^m_xx+u^p, (1.4)

with homogeneous Dirichlet conditions in the critical exponentq=(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])

ut=

u^m_xx+u^p. (1.5)

For the casep > q+ 1, in [4] Chan and Liu continued to study problem (1.2). Under certain conditions, they proved thatx=0 is not a blowup point and the blowup set is a proper compact subset ofD.

In [7], Chen and Xie discussed the following degenerate and singular semilinear para- bolic equation:

u_t−

x^αu_xx= _a

0 fu(x,t)dx, (x,t)∈(0,a)×(0,T), u(0,t)=u(a,t)=0, t∈(0,T),

u(x, 0)=u0(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.

In [6], Chen et al. consider the following degenerate nonlinear reaction-diffusion equation with nonlocal source:

x^qu_t−

x^γu_xx= _a

0u^pdx, (x,t)∈(0,a)×(0,T), u(0,t)=u(a,t)=0, t∈(0,T),

u(x, 0)=u0(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] that under 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. InSection 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. Letb1(x,t) andb2(x,t) be continuous nonnegative functions defined on [0,a]× [0,r] for anyr∈(0,T), and let (u(x,t),v(x,t))∈(C(Ωr)∩C^2,1(Ωr))²satisfy

x^q1ut− x^r1ux

x≥ _a

0b1(x,t)v(x,t)dx, (x,t)∈(0,a)×(0,r], x^q2v_t−

x^r2v_xx≥ _a

0b2(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.

IfW(x,t) andZ(x,t)∈C(Ωr)∩C^2,1(Ωr) satisfy x^q1Wt−

x^r1Wx

x≥ _a

0b1(x,t)Z(x,t)dx, (x,t)∈(0,a)×(0,r], x^q2Zt−

x^r2Zx

x≥ _a

0b2(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 letr1∈(r1, 1),r2∈(r2, 1) be positive constants and

W(x,t)=u(x,t) +η1 +x^r1−r1e^ct, Z(x,t)=v(x,t) +η1 +x^r2−r2e^ct, (2.3) where η >0 is sufficiently small and c is a positive constant to be determined. Then W(x,t)>0,Z(x,t)>0 on the parabolic boundary ofΩr, and in (0,a)×(0,r], we have

x^q1Wt− x^r1Wx

x− _a

0b1(x,t)Z(x,t)dx

≥x^q1η1 +x^r1−r1ce^ct+ r1−r1

1−r1

ηe^ct x^{2−r1 −}

_a

0b1(x,t)η1 +x^r2−r2e^ctdx

≥ηe^ct

cx^q1+

r1−r1 1−r1

x^{2−r1 −}a1 +a^r2−r2 max

(x,t)∈[0,a]×[0,r]b1(x,t)

, x^q2Z_t−

x^r2Z_xx− _a

0b2(x,t)W(x,t)dx

≥ηe^ct

cx^q2+ r2−r2

1−r2

x^{2−r2 −}a1 +a^r1−r1 max

(x,t)_∈[0,a]_×[0,r]b2(x,t)

.

(2.4)

We will prove that the above inequalities are nonnegative in three cases.

Case 1. When

(x,t)_∈max[0,a]_×[0,r]b1(x,t)≤ r1−r1

1−r1

a^3−r11 +a^r2−r2,

(x,t)∈max[0,a]×[0,r]b2(x,t)≤ r2−r2

1−r2

a^3−r21 +a^r1−r1.

(2.5)

It is obvious that

x^q1W_t−

x^r1W_xx− _a

0b1(x,t)Z(x,t)dx≥0, x^q2Z_t−

x^r2Z_xx− _a

0b2(x,t)W(x,t)dx≥0.

(2.6)

Case 2. If

(x,t)∈max[0,a]×[0,r]b1(x,t)>

r1−r1

1−r1

a^3−r11 +a^r2−r2,

(x,t)∈max[0,a]×[0,r]b2(x,t)>

r2−r2

1−r2

a^3−r21 +a^r1−r1.

(2.7)

Letx0andy0be the root of the algebraic equations a1 +a^r2−r2 max

(x,t)∈[0,a]×[0,r]b1(x,t)= r1−r1

1−r1

x^2−r1 , a1 +a^r1−r1 max

(x,t)∈[0,a]×[0,r]b2(x,t)= r2−r2

1−r2

y^2−r2 ,

(2.8)

andC1,C2>0 be sufficient large such that

C1>

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

(x,t)_∈max[0,a]_×[0,r]b1(x,t)

a1 +a^r2−r2 x0^q1

forq1≥0,

(x,t)_∈max[0,a]_×[0,r]b1(x,t)

a1 +a^r2−r2

a^q1 forq1<0,

C2>

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

(x,t)_∈max[0,a]_×[0,r]b2(x,t)

a1 +a^r1−r1 y^q0²

forq2≥0,

(x,t)_∈max[0,a]_×[0,r]b2(x,t)

a1 +a^r1−r1

a^q2 forq2<0.

(2.9)

Setc=max{C1,C2}, then we have x^q1Wt−

x^r1Wx

x− _a

0b1(x,t)Z(x,t)dx

≥

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

ηe^ct r1−r1

1−r1

x^{2−r1 −}a1 +a^r2−r2 max

(x,t)_∈[0,a]_×[0,r]b1(x,t)

forx≤x0, ηe^ct

cx^q1−a1 +a^r2−r2 max

(x,t)_∈[0,a]_×[0,r]b1(x,t)

forx > x0,

≥0, x^q2Zt−

x^r2Zx

x− _a

0b2(x,t)W(x,t)dx

≥

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

ηe^ct r2−r2

1−r2

x^{2−r2 −}a1 +a^r1−r1 max

(x,t)∈[0,a]×[0,r]b2(x,t)

forx≤y0, ηe^ct

cx^q2−a1 +a^r1−r1 max

(x,t)∈[0,a]×[0,r]b2(x,t)

forx > y0,

≥0.

(2.10)

Case 3. When

(x,t)_∈max[0,a]_×[0,r]b1(x,t)≤ r1−r1

1−r1

a^3−r11 +a^r2−r2,

(x,t)_∈max[0,a]_×[0,r]b2(x,t)>

r2−r2

1−r2

a^3−r21 +a^r1−r1,

(2.11)

or

(x,t)_∈max[0,a]_×[0,r]b2(x,t)≤ r2−r2

1−r2

a^3−r21 +a^r1−r1,

(x,t)∈max[0,a]×[0,r]b1(x,t)>

r1−r1 1−r1

a^3−r11 +a^r2−r2.

(2.12)

Combining Cases1with2, it is easy to prove x^q1W_t−

x^r1W_xx− _a

0b1(x,t)Z(x,t)dx≥0, x^q2Z_t−

x^r2Z_xx− _a

0 b2(x,t)W(x,t)dx≥0,

(2.13)

so we omit the proof here.

From the above three cases, we know thatW(x,t)>0, Z(x,t)>0 on [0,a]×[0,r].

Lettingη→0⁺, we haveu(x,t)≥0,v(x,t)≥0 on [0,a]×[0,r]. By the arbitrariness of

r∈(0,T), we complete the proof ofLemma 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 constantt0 (t0< T) such that the problem (1.1) has a supersolution (h1(x,t),h2(x,t))∈(C(Ωt⁰)∩C^2,1(Ωt⁰))².

Proof. Let

ψ(x)= x

a 1−r1

1−x a

+ x

a

(1−r1)/2 1−x

a 1/2

, ϕ(x)=

x a

1₋r2 1−x

a

+ x

a

(1₋r2)/2 1−x

a 1/2

,

(2.14)

and letK0be a positive constant such thatK0ψ(x)≥u0(x),K0ϕ(x)≥v0(x).

Denote the positive constant ₀¹[s^1−r1(1−s) +s^(1−r1)/2(1−s)^1/2]^p2ds by b20 and 1

0[s^1−r2(1−s) +s^(1−r2)/2(1−s)^1/2]^p1ds by b10. Let K10 ∈(0, (1−r1)/(2−r1)), K20 ∈ (0, (1−r2)/(2−r2)) be positive constants such that

K10≤

2^p1+1a^3−r1b10K0^p1−1

_−2/(1−r1)

, K20≤

2^p2+1a^3−r2b20K0^p2−1

₋2/(1₋r2)

.

(2.15)

Let (K1(t),K2(t)) be the positive solution of the following initial value problem:

K1(t)=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

b10K2^p1(t) a^q1−1K10^q1

K10

1−K10

1₋r1

+K10^1/2

1−K10

(1₋r1)/2, q1≥0, b10K2^p1(t)

a^q1−11−K10_q1 K10

1−K101−r¹+K10^1/2

1−K10(1−r¹)/2, q1<0, K1(0)=K0,

K2(t)=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

b20K1^p2(t) a^q2−1K20^q2

K20

1−K20

1₋r2

+K20^1/2

1−K20

(1₋r2)/2, q2≥0, b20K1^p2(t)

a^q2−11−K20

_q2 K20

1−K20

1−r²+K20^1/2

1−K20

(1−r²)/2, q2<0, K2(0)=K0.

(2.16) SinceK1(t),K2(t) are increasing functions, we can chooset0>0 such thatK1(t)≤2K0, K2(t)≤2K0for allt∈[0,t0]. Seth1(x,t)=K1(t)ψ(x),h2(x,t)=K2(t)ϕ(x), thenh1(x,t)≥ 0,h2(x,t)≥0 onΩt⁰. We would like to show that (h1(x,t),h2(x,t)) is a supersolution of (1.1) inΩt0. To do this, let us construct two functionsJ1,J2by

J1=x^q1h1t− x^r1h1x

x− _a

0h2^p1dx, (x,t)∈Ωt0, J2=x^q2h2t−

x^r2h2x

x− _a

0h1^p2dx, (x,t)∈Ωt0.

(2.17)

Then,

J1=x^q1h1t− x^r1h1x

x− _a

0h2^p1dx

=x^q1K1ψ(x)+

2−r1

a^2−r1 +

1−r1

2

4 x^(r1−3)/2(a−x)^1/2+1

2x^(r1−1)/2(a−x)^−1/2 +1

4x^(1+r1)/2(a−x)^−3/2

× 1 a^1−r1/2

K1(t)−ab10K2^p1(t)

≥x^q1K1(t)ψ(x) +x^(r1−1)/2(a−x)^−1/2 K1(t)

2a^1−r1/2−ab10K2^p1(t), J2≥x^q2K2(t)ϕ(x) +x^(r2−1)/2(a−x)^−1/2 K2(t)

2a^1−r2/2−ab20K1^p2(t).

(2.18)

For (x,t)∈(0,aK10)×(0,t0]∪(a(1−K10),a)×(0,t0], by (2.15), we have J1≥x^(r1−1)/2(a−x)^−1/2 K1(t)

2a^1−r1/2−ab10K2^p1(t)

≥

K10^(r1−1)/2

2a^2−r1

K1(t)−ab10K2^p1

t0

≥

K10^(r1−1)/2

2a^2−r1

K0−ab10

2K0

_p1

≥0.

(2.19)

For (x,t)∈(0,aK20)×(0,t0]∪(a(1−K20),a)×(0,t0], by (2.15), we have J2≥

K20^(r2−1)/2

2a^2−r2

K0−ab20

2K0

_p2

≥0. (2.20)

For (x,t)∈[aK10,a(1−K10)]×(0,t0] by (2.16), we have J1≥x^q1K1(t)ψ(x)−ab10K2^p1(t)

≥

⎧⎪

⎪⎨

⎪⎪

⎩

a^q1K10^q1K1(t)K10

1−K10

1₋r1

+K10^1/2

1−K10

(1₋r1)/2

−ab10K2^p1(t), q1≥0, a^q11−K10

_q1

K1(t)K10

1−K10

1₋r1

+K10^1/2

1−K10

(1₋r1)/2

−ab10K2^p1(t), q1<0,

≥0,

(2.21) For (x,t)∈[aK20,a(1−K20)]×(0,t0] by (2.16), we have

J2≥x^q2K2(t)ϕ(x)−ab20K1^p2(t)

≥

⎧⎪

⎪⎨

⎪⎪

⎩

a^q2K20^q2K2(t)K20

1−K20

1−r2

+K20^1/2

1−K20

(1−r2)/2

−ab20K1^p2(t), q2≥0, a^q21−K20

_q2

K2(t)K20

1−K20

1−r2

+K20^1/2

1−K20

(1−r1)/2

−ab20K1^p2(t), q2<0,

≥0.

(2.22) Thus,J1(x,t)≥0,J2(x,t)≥0 inΩt0. It follows fromh1(0,t)=h1(a,t)=h2(0,t)=h2(a,t)=0 andh1(x, 0)=K0ψ(x)≥u0(x),h2(x, 0)=K0ϕ(x)≥v0(x) that (h1(x,t),h2(x,t)) is a super- solution of (1.1) inΩt0. The proof ofLemma 2.2is complete.

To show the existence of the classical solution (u(x,t),v(x,t)) of (1.1), let us intro- duce a cutofffunction ρ(x). By Dunford and Schwartz [8, page 1640], there exists a

nondecreasingρ(x)∈C³(R) such thatρ(x)=0 ifx≤0 andρ(x)=1 ifx≥1. Let 0<

δ <min{(1−r1)/(2−r1)a, (1−r2)/(2−r2)a},

ρδ(x)=

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

0, x≤δ,

ρ x

δ⁻1

, δ < x <2δ,

1, x≥2δ,

(2.23)

andu0δ(x)=ρ_δ(x)u0(x),v0δ(x)=ρ_δ(x)v0(x). We note that

∂u0δ(x)

∂δ ⁼

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

0, x≤δ,

−x δ²ρ

x δ⁻1

u0(x), δ < x <2δ,

0, x≥2δ,

∂v0δ(x)

∂δ ⁼

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

0, x≤δ,

−x δ²ρ

x δ⁻1

v0(x), δ < x <2δ,

0, x≥2δ.

(2.24)

Sinceρis nondecreasing, we have∂u0δ(x)/∂δ≤0,∂v0δ(x)/∂δ≤0. From 0≤ρ(x)≤1, we haveu0(x)≥u0δ(x),v0(x)≥v0δ(x) and limδ→0u0δ(x)=u0(x), limδ→0v0δ(x)=v0(x).

LetD_δ=(δ,a), letw_δ=D_δ×(0,t0], letD_δandw_δbe their respective closures, and let Sδ= {δ,a} ×(0,t0]. We consider the following regularized problem:

x^q1u_δt−

x^r1u_δxx= _a

δ v_δ^p1dx, (x,t)∈w_δ, x^q2vδt−

x^r2vδx

x= _a

δu_δ^p2dx, (x,t)∈wδ, u_δ(δ,t)=u_δ(a,t)=v_δ(δ,t)=v_δ(a,t)=0, t∈

0,t0

, uδ(x, 0)=u0δ(x), vδ(x, 0)=v0δ(x), x∈Dδ.

(2.25)

By using Schauder’s fixed point theorem, we have the following.

Theorem 2.3. The problem (2.25) admits a unique nonnegative solution (u_δ,v_δ)∈ (C^2+α,1+α/2(wδ))². Moreover, 0≤uδ≤h1(x,t), 0≤vδ≤h2(x,t), (x,t)∈wδ, whereh1(x,t), h2(x,t) are given byLemma 2.2.

Proof. By the proof ofLemma 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

X1=

v1∈C^α,α/2wδ

: 0≤v1(x,t)≤h2(x,t), (x,t)∈wδ , X2=

u1∈C^α,α/2w_δ) : 0≤u1(x,t)≤h1(x,t), (x,t)∈w_δ. (2.26)