Global Existence and Finite Time Blow-up for a Reaction-Diffusion System with Three Components
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Global Existence and Finite Time Blow-up for a Reaction-Diffusion System with Three Components Huiling Li1
· Yang Zhang2
Received: 7 July 2016 / Accepted: 2 May 2017 © Springer Science+Business Media Dordrecht 2017
Abstract This paper concerns global existence and finite time blow-up behavior of positive solutions for a nonlinear reaction-diffusion system with different diffusion coefficients. By use of algebraic matrix theory and modern analytical theory, we extend results of Wang (Z. Angew. Math. Phys. 51:160–167, 2000) to a more general system. Furthermore, we give a complete answer to the open problem which was brought forward in Wang (Z. Angew. Math. Phys. 51:160–167, 2000). Keywords Global existence · Finite time blow-up · Structure of the matrix · Reaction-diffusion system · Three components · Different diffusion coefficients Mathematics Subject Classification (2000) 35B40 · 35K55 · 35K61
1 Introduction and Main Results In this paper, global existence and finite time blow-up behaviors of positive solutions for a nonlinear reaction-diffusion system are to be discussed: ⎧ p p p ⎨ uit = di ui + u1 i1 u2 i2 u3 i3 , x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, ui (x, t) = 0, (1.1) ⎩ x ∈ Ω, i = 1, 2, 3, ui (x, 0) = ui0 (x), where Ω is a bounded domain in Rn with smooth boundary ∂Ω, initial values ui0 (x) (1 ≤ i ≤ 3) are non-negative and continuous functions which satisfy compatibility conditions.
B H. Li
[email protected]
B Y. Zhang
[email protected]
1
School of Mathematics, Southeast University, Nanjing 210096, P.R. China
2
Department of Mathematics, Harbin Engineering University, Harbin 150001, P.R. China
H. Li, Y. Zhang
The exponents pij (i, j = 1, 2, 3) are non-negative constants, and diffusion coefficients di are positive constants for all 1 ≤ i ≤ 3. System (1.1) is usually used as a model to describe heat propagation in a three-component combustible mixture (cf. [2]). In this case, u1 , u2 and u3 represent temperatures of the interacting components, and corresponding di (1 ≤ i ≤ 3) are thermal conductivity, which are supposed constant. Written system (1.1) as integral equations, by constructing bounded monotone iterative sequences it can be proved that system (1.1) has a local non-negative solution (cf. [15]). However, uniqueness does not hold (cf. [12]). The comparison principle holds, see Sect. 1 of this paper. For system of two components in a bounded domain Ω of the form ⎧ p p ⎨ uit = ui + u1 i1 u2 i2 , x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, ui (x, t) = 0, (1.2) ⎩ ui (x, 0) = ui0 (x) ≥ 0, x ∈ Ω, i = 1, 2, Chen [4] in 1997 investigated special case: p12 , p21 > 0, p12 > p22 − 1 and p21 > p11 − 1, and critical exponents were proved. Later, Wang [22] considered a general case, and obtained significant results which cover that of [4]. Let λ be the first eigenvalue and ϕ(x) the corresponding eigenfunction of the problem −ϕ(x) = λϕ(x),
x ∈ Ω;
ϕ(x) = 0,
x ∈ ∂Ω.
(1.3)
It is well known that λ > 0, ϕ(x) > 0 in Ω and ∂ϕ/∂η < 0 on ∂Ω, here η is the unit outward normal vector on ∂Ω. Then main results of [22]
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