Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 49, 1-14; http://www.math.u-szeged.hu/ejqtde/

Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems^{1}

Haihua Lu^{a,b}

a. Department of Mathematics, Nantong University, Nantong 226019, P. R. China b. Department of Mathematics, Southeast University, Nanjing 211189, P. R. China

Abstract. This paper deals with positive solutions of some degenerate and quasilinear parabolic systems not in divergence form: u_{1t} = f_{1}(u_{2})(Δu_{1} + a_{1}u_{1}), ⋯, u_{(n−1)t} = f_{n−1}(u_{n})(Δu_{n−1} + a_{n−1}u_{n−1}), u_{nt} = f_{n}(u_{1})(Δu_{n}+a_{n}u_{n}) with homogenous Dirichlet boundary condition and positive initial condition, where a_{i} (i = 1, 2, ···, n) are positive constants and f_{i} (i = 1, 2, ···, n) satisfy some conditions. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that: (i) when min{a_{1}, ⋯, a_{n}} ≤ λ_{1} then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm; (ii) when min{a_{1}, ⋯, a_{n}} > λ_{1}, and the initial datum (u_{10}, ⋯, u_{n0}) satisfies some assumptions, then the positive classical solution is unique and blows up in finite time, where λ_{1} is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition.

Key words: quasilinear parabolic system; global existence; blow up in finite time; not in divergence form.

1 Introduction and main result

In this paper, we consider the following degenerate and quasilinear parabolic systems not in divergence form:

\{ \begin{matrix}u_{it} = f_{i}(u_{i+1})(Δu_{i} + a_{i}u_{i}), & x ∈ Ω, t > 0, i = 1, 2, ⋯, n − 1, \\ u_{nt} = f_{n}(u_{1})(Δu_{n} + a_{n}u_{n}), & x ∈ Ω, t > 0, \\ u_{i}(x, 0) = u_{i0}(x), & x ∈ Ω, i = 1, 2, ⋯, n, \\ u_{i}(x, t) = 0, & x ∈ ∂Ω, t > 0, i = 1, 2, ⋯, n, \end{matrix}  (1.1)

where a_{i} are positive constants, f_{i}, u_{i0}(x), i = 1, 2, ⋯, n, satisfy (H1) u_{i0}(x) ∈ C^{1}(Ω̄), u_{i0}(x) > 0 in Ω, ; (H2) u_{i0}(x) = 0 and \frac{∂u_{i0}}{∂η} < 0 on ∂Ω, where η is the outward normal vector on ∂Ω; (H3) f_{i} ∈ C^{1}([0, ∞)), such that f_{i} > 0 and f′_{i} ≥ 0 on [0, ∞); (H4) there exists 1 ≤ j ≤ n, such that lim inf_{s→∞} \frac{f_{i}(s)}{f_{j}(s)} > 0, i = 1, ⋯, n. This system can be used to describe the development of n groups in the dynamics of biological

groups, where u_{i} are the densities of the different groups.

1This work was supported by the Natural sciences Foundation of Nantong University (08Z05).

It has been shown that classical positive solutions of parabolic problems of single equation

\{ \begin{matrix}u_{t} = u^{p}(Δu + u), & x ∈ Ω, t > 0, \\ u(x, t) = 0, & x ∈ ∂Ω, t > 0, \\ u(x, 0) = u_{0}(x), & x ∈ Ω \end{matrix}  (1.2)

blow up at some finite time T < ∞ if p ≥ 1 and the bounded smooth domain Ω is large enough such that λ_{1} < 1, where λ_{1} is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition. (See [1, 4, 5, 7, 8, 9, 14, 15]).

For the case f_{1}(v) = v^{p}, f_{2}(u) = u^{q} (p, q ≥ 1), (1.1) has been discussed by many authors, see [3, 12] and the references therein. In [12], Wang and Xie discussed the following system

\{ \begin{matrix}u_{t} = v^{p}(Δu + a_{1}u), \\ v_{t} = u^{q}(Δv + a_{2}v), & x ∈ Ω, t > 0, \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ Ω, \\ u(x, t) = v(x, t) = 0, & x ∈ ∂Ω, t > 0. \end{matrix}  (1.3)

They proved that: (i) when min{a, b} ≤ λ_{1} then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm; (ii) when min{a, b} > λ_{1}, there is no global positive classical solution. And if in addition the initial datum (u_{0}, v_{0}) satisfies some assumptions then the positive classical solution is unique and blows up in finite time.

In [13], Wang also considered the problem

\{ \begin{matrix}u_{t} = u^{p}(Δu + a_{1}v), \\ v_{t} = v^{q}(Δv + a_{2}u), & x ∈ Ω, t > 0, \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ Ω, \\ u(x, t) = v(x, t) = 0, & x ∈ ∂Ω, t > 0 \end{matrix}  (1.4)

with p, q > 0. And he shown that all positive solutions of problem (1.4) exist globally if and only if a_{1}a_{2} ≤ λ_{1}^{2}.

In [2], Deng and Xie promoted the problem (1.4) to the following problem

\{ \begin{matrix}u_{t} = f_{1}(u)(Δu + a_{1}v), \\ v_{t} = f_{2}(v)(Δv + a_{2}v), & x ∈ Ω, t > 0, \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ Ω, \\ u(x, t) = v(x, t) = 0, & x ∈ ∂Ω, t > 0. \end{matrix}  (1.5)

Under the proper assumptions, they proved that: the positive solution of (1.5) blows up in finite time if and only if λ_{1}^{2} < a_{1}a_{2} and ∫_{0}^{∞} ds/(sf_{i}(s)) < ∞ for i = 1, 2.

Remark 1. Without loss of generality, by assumptions (H3) and (H4), we may assume that j = n. And for any given δ > 0, there exists a constant K_{0} > 0, such that

f_{i}(s) ≥ K_{0}f_{n}(s), s ≥ δ, i = 1, 2, ⋯, n − 1.  (1.6)

Indeed, let K_{1} = min_{1≤i≤n−1} \{ lim inf_{s→∞} \frac{f_{i}(s)}{f_{n}(s)}\}, then K_{1} > 0. Set K^{∗} = \frac{K_{1}}{2} when K_{1} < ∞ and K^{∗} = 1 when K_{1} = ∞. It is obvious that there exists a constant s_{0} > 0 such that

\frac{f_{i}(s)}{f_{n}(s)} ≥ K^{∗}, i = 1, 2, ⋯, n − 1, s > s_{0}.

Furthermore, by (H3) and (H4), denoting K_{2} = min_{1≤i≤n−1} min_{δ≤s≤s_{0}} \frac{f_{i}(s)}{f_{n}(s)}, we have 0 < K_{2} < ∞. Hence, let K_{0} = min{K^{∗}, K_{2}}, which shows (1.6).

Remark 2. By change the order of i, we may assume a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{n}.

Remark 3. If n = 2, f_{1}(s) = s^{p} and f_{2}(s) = s^{q}, where p, q ≥ 1, then the assumptions (H3) and (H4) hold automatically. So our present results develop the work of [12].

This paper is organized as follows. In section 2, we first give a Maximum Principle for degenerate parabolic systems with unbounded coefficients, which is useful in the following arguments, and the local existence of positive classical solution is proved. In sections 3 and 4, we discuss the cases min{a_{1}, ⋯, a_{n}} ≤ λ_{1} and min{a_{1}, ⋯, a_{n}} > λ_{1}, respectively. By above Remarks, we may assume f_{i}(s) ≥ f_{n}(s), i = 1, 2, ⋯, n − 1, and a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{n} throughout this paper.

2 Local existence

We first give a maximum principle, the proof is standard and we omit it (see [12]).

Proposition 1. Let a_{i}(x, t), b_{i}(x, t), c_{i}(x, t), i = 1, 2 · ··, n, be continuous functions in Ω × (0, T). Assume that a_{i}(x, t), c_{i}(x, t) ≥ 0 in Ω × (0, T) and b_{i}(x, t), c_{i}(x, t) are bounded on Ω̄ × [0, T_{0}] for any T_{0} < T. If functions u_{i} ∈ C^{2,1}(Ω × (0, T)) ∩ C(Ω̄ × [0, T]), i = 1, 2, ⋯, n, and satisfy

\{ \begin{matrix}u_{it} ≤ (≥)a_{i}Δu_{i} + b_{i}u_{i} + c_{i}u_{i+1}, & i = 1, ⋯, n − 1, \\ u_{nt} ≤ (≥)a_{n}Δu_{n} + b_{n}u_{n} + c_{n}u_{1}, & x ∈ Ω, 0 < t < T, \\ u_{i}(x, 0) ≤ (≥)0, & x ∈ Ω, \\ u_{i}(x, t) ≤ (≥)0, & x ∈ ∂Ω, 0 < t < T, \end{matrix}  (2.1)

then

u_{i}(x, t) ≤ (≥)0, ∀(x, t) ∈ Ω̄ × [0, T), i = 1, 2, ⋯, n.

Since u_{i} = 0 (i = 1, 2, ···, n) on the boundary ∂Ω, the equation of (1.1) is not strictly parabolic type. The standard parabolic theory cannot be used directly to prove the local existence of solution to problem (1.1). To overcome this difficultly we will modify both different equations and boundary conditions. For ε > 0, we consider the following approximate problem

\{ \begin{matrix}(u_{iε})_{t} = g_{iε}(u_{(i+1)ε})(Δu_{iε} + a_{i}u_{iε} − a_{i}ε), & i = 1, ⋯, n − 1, \\ (u_{nε})_{t} = g_{nε}(u_{1ε})(Δu_{nε} + a_{n}u_{nε} − a_{n}ε), & x ∈ Ω, t > 0, \\ u_{iε}(x, 0) = u_{i0}(x) + ε, & x ∈ Ω, \\ u_{iε}(x, t) = ε, & x ∈ ∂Ω, t > 0, \end{matrix}  (2.2)

where

g_{iε}(u_{i+1}) = \{ \begin{matrix}f_{i}(u_{i+1}), & u_{i+1} ≥ ε, i = 1, ⋯, n − 1, \\ f_{n}(u_{1}), & u_{1} ≥ ε, \\ f_{i}(\frac{ε}{2}), & u_{i+1} < ε, i = 1, ⋯, n − 1, \\ f_{n}(\frac{ε}{2}), & u_{1} < ε. \end{matrix}

By the standard parabolic theory, it is easy to prove that u_{iε} ≥ ε, i = 1, ⋯, n. The fact that u_{iε} ≥ ε shows g_{iε} = f_{i}, i = 1, 2, ⋯, n, and hence (u_{1ε}, u_{2ε}, ⋯, u_{nε}) solves the following problem

\{ \begin{matrix}(u_{iε})_{t} = f_{i}(u_{(i+1)ε})(Δu_{iε} + a_{i}u_{iε} − a_{i}ε), & i = 1, ⋯, n − 1, \\ (u_{nε})_{t} = f_{n}(u_{1ε})(Δu_{nε} + a_{n}u_{nε} − a_{n}ε), & x ∈ Ω, t > 0, \\ u_{iε}(x, 0) = u_{i0}(x) + ε, & x ∈ Ω, \\ u_{iε}(x, t) = ε, & x ∈ ∂Ω, t > 0, \end{matrix}  (2.3)

where ε ∈ (0, 1]. By the classical parabolic theory, under hypothesis (H1)-(H4), (2.3) admits a unique positive solution (u_{1ε}, ⋯, u_{nε}) ∈ [C(Ω̄_{T}) ∩ C^{2,1}(Ω_{T})]^{n} for 0 < T < T (ε), where T (ε) is the maximal existence time.

Now, estimate the lower and upper bounds of (u_{1ε}, u_{2ε}, ⋯, u_{nε}).

Let λ_{1}, φ(x) > 0 be the first eigenvalue and the corresponding eigenfunction of the following eigenvalue problem

−Δφ = λφ in Ω; φ = 0 on ∂Ω  (2.4)

and think that max_{Ω̄} φ(x) = 1, then λ_{1} > 0 and \frac{∂φ}{∂η} < 0 on ∂Ω, where η is the outward normal vector on ∂Ω. By (H_{1}) and (H_{2}), there exist positive constants k_{1} and k_{2} such that

k_{1}φ(x) ≤ u_{i0}(x) ≤ k_{2}φ(x), x ∈ Ω̄, i = 1, 2, ⋯, n.  (2.5)

Let M = max_{1≤i≤n} max_{Ω̄} u_{i0}(x) and (g_{1}(t), ⋯, g_{n}(t)) be the unique solution of the following ODE

\{ \begin{matrix}g′_{i} = a_{i}g_{i}f_{i}(g_{i+1}), & i = 1, ⋯, n − 1, \\ g′_{n} = a_{n}g_{n}f_{n}(g_{1}), & t > 0, \\ g_{i}(0) = M + 1, & i = 1, 2, ⋯, n, \end{matrix}  (2.6)

where f_{1}, ···, f_{n} are given as above. Then g_{i}(t) ≥ M +1, i = 1, 2, ···, n. Denote by T^{∗}, 0 < T^{∗} < ∞, its maximum existence time (note that T^{∗} < ∞ must hold because that (g_{1}, ⋯, g_{n}) blow up in finite time).

Applying the standard comparison principle for parabolic system, we have the following Lemmas (see [13, 14]).

Lemma 1. Assume that u_{i}(x, t) ∈ C(Ω̄ × [0, T (ε))) ∩ C^{2,1}(Ω × (0, T (ε)))(i = 1, ⋯, n) is a lower (or upper) solution of (2.3), then (u_{1}, ⋯, u_{n}) ≤ (≥)(u_{1ε}, ⋯, u_{nε}) on Ω̄ × [0, T (ε)).

Lemma 2. If ε_{1} < ε_{2}, then (u_{1ε_{1}}(x, t), ···, u_{nε_{1}}(x, t)) ≤ (u_{1ε_{2}}(x, t), ···, u_{nε_{2}}(x, t)) on Ω̄×[0, T (ε_{2})) and T (ε_{1}) ≥ T (ε_{2}).

Lemma 3. Let ε < 1, (u_{1ε}, ⋯, u_{nε}) be the solution of (2.3), then for any fixed T : 0 < T < min{T (ε), T^{∗}},

u_{iε}(x, t) ≤ g_{i}(t), ∀ (x, t) ∈ Ω̄ × [0, T], i = 1, 2, ⋯, n,

which implies that T (ε) ≥ T^{∗} for all ε < 1.

Proof. Set w_{i}(x, t) = g_{i}(t) − u_{iε}(x, t), then we have

w_{1t} = g′_{1} − (u_{1ε})_{t}
= a_{1}g_{1}f_{1}(g_{2}) − f_{1}(u_{2ε})(Δu_{1ε} + a_{1}u_{1ε} − a_{1}ε)
= a_{1}g_{1}f_{1}(g_{2}) − f_{1}(u_{2ε})(−Δw_{1} + a_{1}g_{1} − a_{1}w_{1} − a_{1}ε)
= f_{1}(u_{2ε})(Δw_{1} + a_{1}w_{1}) + a_{1}g_{1}(f_{1}(g_{2}) − f_{1}(u_{2ε})) + a_{1}εf_{1}(u_{2ε})
> f_{1}(u_{2ε})(Δw_{1} + a_{1}w_{1}) + a_{1}g_{1}( ∫_{0}^{1} f′_{1}(u_{2ε} + s(g_{2} − u_{2ε}))ds)w_{2},
· · · · · · · · · · · · · · · · · · · · · · · · · · · ,
w_{(n−1)t} > f_{n−1}(u_{nε})(Δw_{n−1} + a_{n−1}w_{n−1}) + a_{n−1}g_{n−1}( ∫_{0}^{1} f′_{n−1}(u_{nε} + s(g_{n} − u_{nε}))ds)w_{n},
w_{nt} > f_{n}(u_{1}ε)(Δw_{n} + a_{n}w_{n}) + a_{n}g_{n}( ∫_{0}^{1} f′_{n}(u_{1ε} + s(g_{1} − u_{1ε}))ds)w_{1}, x ∈ Ω, 0 < t ≤ T,
w_{i}(x, 0) = M + 1 − u_{i0} − ε > 0, x ∈ Ω, i = 1, 2, ⋯, n,
w_{i}(x, t) = g_{i}(t) − ε > 0, x ∈ ∂Ω, 0 < t ≤ T, i = 1, 2, ⋯, n.

Proposition 1 implies that w_{i} ≥ 0 (i = 1, 2, ⋯, n) and hence the result of this Lemma holds.

In the following we denote T_{∗} = T^{∗}/2.

Lemma 4. Let ε < 1, (u_{1ε}, ⋯u_{nε}) be the solution of (2.3), and the positive constant k_{1} satisfies (2.5), then we have the following estimates:

(i) if λ_{1} ≤ a_{1} ≤ ⋯ ≤ a_{n} then

u_{iε}(x, t) ≥ k_{1}φ(x) + ε, ∀(x, t) ∈ Ω̄ × [0, T (ε)), i = 1, 2, ⋯, n;

(ii) if there exists j : 1 ≤ j ≤ n − 1, such that a_{j} < λ_{1} ≤ a_{j+1}, then we have, for ρ = max{(λ_{1} − a_{1})f_{1}(g_{2}(T_{∗})), ⋯, (λ_{1} − a_{j})f_{j}(g_{j+1}(T_{∗})},

u_{iε}(x, t) ≥ \{ \begin{matrix}k_{1}φ(x)e^{−ρt} + ε, 1 ≤ i ≤ j, ∀(x, t) ∈ Ω̄ × [0, T_{∗}], \\ k_{1}φ(x) + ε, j + 1 ≤ i ≤ n, ∀(x, t) ∈ Ω̄ × [0, T_{∗}]; \end{matrix}

(iii) if a_{1} ≤ ⋯ ≤ a_{n} < λ_{1}, then we have, for r = max {(λ_{1} − a_{1})f_{1}(g_{2}(T_{∗})), ⋯, (λ_{1} − a_{n−1})f_{n−1}(g_{n}(T_{∗})), (λ_{1} − a_{n})f_{n}(g_{1}(T_{∗}))},

u_{iε}(x, t) ≥ k_{1}φ(x)e^{−rt} + ε, ∀(x, t) ∈ Ω̄ × [0, T_{∗}], i = 1, 2 · ··, n.

Proof. (i) when λ_{1} ≤ a_{1} ≤ ⋯ ≤ a_{n}, set

w_{i}(x, t) = u_{iε} − (k_{1}φ(x) + ε), i = 1, 2, ⋯, n.

Then we have, by (2.3),

w_{1t} = u_{1εt} = f_{1}(u_{2ε})(Δu_{1ε} + a_{1}u_{1ε} − a_{1}ε)
= f_{1}(u_{2ε})(Δw_{1} + k_{1}Δφ + a_{1}w_{1} + a_{1}k_{1}φ)
= f_{1}(u_{2ε})(Δw_{1} + a_{1}w_{1}) + (a_{1} − λ_{1})k_{1}φf_{1}(u_{2ε}),
≥ f_{1}(u_{2ε})(Δw_{1} + a_{1}w_{1}), x ∈ Ω, 0 < t < T (ε),
w_{1}(x, 0) = u_{10}(x) − k_{1}φ(x) ≥ 0, x ∈ Ω,
w_{1}(x, t) = 0, x ∈ ∂Ω, 0 < t < T (ε).

Applying Proposition 1 we see that w_{1}(x, t) ≥ 0 , i.e. u_{1ε} ≥ k_{1}φ + ε. Similarly we have u_{iε} ≥ k_{1}φ + ε, i = 2, ⋯, n.

(ii) when there exists j : 1 ≤ j ≤ n − 1, such that a_{j} < λ_{1} ≤ a_{j+1}, set

w_{i}(x, t) = u_{iε} − (k_{1}φ(x)e^{−ρt} + ε), 1 ≤ i ≤ j.

A routine calculation yields

w_{1t} = u_{1εt} + ρk_{1}φe^{−ρt} = f_{1}(u_{2ε})(Δu_{1ε} + a_{1}u_{1ε} − a_{1}ε) + ρk_{1}φe^{−ρt}
= f_{1}(u_{2ε})(Δw_{1} + k_{1}Δφe^{−ρt} + a_{1}w_{1} + a_{1}k_{1}φe^{−ρt}) + ρk_{1}φe^{−ρt}
= f_{1}(u_{2ε})(Δw_{1} + a_{1}w_{1}) + (a_{1} − λ_{1})f_{1}(u_{2ε})k_{1}φe^{−ρt} + ρk_{1}φe^{−ρt}
= f_{1}(u_{2ε})(Δw_{1} + a_{1}w_{1}) + k_{1}φ((a_{1} − λ_{1})f_{1}(u_{2ε}) + ρ)e^{−ρt},
· · · · · · · · · · · · · · · · · · · · · · · · · · · ,
w_{jt} = f_{j}(u_{(j+1)ε})(Δw_{j} + a_{j}w_{j}) + k_{1}φ((a_{j} − λ_{1})f_{j}(u_{(j+1)ε}) + ρ)e^{−ρt}.

Since u_{iε}(x, t) ≤ g_{i}(t) ≤ g_{i}(T_{∗}), 1 ≤ i ≤ n for all (x, t) ∈ Ω̄ × [0, T_{∗}], it follows that

(a_{i} − λ_{1})f_{i}(u_{(i+1)ε}) + ρ ≥ ρ − (λ_{1} − a_{i})f_{i}(g_{i+1}(T_{∗})) ≥ 0, 1 ≤ i ≤ j.

Therefore

w_{it} ≥ f_{i}(u_{(i+1)ε})(Δw_{i} + a_{i}w_{i}), 1 ≤ i ≤ j.

It is obvious that

w_{i}(x, 0) = u_{i0}(x) − k_{1}φ(x) ≥ 0, x ∈ Ω, i = 1, 2, ⋯, j,
w_{i}(x, t) = 0,  x ∈ ∂Ω, 0 < t ≤ T_{∗}, i = 1, 2, ⋯, j.

Proposition 1 asserts that w_{i} ≥ 0, 1 ≤ i ≤ j, i.e. u_{iε} ≥ k_{1}φ(x)e^{−ρt} + ε, 1 ≤ i ≤ j. The proof of the second one is the same as that of (i).

(iii) when a_{1} ≤ ⋯ ≤ a_{n} < λ_{1}, the proof is the same as that of (ii). The proof is completed.

It follows from Lemma 2 that T_{1} := T (1) < T (ε) for all ε ∈ (0, 1), and there exists u_{i}(x, t) ∈ L^{∞}(Ω × (0, T_{1})) such that (u_{1ε}, ⋯, u_{nε}) → (u_{1}, ⋯, u_{n}) as ε → 0^{+}. Furthermore, making use of the fact that on any smooth sub-domain Ω′ ⊂⊂ Ω, by Lemma 4 we obtain that u_{i}(x, t) (i = 1, 2, ⋯, n) has a positive lower bound in Ω′ × [0, T_{1}] which is independent of ε.

From the fact just proved it follows in turn by the interior parabolic Hölder estimates, which can obtained by the same argument as that of Theorem VII 3.1 in [6] that for each τ > 0 and Ω′ ⊂⊂ Ω, there is α > 0 such that

∥u_{i}∥_{C^{α, α/2}(Ω′×[τ, T_{1}])} ≤ C(Ω′, τ)

and then the local Hölder continuity of f_{i} (i = 1, 2, ···, n) on (0, ∞) together with Schauder estimates and diagonal methods we have that there exist subsequence {ε′} of {ε} and u_{i} ∈ C_{loc}^{2+α, 1+α/2}(Ω × (0, T̃]), where T̃ := min{T_{∗}, T_{1}}, such that

(u_{1ε′}, ⋯, u_{nε′}) ⟶ (u_{1}, ⋯, u_{n}) in [C^{2+α, 1+α/2}(Ω̄′ × [τ, T̃])]^{n} as ε′ → 0^{+}

for any Ω′ ⊂⊂ Ω and 0 < τ < T̃. And hence (u_{1}, ⋯, u_{n}) satisfies the problem (1.1). Fix ε_{0} : 0 < ε_{0} ≪ 1. For any Ω′ ⊂⊂ Ω and 0 < ε′ < ε_{0}, thanks to Lemma 4 and

u_{iε′}(x, t) ≤ g_{i}(t) on Ω̄′ × [0, T̃],

the L^{p} theory and imbedding theorem show that the C^{α, α/2}(Ω̄′×[0, T̃]) norms of u_{iε′} (i = 1, 2, ···, n) are uniformly bounded for all ε′ < ε_{0}. And hence

(u_{1ε′}, ⋯, u_{nε′}) ⟶ (u_{1}, ⋯, u_{n}) in [C^{β, β/2}(Ω̄′ × [0, T̃])]^{n} (0 < β < α) as ε′ → 0^{+},

which implies that u_{i} ∈ C(Ω × [0, T̃]), i = 1, 2, ⋯, n. Similar to the arguments of [4, 15] we can prove that (u_{1}, ⋯, u_{n}) is continuous on ∂Ω × (0, T̃]. Using the initial and boundary condition of (2.3) we see that (u_{1}, ⋯, u_{n}) satisfies the initial and boundary conditions of (1.1), i.e.

(u_{1}, ⋯, u_{n}) ∈ [C_{loc}^{2+β,1+β/2}(Ω × (0, T̃]) ∩ C(Ω̄ × [0, T̃])]^{n}

is a classical solution of (1.1).

Theorem 1. Problem (1.1) has a positive classical solution (u_{1}, ···, u_{n}) ∈ [C_{loc}^{2+β,1+β/2}(Ω×(0, T])∩ C(Ω̄ × [0, T])]^{n} for some β : 0 < β < 1 and T < ∞.

3 The case a_{1} ≤ λ_{1}: global existence

In this section we will prove the global existence of positive classical solution. Theorem 2. If a_{1} ≤ λ_{1} then problem (1.1) has positive global classical solution (u_{1}, ⋯, u_{n}). Moreover, all positive classical solutions must satisfy the following estimates.

(i) if a_{n} ≤ λ_{1}, then

u_{i}(x, t) ≤ k_{2}φ(x) on Ω̄ × [0, ∞), i = 1, ⋯, n;

(ii) if there exists j : 1 ≤ j ≤ n − 1 such that a_{j} ≤ λ_{1} < a_{j+1}, then

u_{i}(x, t) ≤ \{ \begin{matrix}k_{2}φ(x), & i = 1, ⋯, j, \\ k_{2}e^{a_{i}f_{i}(F_{i+1})t}, & i = j + 1, ⋯, n, \end{matrix}

where F_{n+1} = k_{2}, F_{i} = k_{2}e^{a_{i}f_{i}(F_{i+1})t}, j + 1 ≤ i ≤ n, and the positive constant k_{2} is given by (2.5).

Proof. For any given ε : 0 < ε < 1, let (u_{1ε}, ⋯, u_{nε}) be the unique positive classical solution of (2.3) which is defined on Ω̄ × [0, T (ε)) with T (ε) ≤ ∞, and the positive constants k_{1} and k_{2} are given by (2.5).

Step 1: upper bounds of (u_{1ε}, ⋯, u_{nε}).

(i) If a_{n} ≤ λ_{1}, let w_{i}(x, t) = k_{2}φ(x) + ε − u_{iε}(x, t), i = 1, ⋯, n, then we have

w_{1t} = −f_{1}(u_{2ε})(Δu_{1ε} + a_{1}u_{1ε} − a_{1}ε)
= f_{1}(u_{2ε})(Δw_{1} + a_{1}w_{1}) + k_{2}f_{1}(u_{2ε})(λ_{1} − a_{1})φ
≥ f_{1}(u_{2ε})(Δw_{1} + a_{1}w_{1}) (x, t) ∈ Ω × (0, T (ε)),
w_{1}(x, t) = 0,  (x, t) ∈ ∂Ω × (0, T (ε)),
w_{1}(x, 0) = k_{2}φ(x) − u_{0}(x) ≥ 0, x ∈ Ω.

Proposition 1 implies that w_{1} ≥ 0, i.e. u_{1ε}(x, t) ≤ k_{2}φ(x) + ε on Ω̄ × (0, T (ε)). By the same way we can prove that u_{iε}(x, t) ≤ k_{2}φ(x) + ε on Ω̄ × (0, T (ε)), 2 ≤ i ≤ n.

(ii) if there exists j : 1 ≤ j ≤ n − 1 such that a_{j} ≤ λ_{1} < a_{j+1}, then similar to (i), u_{iε}(x, t) ≤ k_{2}φ(x) + ε on Ω̄ × (0, T (ε)), i = 1, ⋯, j. Let

w_{i} = k_{2}e^{a_{i}f_{i}(F_{(i+1)ε})t} + ε − u_{iε}, i = j + 1, ⋯, n,

where F_{(n+1)ε} = k_{2} + ε, F_{nε} = k_{2}e^{a_{n}f_{n}(k_{2}+ε)t} + ε, F_{iε} = k_{2}e^{a_{i}f_{i}(F (i+1)ε)t} + ε, j + 1 ≤ i ≤ n − 1. in view of u_{iε}(x, t) ≤ k_{2}φ(x) + ε ≤ k_{2} + ε, 1 ≤ i ≤ j, we have that

w_{nt} = k_{2}a_{n}f_{n}(k_{2} + ε)e^{a_{n}f_{n}(k_{2}+ε)t} − (u_{nε})_{t}
= k_{2}a_{n}f_{n}(k_{2} + ε)e^{a_{n}f_{n}(k_{2}+ε)t} − f_{n}(u_{1ε})(Δu_{nε} + a_{n}u_{nε} − a_{n}ε)
= k_{2}a_{n}f_{n}(k_{2} + ε)e^{a_{n}f_{n}(k_{2}+ε)t} − f_{n}(u_{1ε})(−Δw_{n} + a_{n}k_{2}e^{a_{n}f_{n}(k_{2}+ε)t} − a_{n}w_{n})
= f_{n}(u_{1ε})(Δw_{n} + a_{n}w_{n}) + a_{n}k_{2}e^{a_{n}f_{n}(k_{2}+ε)t}(f_{n}(k_{2} + ε) − f_{n}(u_{1ε}))
≥ f_{n}(u_{1ε})(Δw_{n} + a_{n}w_{n}), (x, t) ∈ Ω × (0, T (ε)),
w_{n}(x, t) = k_{2}e^{a_{n}f_{n}(k_{2}+ε)t} + ε − ε > 0, (x, t) ∈ ∂Ω × (0, T (ε)),
w_{n}(x, 0) = k_{2} − u_{n0} > 0, x ∈ Ω.

Proposition 1 implies that w_{n} ≥ 0, i.e. u_{nε} ≤ k_{2}e^{a_{n}f_{n}(k_{2}+ε)t} + ε on Ω̄ × (0, T (ε)). Similarly, we have

w_{(n−1)t} = f_{n−1}(u_{nε})(−Δw_{n−1} + a_{n−1}w_{n−1}) + a_{n−1}k_{2}e^{a_{n−1}f_{n−1}(F_{nε})t}(f_{n−1}(F_{nε}) − f_{n−1}(u_{nε}))
≥ f_{n−1}(u_{nε})(−Δw_{n−1} + a_{n−1}w_{n−1}), (x, t) ∈ Ω × (0, T (ε)),
w_{n−1}(x, t) = k_{2}e^{a_{n−1}f_{n−1}(F_{nε})t} > 0, (x, t) ∈ ∂Ω × (0, T (ε)),
w_{n−1}(x, 0) = k_{2} − u_{(n−1)0} > 0, x ∈ Ω.

Proposition 1 also implies that w_{n−1} ≥ 0, i.e. u_{(n−1)ε} ≤ k_{2}e^{a_{n−1}f_{n−1}(F_{nε})t} + ε on Ω̄ × (0, T (ε)). Similarly, we can get u_{iε} ≤ k_{2}e^{a_{i}f_{i}(F_{i+1})t} + ε on Ω̄ × (0, T (ε)), j + 1 ≤ i ≤ n − 2.

Step 2: lower bounds of (u_{1ε}, ⋯, u_{nε}).

(i) if a_{1} = λ_{1}, similar to the proof of (i) of Lemma 4 we have

u_{iε}(x, t) ≥ k_{1}φ(x) + ε, (x, t) ∈ Ω × (0, T (ε));

(ii) if there exists j : 1 ≤ j ≤ n − 1, such that a_{j} < λ_{1} ≤ a_{j+1}, then

u_{iε}(x, t) ≥ \{ \begin{matrix}k_{1}φ(x) exp\{− ∫_{0}^{t}(λ_{1} − a_{i})f_{i}(k_{2}e^{a_{i+1}f_{i+1}(F_{(i+2)ε})s} + ε)ds\} + ε, (x, t) ∈ \overline{Ω} × [0, T (ε)), 1 ≤ i ≤ j, \\ k_{1}φ(x) + ε, (x, t) ∈ [0, T (ε)), j + 1 ≤ i ≤ n. \end{matrix}

In fact, similar to the proof of (ii) of Lemma 4 we know that u_{iε} ≥ k_{1}φ(x)+ε on Ω̄×[0, T_{ε}), j + 1 ≤ i ≤ n. Denote

h_{i}(t) = (λ_{1} − a_{i})f_{i}(k_{2}e^{a_{i+1}f_{i+1}(F_{(i+2)ε})t} + ε), i = 1, 2, ⋯, j,

where F_{(n+1)ε} = k_{2} + ε, F_{iε} = k_{2}e^{a_{i}f_{i}(F (i+1)ε)t} + ε, i = 1, 2, ⋯, j. And let

w_{i} = u_{iε} − k_{1}φ(x) exp { − ∫_{0}^{t} h_{i}(s)ds} − ε, i = 1, 2, ⋯, j.

Note that u_{(j+1)ε} ≤ k_{2}e^{a_{j+1}f_{j+1}(F_{(j+2)ε})t} + ε on Ω̄ × [0, T_{ε}), and by the direct computations we can see that

w_{jt} = (u_{jε})_{t} + k_{1}φ(x)h_{j}(t)e^{− ∫_{0}^{t} h_{j}(s)ds}
= f_{j}(u_{(j+1)ε})(Δw_{j} + a_{j}w_{j}) + k_{1}φ(x)(a_{j} − λ_{1})f_{j}(u_{(j+1)ε})e^{− ∫_{0}^{t} h_{j}(s)ds} + k_{1}φ(x)h_{j}(t)e^{− ∫_{0}^{t} h_{j}(s)ds}
≥ f_{j}(u_{(j+1)ε})(Δw_{j} + a_{j}w_{j}) + k_{1}φ(x)(h_{j}(t) − (λ_{1} − a_{j})f_{j}(u_{(j+1)ε}))e^{− ∫_{0}^{t} h_{j}(s)ds}
≥ f_{j}(u_{(j+1)ε})(Δw_{j} + a_{j}w_{j}), (x, t) ∈ Ω × (0, T (ε)),
w_{j}(x, t) = ε − ε = 0, (x, t) ∈ ∂Ω × (0, T (ε)),
w_{j}(x, 0) = u_{j0} − k_{1}φ(x) ≥ 0, x ∈ Ω.

By Proposition 1 we have w_{j} ≥ 0, i.e. u_{jε} ≥ k_{1}φ(x)e^{− ∫_{0}^{t} h_{j}(s)ds} + ε on Ω̄ × (0, T (ε)). If 1 ≤ i ≤ j, similar as above we also can prove

u_{iε} ≥ k_{1}φ(x)e^{− ∫_{0}^{t} h_{i}(s)ds} + ε, on Ω̄ × [0, T (ε)), 1 ≤ i ≤ j.

(iii) If a_{n} ≤ λ_{1}, then for i = 1, 2, ⋯, n, then

u_{iε}(x, t) ≥ k_{1}φ(x) exp { − ∫_{0}^{t}(λ_{1} − a_{i})f_{i}(k_{2}e^{a_{i+1}f_{i+1}(F_{(i+2)ε})s} + ε)ds} + ε, (x, t) ∈ Ω̄ × [0, T_{ε}),

where f_{n+1} = f_{1} and F_{i} given above.

The proof is similar as (ii).

Step 3: the upper bounds of (u_{1ε}, ⋯, u_{nε}) obtained by Step 1 show that (u_{1ε}, ⋯, u_{nε}) exists globally, i.e. T (ε) = ∞ for all 0 < ε < 1. For any Ω_{n} ⊂⊂ Ω and 0 < τ < T_{n} < ∞, Step 1 and 2 show that there exist positive constants σ(n, τ) and M(n, τ) such that

σ(n, τ) ≤ u_{iε} ≤ M(n, τ) on Ω̄_{n} × [τ, T_{n}], i = 1, 2, ⋯, n

for all 0 < ε < 1. Applying the standard local Schauder estimates and diagonal method we have that there exist subsequence {ε′} of {ε} and u_{i} ∈ C_{loc}^{2+α,1+α/2}(Ω × (0, ∞)) such that

(u_{1ε′}, ⋯, u_{nε′}) ⟶ (u_{1}, ⋯, u_{n}) in [C_{loc}^{2+α,1+α/2}(Ω̄_{∗} × [τ, T_{0}])]^{n} as ε′ → 0^{+}

for any Ω_{∗} ⊂⊂ Ω and 0 < τ < T_{0} < ∞. And hence (u_{1}, ⋯, u_{n}) satisfies the problem (1.1) in Ω × (0, ∞). Similar to the arguments of §2 we see that

(u_{1}, ⋯, u_{n}) ∈ [C_{loc}^{2+α,1+α/2}(Ω × (0, ∞)) ∩ C(Ω̄ × [0, ∞))]^{n}

is a classical solution of (1.1).

Estimates (i) and (ii) can be proved in the similar way to that of Step 1. The proof is completed.

4 The case a_{1} > λ_{1}: blow up result

In this section we will prove the blow up result of problem (1.1). Let G be a bounded domain of R^{N}, λ_{1}(G) be the first eigenvalue of −Δ on G with homogeneous boundary condition. And we consider the following initial-boundary problem:

\{ \begin{matrix}w_{t} = Af(w)(Δw + Bw), & s ∈ G, t > 0, \\ w(x, 0) = C, & x ∈ G, \\ w(x, t) = C, & x ∈ ∂Ω, t ≥ 0. \end{matrix}  (4.1)

where constants A, B, C > 0 and f(w) satisfy (H3). By the standard method (see [10]), it follows that (4.1) has a unique classical solution w(x, t) and w(x, t) ≥ C. And we can get following Lemma (see [2]):

Lemma 5. Let f satisfy (H3), then for any A > 0 and C > 0, the unique (local) solution w(x, t) of (4.1) blow up in finite time T if ∫_{0}^{∞} 1/(sf(s))ds < ∞ and λ_{1} < B, while the solution w(x, t) exists globally if ∫_{0}^{∞} 1/(sf(s))ds = ∞.

Theorem 3. Assume that a_{1} > λ_{1}, the initial datum (u_{10}, ⋯, u_{n0}) ∈ [C^{4}(Ω̄)]^{n} and (H1) − (H4) hold. If Δu_{i0} +a_{i}u_{i0} ≥ 0 in Ω and Δu_{i0} = 0 on ∂Ω, i = 1, ···, n, then the positive classical solution of (1.1) is unique and blow up in the finite time.

Proof. Step 1: monotonicity of (u_{1}, ⋯, u_{n}) in t. Let ε ∈ (0, 1) and (u_{1ε}, ⋯, u_{nε}) be the solution of (2.3), then

(u_{1ε}, ⋯, u_{nε}) ∈ [C^{2+α,1+α/2}(Ω × (0, T (ε))) × C^{2,1}(Ω̄ × [0, T (ε)))]^{n}

since Δu_{i0} = 0, i = 1, 2, ···, n on ∂Ω ( see Theorem 7.1 of chap.7 of [6]). Let w_{i} = u_{iεt}, i = 1, 2, ···, n, then we have,

w_{1t} = (f_{1}(u_{2ε})(Δu_{1ε} + a_{1}u_{1ε} − a_{1ε}))_{t}

= f_{1}(u_{2ε})(Δw_{1} + a_{1}w_{1}) + f_{1t}(u_{2ε})w_{2}(Δu_{1ε} + a_{1}u_{1ε} − a_{1}ε)

= f_{1}(u_{2ε})(Δw_{1} + a_{1}w_{1}) + [\frac{f_{1t}(u_{2ε})}{f_{1}(u_{2ε})}w_{2}]w_{1}, (x, t) ∈ Ω × (0, T (ε)),

· · · · · · · · · · · · · · · · · · · · · · · · · · · ,

w_{(n−1)t} = f_{n−1}(u_{nε})(Δw_{n−1} + a_{n−1}w_{n−1}) + [\frac{f_{(n−1)t}(u_{nε})}{f_{n−1}(u_{nε})}w_{n}]w_{n−1}, (x, t) ∈ Ω × (0, T (ε)),
w_{nt} = f_{n}(u_{1ε})(Δw_{n} + a_{n}w_{n}) + [\frac{f_{nt}(u_{1ε})}{f_{n}(u_{1ε})}w_{1}]w_{n}, (x, t) ∈ Ω × (0, T (ε)),
w_{i}(x, 0) = f_{i}(u_{i0}(x) + ε)(Δu_{i0}(x) + a_{i}u_{i0}(x)) ≥ 0, x ∈ Ω, i = 1, ⋯, n,
w_{i}(x, t) = 0, (x, t) ∈ Ω × (0, T (ε)), i = 1, ⋯, n.

In view of u_{iε} ≥ ε and w_{i} ∈ C(Ω̄ × [0, T (ε))), i = 1, ⋯, n, the L^{p}−theory and Schauder-Theory implies that (w_{1}, ⋯, w_{n}) is a classical solution, i.e. w_{i} ∈ C(Ω̄ × [0, T (ε))) ∩ C^{2,1}(Ω × (0, T (ε))). Proposition 1 shows that w_{i} ≥ 0, i.e. u_{iεt} ≥ 0, i = 1, 2, ⋯, n. since

(u_{1ε}, ⋯, u_{nε}) ⟶ (u_{1}, ⋯, u_{n}) in [C_{loc}^{2+α,1+α/2}(Ω × (0, T)]^{n} as ε → 0^{+},

we know that u_{it} ≥ 0, i = 1, ⋯, n and hence u_{i} ≥ u_{i0}(x), i = 1, ⋯, n, in Ω × (0, T). Step 2: the uniqueness. Let (u_{1}, ⋯, u_{n}) be the solution of (1.1) obtained by §2, then by step 1 u_{it} ≥ 0, i = 1, ⋯, n,

which implies Δu_{i} + a_{i}u_{i} ≥ 0. Let (ũ_{1}, ⋯, ũ_{n}), which defined on Ω̄ × [0, T̃), be another positive solution of (1.1) with the same initial datum (u_{10}, ⋯, u_{n0}), and set w_{i} = ũ_{i} − u_{i}, i = 1, ⋯, n, then we have, for any 0 < T_{0} < min{T, T̃},

w_{1t} = ũ_{1t} − u_{1t} = f_{1}(ũ_{2})(Δũ_{1} + a_{1}ũ_{1}) − f_{1}(u_{2})(Δu_{1} + a_{1}u_{1})
= f_{1}(ũ_{2})(Δw_{1} + Δu_{1} + a_{1}w_{1} + a_{1}u_{1}) − f_{1}(u_{2})(Δu_{1} + a_{1}u_{1})
= f_{1}(ũ_{2})(Δw_{1} + a_{1}w_{1}) + (f_{1}(ũ_{2}) − f_{1}(u_{2}))(Δu_{1} + a_{1}u_{1})
= f_{1}(ũ_{2})(Δw_{1} + a_{1}w_{1}) + ( ∫_{0}^{1} f′_{1}(u_{2} + s(ũ_{2} − u_{2}))ds(Δu_{1} + a_{1}u_{1}))w_{2}, (x, t) ∈ Ω × (0, T_{0}),
· · · · · · · · · · · · · · · · · · · · · · · · · · · ,
w_{(n−1)t} = f_{n−1}(ũ_{n})(Δw_{n−1} + a_{n−1}w_{n−1})
+( ∫_{0}^{1} f′_{1}(u_{n} + s(ũ_{n} − u_{n}))ds(Δu_{n−1} + a_{n−1}u_{n−1}))w_{n}, (x, t) ∈ Ω × (0, T_{0}),
w_{nt} = f_{n}(ũ_{1})(Δw_{n} + a_{n}w_{n}) + ( ∫_{0}^{1} f′_{1}(u_{1} + s(ũ_{1} − u_{1}))ds(Δu_{n} + a_{n}u_{n}))w_{1}, (x, t) ∈ Ω × (0, T_{0}),
w_{i} = 0, (x, t) ∈ ∂Ω × (0, T_{0}) ∪ Ω × {0}, i = 1, 2 · ··, n.

Since Δu_{i} + a_{i}u_{i} ≥ 0, i = 1, 2 · ··, n, Proposition 1 implies that w_{i}(x, t) ≡ 0, i = 1, 2, ⋯, n, i.e. ũ_{i}(x, t) ≡ u_{i}(x, t), i = 1, 2, ⋯, n. The uniqueness is proved.

Step 3: (u_{1}, ⋯, u_{n}) blow up in the finite time.

Since a_{1} > λ_{1}, ∃Ω′ : Ω′ ⊂⊂ Ω, s.t. the first eigenvalue λ′_{1} of −Δ in Ω′ with homogeneous Dirichlet boundary condition satisfies a_{1} > λ′_{1}. Applying u_{i} ≥ u_{i0} > 0 (i = 1, 2, ⋯, n) in Ω we know that

u_{i}(x, t) ≥ σ, i = 1, 2, ⋯, n on Ω̄′ × (0, T)

for some positive constant σ. Next, we consider the following system

\{ \begin{matrix}\underline{u}_{it} = f_{i}(\underline{u}_{i+1})(Δ\underline{u}_{i} + a_{i}\underline{u}_{i}), & i = 1, ⋯, n − 1, \\ \underline{u}_{nt} = f_{n}(\underline{u}_{1})(Δ\underline{u}_{n} + a_{n}\underline{u}_{n}), & x ∈ Ω′, t > 0, \\ \underline{u}_{i}(x, t) = σ, & x ∈ ∂Ω′, i = 1, 2, ⋯, n, \\ \underline{u}_{i}(x, 0) = σ, & x ∈ Ω′, i = 1, 2, ⋯, n. \end{matrix}  (4.2)

By the classical parabolic theory, there exists a nonnegative classical solution (\underline{u}_{1}, ⋯, \underline{u}_{n}) for (x, t) ∈ Ω′×(0, T′), where T′ denotes the maximal existence time of (4.2). The standard comparison principle for parabolic system implies that T′ ≥ T and

u_{i}(x, t) ≥ \underline{u}(x, t), (x, t) ∈ Ω̄′ × [0, T), i = 1, 2, ⋯, n.

If we can prove that the solution (\underline{u}_{1}(x, t), ⋯, \underline{u}_{n}(x, t)) of (4.2) blow up in finite time, So does (u_{1}, ⋯ u_{n}).

Since the initial data (σ, ···, σ) is a lower solution of (4.2), the standard upper and lower solutions method asserts that \underline{u}_{it} ≥ 0, i = 1, 2, ⋯, n, which implies that Δ\underline{u}_{i} + a_{i}\underline{u}_{i} ≥ 0, i = 1, 2, ⋯, n. And hence

\underline{u}_{i}(x, t) ≥ σ, i = 1, 2, ⋯, n, (x, t) ∈ Ω̄′ × [0, T′).

Furthermore, (\underline{u}_{1}, ⋯ \underline{u}_{n}) satisfies, by (1.6),

\{ \begin{matrix}\underline{u}_{it} & ≥ & K_{0}f_{n}(\underline{u}_{i+1})(Δ\underline{u}_{i} + a_{i}\underline{u}_{i}), i = 1, 2, ⋯, n − 1, \\ \underline{u}_{nt} & = & f_{n}(\underline{u}_{1})(Δ\underline{u}_{n} + a_{n}\underline{u}_{n}),  x ∈ Ω′, 0 < t < T′. \end{matrix}  (4.3)

Choose k = min{1, K_{0}}, and denote by z(x, t) the unique positive solution of the following problem:

\{ \begin{matrix}z_{t} = kf_{n}(z)(Δz + a_{1}z), & x ∈ Ω′, t > 0, \\ z(x, 0) = σ, & x ∈ Ω̄′, \\ z(x, t) = σ, & x ∈ ∂Ω′, t > 0. \end{matrix}  (4.4)

By Lemma 5 it comes that z(x, t) blow up in finite time T_{0} < ∞. Moreover z_{t} ≥ 0, i.e. Δz+a_{1}z ≥ 0, since the initial data is a lower solution of (4.4). Next we will prove z(x, t) ≤ \underline{u}_{i}(x, t), i = 1, 2, ···, n. Let

w_{i}(x, t) = \underline{u}_{i}(x, t) − z(x, t), i = 1, 2, ⋯, n,

then

w_{1t} = \underline{u}_{1t} − z_{t} ≥ kf_{n}(\underline{u}_{2})(Δ\underline{u}_{1} + a_{1}\underline{u}_{1}) − kf_{n}(z)(Δz + a_{1}z)
≥ kf_{n}(\underline{u_{2}})(Δw_{1} + Δz + a_{1}w_{1} + a_{1}z) − kf_{n}(z)(Δz + a_{1}z)
= kf_{n}(\underline{u_{2}})(Δw_{1} + a_{1}w_{1}) + k(Δz + a_{1}z)(f_{n}(\underline{u}_{2}) − f_{n}(z))
= kf_{n}(\underline{u_{2}})(Δw_{1} + a_{1}w_{1}) + [k(Δz + a_{1}z) ∫_{0}^{1} f′_{n}(z + s(\underline{u}_{2} − z))ds]w_{2}, x ∈ Ω′, 0 < t < T_{0},
w_{1}(x, 0) = 0,  x ∈ Ω̄′,
w_{1}(x, t) = 0,  x ∈ ∂Ω̄′, 0 < t < T_{0}.

Proposition 1 implies w_{1} ≥ 0, i.e. \underline{u}_{1}(x, t) ≥ z(x, t). similarly we can prove \underline{u}_{i}(x, t) ≥ z(x, t), i = 2, ⋯, n. which also implies that (\underline{u}_{1}, ⋯, \underline{u}_{n}) blow up in finite time, and so does the solution (u_{1}, ⋯, u_{n}) of (1.1). The proof is completed.

5 Acknowledgement

The author would like to express his sincere gratitude to the valuable suggestions from Professor Mingxin Wang.

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(Received June 8, 2009)