Blowup Estimates for a Mutualistic Model in Ecology ^{∗}

Zhigui Lin

Department Mathematics, Yangzhou University Yangzhou 225002, P. R. China e-mail: zglin68@hotmail.com

Abstract. The cooperating two-species Lotka-Volterra model is discussed. We study the blowup properties of the solution to a parabolic system with homogeneous Dirichlet boundary conditions. The upper and lower bounds of blowup rate are obtained.

Key words: reaction diffusion system, blowup estimates, upper and lower bounds.

AMS subject classifications: 35K15, 35K65.

1 Introduction and main results

The well-known Lotka-Volterra ecological model, which involves a coupled system of two ordinary differential equations, has been given an enormous attention in the past decades. When the effect of dispersion is taken into consideration the densities u, v of the species are governed by

\{ \begin{matrix}u_{t} − d_{1}Δu = u(a_{1} − b_{1}u − c_{1}v), & x ∈ Ω, t > 0, \\ v_{t} − d_{2}Δv = v(a_{2} − b_{2}u − c_{2}v), & x ∈ Ω, t > 0, \\ u(x) = v(x) = 0, & x ∈ ∂Ω, t > 0, \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ \overline{Ω}, \end{matrix}  (1.1)

where Δ is the Laplacian operator, Ω is a bounded domain in R^{N} with ∂Ω uniformly C^{2+α}-smooth, u_{0}(x) and v_{0}(x) are nonnegative smooth functions with u_{0}(x) = v_{0}(x) = 0 on ∂Ω. d_{i}, a_{i}, b_{i} and c_{i} (i = 1, 2) are positive constants. d_{i} represents its respective diffusion rate and the real number a_{i}, its net birth rate. b_{1} and c_{2} are the coefficients of intra-specific competitions and b_{2}, c_{1} are that of inter-specific competitions. Here we consider the case with homogeneous Dirichlet boundary conditions, which implies that the habitat is surrounded by a totally hostile environment.

∗The work is partially supported by PRC grant NSFC 10171088 and CSC Foundation.

If the presence of one species encourages the growth of the other species then the system (1.1) becomes so-called mutualistic model:

\{ \begin{matrix}u_{t} − d_{1}Δu = u(a_{1} − b_{1}u + c_{1}v), & x ∈ Ω, t > 0, \\ v_{t} − d_{2}Δv = v(a_{2} + b_{2}u − c_{2}v), & x ∈ Ω, t > 0, \\ u(x) = v(x) = 0, & x ∈ ∂Ω, t > 0, \\ u(x, 0) = u_{0}(x), v(x, 0) = v_{0}(x), & x ∈ \overline{Ω}. \end{matrix}  (1.2)

Because of the quasimonotone nondecreasing of reaction functions in (1.2), there is a quite different behavior of solutions compared with the solutions of (1.1). The solution of (1.1) with any nonnegative initial data is unique and global, while the blowup solutions are possible when the two species are strongly mutualistic (b_{2}c_{1} > b_{1}c_{2}), which means that the geometric mean of the interaction coefficients exceeds that of population regulation coefficients. Here we give only the related result of Pao [20].

Theorem 1.1 (i) If b_{2}c_{1} < b_{1}c_{2}, the problem (1.2) has a unique global solution (u, v), which is uniformly bounded in [0, ∞) × \overline{Ω};

(ii) If b_{2}c_{1} > b_{1}c_{2} and a_{1} ≥ λ_{1}, a_{2} ≥ λ_{2}, there exists a finite time T such that the unique solution to (1.2) exists in [0, T)×\overline{Ω} and blows up in the meaning that lim_{t→T} max(|u(x, t)| + |v(x, t)|) = ∞;

(iii) If b_{2}c_{1} > b_{1}c_{2}, the solution will blow up for any a_{1} ≥ 0 and a_{2} ≥ 0 under suitable initial data.

Based on the above result, we are chiefly interested in studying the blowup properties of the solution. We derive the upper and lower bounds of blowup rate, that is, there are positive constants c and C such that

c(T − t)^{−1} ≤ max_{\overline{Ω}} u(x, t) ≤ C(T − t)^{−1}, c(T − t)^{−1} ≤ max_{\overline{Ω}} v(x, t) ≤ C(T − t)^{−1}

for t ∈ (0, T) if N = 1.

There are some related results on the blowup of solutions to nonlinear parabolic systems, see for example [19] and [24]. In a recent paper, Lou etc. in [18] considered (1.2) with homogeneous Neumann boundary conditions and gave a sufficient condition on the initial data for the solution to blow up in finite time. For the blowup estimates, as we know, no result has been given owing to the cross-coupled reactions.

For the related elliptic systems, there is an extensive literature regarding the existence and uniqueness of positive solutions, the reader can see [1, 10, 12, 14, 15, 16, 17, 20, 23] and the references therein.

The paper is arranged as follows. In §2 the comparison principles for bounded and unbounded domains are given. In §3, we derive the lower bound of blowup rate and §4 deals with its upper bound.

2 Comparison principles

In this section, we show the comparison principle for unbounded domains, which will be used in the sequel. For completeness, we also give the comparison principle for bounded domains.

Lemma 2.1 Let Ω be a bounded domain with smooth boundary ∂Ω. u_{i}, v_{i} ∈ C(\overline{Ω} × [0, T)) ⋂ C^{2,1}(Ω × (0, T)) (i=1,2) and satisfy

\{ \begin{matrix}u_{1t} − d_{1}Δu_{1} ≥ u_{1}(a_{1} − b_{1}u_{1} + c_{1}v_{1}), & x ∈ Ω, t > 0, \\ v_{1t} − d_{2}Δv_{1} ≥ v_{1}(a_{2} + b_{2}u_{1} − c_{2}v_{1}), & x ∈ Ω, t > 0, \\ u_{2t} − d_{1}Δu_{2} ≤ u_{2}(a_{1} − b_{1}u_{2} + c_{1}v_{2}), & x ∈ Ω, t > 0, \\ v_{2t} − d_{2}Δv_{2} ≤ v_{2}(a_{2} + b_{2}u_{2} − c_{2}v_{2}), & x ∈ Ω, t > 0, \\ u_{1}(x, t) ≥ u_{2}(x, t), v_{1}(x, t) ≥ v_{2}(x, t), & x ∈ ∂Ω, t > 0, \\ u_{1}(x, 0) ≥ u_{2}(x, 0), v_{1}(x, 0) ≥ v_{2}(x, 0), & x ∈ \overline{Ω}. \end{matrix}  (2.1)

Then u_{1}(x, t) ≥ u_{2}(x, t) and v_{1}(x, t) ≥ v_{2}(x, t) in \overline{Ω} × [0, T). Moreover, if u_{2}(x, 0) ≢ u_{1}(x, 0) ≥ u_{2}(x, 0) and v_{2}(x, 0) ≢ v_{1}(x, 0) ≥ v_{2}(x, 0), then u_{1}(x, t) > u_{2}(x, t) and v_{1}(x, t) > v_{2}(x, t) in Ω × (0, T).

Lemma 2.2 Let Ω_{u} be a unbounded domain with boundary ∂Ω ∈ C^{2+α}. u_{i}, v_{i} ∈ C(\overline{Ω}_{u} × [0, T)) ⋂ C^{2,1}(Ω_{u} × (0, T)) (i=1,2) and satisfy

\{ \begin{matrix}u_{1t} − d_{1}Δu_{1} ≥ u_{1}(a_{1} − b_{1}u_{1} + c_{1}v_{1}), & x ∈ Ω_{u}, t > 0, \\ v_{1t} − d_{2}Δv_{1} ≥ v_{1}(a_{2} + b_{2}u_{1} − c_{2}v_{1}), & x ∈ Ω_{u}, t > 0, \\ u_{2t} − d_{1}Δu_{2} ≤ u_{2}(a_{1} − b_{1}u_{2} + c_{1}v_{2}), & x ∈ Ω_{u}, t > 0, \\ v_{2t} − d_{2}Δv_{2} ≤ v_{2}(a_{2} + b_{2}u_{2} − c_{2}v_{2}), & x ∈ Ω_{u}, t > 0, \\ u_{1}(x, t) ≥ u_{2}(x, t), v_{1}(x, t) ≥ v_{2}(x, t), & x ∈ ∂Ω_{u}, t > 0, \\ u_{1}(x, 0) ≥ u_{2}(x, 0), v_{1}(x, 0) ≥ v_{2}(x, 0), & x ∈ \overline{Ω}_{u} \end{matrix}  (2.2)

and there exist positive constants A and γ such that

\begin{matrix}|u_{i}(x, t)| ≤ A exp(γ|x|^{2}) \\ |v_{i}(x, t)| ≤ A exp(γ|x|^{2}) \end{matrix} as |x| → ∞ (0 < t < T).  (2.3)

Then u_{1}(x, t) ≥ u_{2}(x, t) and v_{1}(x, t) ≥ v_{2}(x, t) in \overline{Ω} × [0, T).

Lemma 2.1 is followed by the strong Maximum principle and Lemma 2.2 is followed by the Phragman-Lindelöf principle ( see [21], [22]).

Remark 2.1 When Ω = R^{N}, the boundary inequality in 2.2 is redundant. The condition in 2.3 is called the growth condition.

Remark 2.2 Since (0, 0) is unique solution of (1.2) with u(x, 0) ≡ 0 and v(x, 0) ≡ 0. Lemma 2.1 implies that if (u, v) be the nonnegative solution of (1.2), then u, v ≡ 0 or u, v > 0 in Ω × (0, T).

Remark 2.3 Lemmas 2.1 and 2.2 hold for the more general case. For example, for the system

\{ \begin{matrix}u_{t} − d_{1}Δu = f(x, t, u, v), & x ∈ Ω, t > 0, \\ v_{t} − d_{2}Δv = g(x, t, u, v), & x ∈ Ω, t > 0, \end{matrix}  (2.4)

Lemmas 2.1 and 2.2 hold if f, g are quasi-monotone nondecreasing, i.e. f is nondecreasing with respect to the component of v and g is nondecreasing with respect to the component of u, see [21] in detail.

3 Lower blowup estimate

We first establish the relationship between u and v as the solution (u, v) of (1.2) near the blow-up time.

Lemma 3.1 Let (u, v) be the nonnegative solution of (1.2), which blows up at t = T. Then there exists δ such that

δ max_{\overline{Ω}×[0,t]} v(x, τ) ≤ max_{\overline{Ω}×[0,t]} u(x, τ) ≤ \frac{1}{δ} max_{\overline{Ω}×[0,t]} v(x, τ), t ∈ (T/2, T).  (3.1)

Proof: Let

U(t) = max_{\overline{Ω}×[0,t]} u(x, τ), V (t) = max_{\overline{Ω}×[0,t]} v(x, τ).

As in [2] or [3], we argue by contradiction. Without loss of generality we may assume that there exists a sequence {t_{n}} with t_{n} → T as n → ∞ such that

V (t_{n})U^{−1}(t_{n}) → 0.  (3.2)

For each t_{n}, there exists

(x̂_{n}, t̂_{n}) ∈ Ω × (0, t_{n}] such that u(x̂_{n}, t̂_{n}) = U(t_{n}).  (3.3)

Since (u, v) blows up, we have that U(t_{n}) → ∞ as t_{n} → T and t̂_{n} → T as n → ∞. Let d_{n} denote the distant of x̂_{n} to ∂Ω. Similarly as in [4], we distinguish two cases:

(i) lim sup_{n→∞} \frac{d_{n}}{λ_{n}} = ∞ and (ii) lim sup_{n→∞} \frac{d_{n}}{λ_{n}} < ∞.

Case (i) Choose a subsequence (denoted again by {t_{n}}) such that

lim_{n→∞} \frac{d_{n}}{λ_{n}} = ∞.

We now introduce the scaling argument inspired by [9]. Let

λ_{n} := λ(t_{n}) := U^{−1/2}(t_{n}),  (3.4)
ϕ^{λ_{n}}(y, s) := λ_{n}^{2}u(λ_{n}y + x̂_{n}, λ_{n}^{2}s + t̂_{n}), (y, s) ∈ \overline{Ω_{n}} × I_{n}(T),  (3.5)
ψ^{λ_{n}}(y, s) := λ_{n}^{2}v(λ_{n}y + x̂_{n}, λ_{n}^{2}s + t̂_{n}), (y, s) ∈ \overline{Ω_{n}} × I_{n}(T),  (3.6)

where

I_{n}(t) := (−λ_{n}^{−2}t̂_{n}, λ_{n}^{−2}(t − t̂_{n})), Ω_{n} := {y : λ_{n}y + x̂_{n} ∈ Ω}.

Clearly, λ_{n} → 0 as n → ∞ and (ϕ^{λ_{n}}, ψ^{λ_{n}}) solves

ϕ_{s} − d_{1}Δϕ = ϕ(a_{1}λ_{n}^{2} − b_{1}ϕ + c_{1}ψ), y ∈ Ω_{n}, s ∈ I_{n}(T),
ψ_{s} − d_{2}Δψ = ψ(a_{2}λ_{n}^{2} + b_{2}ϕ − c_{2}ψ), y ∈ Ω_{n}, s ∈ I_{n}(T)

and satisfies

ϕ^{λ_{n}}(0, 0) = 1,
0 ≤ ϕ^{λ_{n}} ≤ 1,  y ∈ Ω_{n}, s ∈ (−λ_{n}^{−2}t̂_{n}, 0],
0 ≤ ψ^{λ_{n}} ≤ V (t_{n})U^{−1}(t_{n}), y ∈ Ω_{n}, s ∈ (−λ_{n}^{−2}t̂_{n}, 0].

It follows from the parabolic estimates [11] that there is a μ ∈ (0, 1) such that for any K > 0,

||ϕ^{λ_{n}}||_{C^{2+μ,1+μ/2}(\overline{Ω}_{n}∩|y|≤K×[−K,0])} ≤ C_{K},

||ψ^{λ_{n}}||_{C^{2+μ,1+μ/2}(\overline{Ω}_{n}∩|y|≤K×[−K,0])} ≤ C_{K},

where the constant C_{K} is independent of n. Hence, we obtain a sequence converging to a solution (ϕ, ψ) of

ϕ_{s} − d_{1}Δϕ = ϕ(−b_{1}ϕ + c_{1}ψ), y ∈ R^{N}, s ∈ (−∞, 0],  (3.7)
ψ_{s} − d_{2}Δψ = ψ(b_{2}ϕ − c_{2}ψ),  y ∈ R^{N}, s ∈ (−∞, 0]  (3.8)

such that ϕ(0, 0) = 1 and ϕ ≤ 1, ψ ≡ 0, which leads to a contradiction. In fact, ϕ achieves its maximum at (0, 0); therefore [ϕ_{s} − d_{1}Δϕ](0, 0) ≥ 0, but [ϕ(−b_{1}ϕ + c_{1}ψ)](0, 0) = −b_{1} < 0. This proves (3.1) in Case (i). Case (ii) Choose a subsequence (denoted again by {t_{n}}) such that

lim_{n→∞} \frac{d_{n}}{λ_{n}} = c ≥ 0.

Let x̃_{n} ∈ ∂Ω such that d_{n} = |x̂_{n} − x̃_{n}| and let R_{n} be an orthonormal transformation in R^{N} that maps −e_{1} := (−1, 0, ⋯, 0) onto the outer normal vector to ∂Ω at x̃_{n}. We now introduce the new scaling. Let

ϕ^{λ_{n}}(y, s) := λ_{n}^{2}u(λ_{n}R_{n}y + x̂_{n}, λ_{n}^{2}s + t̂_{n}), (y, s) ∈ \overline{Ω_{n}} × I_{n}(T), (3.9)
ψ^{λ_{n}}(y, s) := λ_{n}^{2}v(λ_{n}R_{n}y + x̂_{n}, λ_{n}^{2}s + t̂_{n}), (y, s) ∈ \overline{Ω_{n}} × I_{n}(T), (3.10)

where

I_{n}(t) := (−λ_{n}^{−2}t̂_{n}, λ_{n}^{−2}(t − t̂_{n})), Ω_{n} := {y : λ_{n}R_{n}y + x̂_{n} ∈ Ω}.

Clearly, λ_{n} → 0 as n → ∞, Ω_{n} approaches the halfspace H_{c} = {y_{1} > −c} as n → ∞ and (ϕ^{λ_{n}}, ψ^{λ_{n}}) solves

ϕ_{s} − d_{1}Δϕ = ϕ(a_{1}λ_{n}^{2} − b_{1}ϕ + c_{1}ψ), y ∈ Ω_{n}, s ∈ I_{n}(T),
ψ_{s} − d_{2}Δψ = ψ(a_{2}λ_{n}^{2} + b_{2}ϕ − c_{2}ψ), y ∈ Ω_{n}, s ∈ I_{n}(T),
ϕ = ψ = 0,  y ∈ ∂Ω_{n}, s ∈ I_{n}(T)

and satisfies

ϕ^{λ_{n}}(0, 0) = 1,
0 ≤ ϕ^{λ_{n}} ≤ 1, y ∈ Ω_{n}, s ∈ (−λ_{n}^{−2}t̂_{n}, 0],
0 ≤ ψ^{λ_{n}} ≤ V (t_{n})U^{−1}(t_{n}), y ∈ Ω_{n}, s ∈ (−λ_{n}^{−2}t̂_{n}, 0].

Noticing that ∂Ω is of C^{2+α}, then uniform Schauder’s estimates for ϕ^{λ_{n}}, ψ^{λ_{n}} yield a subsequence converging to a solution (ϕ, ψ) of

ϕ_{s} − d_{1}Δϕ = ϕ(−b_{1}ϕ + c_{1}ψ), y ∈ H_{c}, s ∈ (−∞, 0],  (3.11)
ψ_{s} − d_{2}Δψ = ψ(b_{2}ϕ − c_{2}ψ),  y ∈ H_{c}, s ∈ (−∞, 0],  (3.12)
ϕ = ψ = 0,  y_{1} = −c, s ∈ (−∞, 0]  (3.13)

such that ϕ(0, 0) = 1 and ϕ ≤ 1, ψ ≡ 0, which leads to a contradiction as in Case (i). This prove (3.1) in Case (ii).

Remark 3.1 We claim from (3.1) that u and v blow up at the same finite time T if (u, v) solves (1.2), that is

lim_{t→T} sup u(x, t) = lim_{t→T} sup v(x, t) = ∞.

Now we first give the lower bound of the blowup rate using the integral equation.

Theorem 3.1 Let (u, v) be the nonnegative solution of (1.2), which blows up at t = T. Then there exists a constant c such that

max_{\overline{Ω}×[0,t]} u(x, τ) ≥ c(T − t)^{−1}, 0 < t < T,

max_{\overline{Ω}×[0,t]} v(x, τ) ≥ c(T − t)^{−1}, 0 < t < T.

Proof: Let G_{i}(x, t; y, τ)(i = 1, 2) be the Green’s function of the parabolic operator (∂/∂t−d_{i}Δ) in the bounded domain Ω×(0, T] under the homogeneous Dirichlet boundary condition on ∂Ω × (0, T]. Then we have the representation formula of (1.2):

u(x, t) = ∫_{Ω} G_{1}(x, t; y, z)u(y, z)dy
+ ∫_{z}^{t} ∫_{Ω} u(a_{1} − b_{1}u + c_{1}v)G_{1}(x, t; y, τ)dydτ,
v(x, t) = ∫_{Ω} G_{2}(x, t; y, z)v(y, z)dy
+ ∫_{z}^{t} ∫_{Ω} v(a_{2} + b_{2}u − c_{2}v)G_{2}(x, t; y, τ)dydτ

for 0 < z < t < T and x ∈ \overline{Ω}.

Noticing that ∫_{Ω} G_{i}(x, t; y, τ)dy ≤ 1 and the relationship (3.1), we have

U(t) ≤ U(z) + ∫_{z}^{t} U(a_{1} + b_{1}U + c_{1}V)(τ)dτ
≤ U(z) + (t − z)U(a_{1} + b_{1}U + c_{1}V)(t)
≤ U(z) + (T − z)U(a_{1} + b_{1}U + c_{1}V)(t)
≤ U(z) + (T − t)U(a_{1} + b_{1}U + \frac{c_{1}}{δ} U)(t),
V (t) ≤ V (z) + (T − t)V (a_{2} + \frac{b_{2}}{δ} V + c_{2}V)(t).

Next we use the argument as in [9]. By assumption, T is the blowup time, so U(t) → +∞ as t → T^{−}. Then we can choose z < t < T such that U(t) = 2U(z), and hence the above inequality for U becomes

2U(z) ≤ U(z) + (T − z)2U(a_{1} + 2b_{1}U + 2\frac{c_{1}}{δ} U)(z),

which implies that

(T − z) ≥ (4a_{1})^{−1} or U(z) ≥ (2b_{1} + 2\frac{c_{1}}{δ})^{−1}(T − z)^{−1}, 0 < z < T.

Take c such that c ≤ (2b_{1} + 2\frac{c_{1}}{δ})^{−1} and c ≤ \frac{1}{4a_{1}} max_{Ω} u_{0}; then

U(t) ≥ c(T − t)^{−1}, 0 < t < T.

The proof for V (t) is similar.

4 Upper blowup estimate

For the upper bound of the blowup rate, we assume that b_{2}c_{1} > b_{1}c_{2} and N = 1. The former assumption b_{2}c_{1} > b_{1}c_{2} is the sufficient condition for the solution of (1.2) to have a finite time blowup, see Theorem 1.1 and the latter N = 1 is restriction for the solution of the related scalar problem to blow up in a finite time, see Lemma 4.3.

Theorem 4.1 Let (u, v) be the nonnegative solution of (1.2), which blows up at t = T. If b_{2}c_{1} > b_{1}c_{2} and N = 1, then there exists a constant C such that

max_{\overline{Ω}×[0,t]} u(x, τ) ≤ C(T − t)^{−1}, 0 < t < T,

max_{\overline{Ω}×[0,t]} v(x, τ) ≤ C(T − t)^{−1}, 0 < t < T.

Proof: From Lemma 3.1 we only need to prove that U(t) ≤ C(T − t)^{−1}. We use a scaling argument inspired by [8]. Noticing that U(t) → ∞ as t → T, for any given t_{0} ∈ (\frac{T}{2} , T) we can define

t_{0}^{+} := t^{+}(t_{0}) := max{t ∈ (t_{0}, T) : U(t) = 2U(t_{0})}.

Choose λ_{0} = λ(t_{0}) = U^{−1/2}(t_{0}) as before. We claim that

λ^{−2}(t_{0})(t_{0}^{+} − t_{0}) ≤ D, t_{0} ∈ (\frac{T}{2} , T),  (4.1)

where the constant D depends only N (it is independent of t_{0} ).

Suppose that (4.1) is not true, then there exists t_{n} → T such that

λ_{n}^{−2}(t_{n})(t_{n}^{+} − t_{n}) → ∞.

For each t_{n}, choose (x̂_{n}, t̂_{n}) as in (3.3) and let d_{n} denote the distant of x̂_{n} to ∂Ω. Similarly as in [4], we distinguish two cases:

(i) lim sup_{n→∞} \frac{d_{n}}{λ_{n}} = ∞ and (ii) lim sup_{n→∞} \frac{d_{n}}{λ_{n}} < ∞.

Case (i) Choose a subsequence (denoted again by {t_{n}}) such that

lim_{n→∞} \frac{d_{n}}{λ_{n}} = ∞.

We introduce the scaling functions as before. Let

λ_{n} := λ(t_{n}) := U^{−1/2}(t_{n}),  (4.2)
ϕ^{λ_{n}}(y, s) := λ_{n}^{2}u(λ_{n}y + x̂_{n}, λ_{n}^{2}s + t̂_{n}), (y, s) ∈ \overline{Ω_{n}} × I_{n}(T), (4.3)
ψ^{λ_{n}}(y, s) := λ_{n}^{2}v(λ_{n}y + x̂_{n}, λ_{n}^{2}s + t̂_{n}), (y, s) ∈ \overline{Ω_{n}} × I_{n}(T), (4.4)

where

I_{n}(t) := (−λ_{n}^{−2}t̂_{n}, λ_{n}^{−2}(t − t̂_{n})), Ω_{n} := {y : λ_{n}y + x̂_{n} ∈ Ω}.

Clearly, (ϕ^{λ_{n}}, ψ^{λ_{n}}) has a sequence converging to a solution (ϕ, ψ) of

ϕ_{s} − d_{1}Δϕ = ϕ(−b_{1}ϕ + c_{1}ψ), y ∈ R^{N}, s ∈ (−∞, ∞),  (4.5)
ψ_{s} − d_{2}Δψ = ψ(b_{2}ϕ − c_{2}ψ),  y ∈ R^{N}, s ∈ (−∞, ∞)  (4.6)

such that ϕ(0, 0) = 1 and ϕ ≤ 1, ψ ≤ \frac{1}{δ}. Moreover, since that ϕ achieves its maximum at (0, 0), ψ must be nontrivial as in Lemma 3.1. Therefore ϕ and ψ are nontrivial nonnegative bounded functions, which leads to a contradiction to the following Theorem 4.2 if N ≤ 2. This prove (4.1) in Case (i).

For the Case (ii), it is easy to show as in Case (i) that there is nontrivial nonnegative solution (ϕ, ψ) of

ϕ_{s} − d_{1}Δϕ = ϕ(−b_{1}ϕ + c_{1}ψ), y ∈ H_{c}, s ∈ (−∞, ∞),  (4.7)
ψ_{s} − d_{2}Δψ = ψ(b_{2}ϕ − c_{2}ψ),  y ∈ H_{c}, s ∈ (−∞, ∞),  (4.8)
ϕ = ψ = 0,  y_{1} = −c, s ∈ (−∞, ∞)  (4.9)

such that ϕ(0, 0) = 1 and ϕ ≤ 1, ψ ≤ \frac{1}{δ}, which leads to a contradiction to Theorem 4.3 if N = 1. This prove (4.1) in Case (ii). Thus (4.1) is established. Step 3 of proof of Theorem 2.1 in [8] shows that (4.1) implies that U(t) ≤ C(T − t)^{−1} for 0 ≤ t < T.

Theorem 4.2 If b_{2}c_{1} > b_{1}c_{2} and N ≤ 2, then any nontrivial nonnegative solution of

\{ \begin{matrix}u_{t} − d_{1}Δu = u(−b_{1}u + c_{1}v), & x ∈ R^{N}, t > 0, \\ v_{t} − d_{2}Δv = v(b_{2}u − c_{2}v), & x ∈ R^{N}, t > 0, \\ u(x, 0) ≥ 0, v(x, 0) ≥ 0, & x ∈ R^{N}, \\ u(x, 0), v(x, 0) ∈ L^{∞}(R^{N}) \end{matrix}  (4.10)

is nonglobal.

To prove Theorem 4.2, it suffices to find a lower solution of (4.10) that blows up at a finite time T_{0}. First we show the following three useful Lemmas:

Lemma 4.1 Any nontrivial nonnegative solution of (4.10) is positive for t > 0.

Proof: If there exist x_{0} ∈ R^{N} and t_{0} > 0 such that u(x_{0}, t_{0}) = 0, then there exist R > 0 and T_{1} with t_{0} < T_{1} < T such that (x_{0}, t_{0}) ∈ B_{R} × (0, T_{1}) and u(x, t) ≢ 0 in \overline{B}_{R} × [0, T_{1}]. Now let B = b_{1} max_{\overline{B}_{R}×[0,T_{1}]} u(x, t) and define the function

w(x, t) = u(x, t)e^{Bt}.

We find from a straightforward computation that

\{ \begin{matrix}w_{t} − d_{1}Δw = w[−b_{1}u + c_{1}v + B] ≥ 0, & x ∈ B_{R}, 0 < t ≤ T_{1}, \\ w(x, 0) ≥ 0, & x ∈ B_{R}. \end{matrix}

It follows form the strong maximum principle that w ≡ 0 in B_{R} × [0, T_{1}] or w > 0 in B_{R} × (0, T]. It leads to a contradiction. So u(x, t) > 0 for t > 0 and also v(x, t) > 0 for t > 0 similarly.

Lemma 4.2 Let w(x, t) be a nontrivial nonnegative solution of

\{ \begin{matrix}dw_{t} − Δw = bw^{2}, & x ∈ R^{N}, t > 0, \\ w(x, 0) ≥ 0, & x ∈ R^{N}, \\ w(x, 0) ∈ L^{∞}(R^{N}). \end{matrix}  (4.11)

(i) If Δw(x, 0) + bw^{2}(x, 0) ≥ 0, then w_{t}(x, t) ≥ 0 in R^{n} × (0, T); (ii) If w(x, 0) is radially symmetric, then w(x, t) is radial. Moreover, if

\frac{∂w(r,0)}{∂r} ≤ 0 for r ≥ 0, then \frac{∂w(r,t)}{∂r} ≤ 0 for r ≥ 0, t > 0, where r = \sqrt{x_{1}^{2} + x_{2}^{2} + ⋯ + x_{N}^{2}}.

Proof: Since w satisfy the growth condition, using the comparison principle (see Lemma 2.2 for the system) and the assumptions on w(x, 0) yield w(x, t) ≥ w(x, 0) in R^{N} × (0, T). Using again the comparison principle gives that w(x, t + ε) ≥ w(x, t) in R^{N} × (0, T − ε) for ε > 0 arbitrarily small. Hence w_{t}(x, t) ≥ 0 in R^{N} × (0, T).

The result that the solution is radial follows by the uniqueness and the rotation invariance of problem (4.11) in the case that w(x, 0) is radial. Furthermore, if the initial data w(x, 0) is radially nonincreasing, then the solution w(x, t) is also radially nonincreasing.

Lemma 4.3 All nontrivial nonnegative solutions of

\{ \begin{matrix}dw_{t} − Δw = bw^{2}, & x ∈ R^{N}, t > 0, \\ w(x, 0) ≥ 0, & x ∈ R^{N} \end{matrix}  (4.12)

are nonglobal if N ≤ 2; all nontrivial nonnegative solutions of

\{ \begin{matrix}dw_{t} − Δw = bw^{2}, & x ∈ H_{c}, t > 0, \\ w(x, t) = 0, & x_{1} = 0, t > 0, \\ w(x, 0) ≥ 0, & x ∈ H_{c} \end{matrix}  (4.13)

are nonglobal if N = 1, where H_{c} := {x_{1} > −c}.

The former blowup result is followed from the well-known result of the general case w_{t} − Δw = w^{p} shown in [6] for 1 < p < 1 + \frac{2}{N} and [7] for p = 1 + \frac{2}{N}, the latter is followed from the result of the general case w_{t} − Δw = w^{p} shown in [13] for 1 < p ≤ 1 + \frac{2}{N+1}.

Proof of Theorem 4.2 We look for a lower solution (\underline{u}, \underline{v}) of (4.10) such that (\underline{u}, \underline{v}) blow up in finite time. Let (\underline{u}, \underline{v}) = (δ_{1}w, δ_{2}w), where δ_{1} and δ_{2} are some positive constants to be chosen later and w is a nonnegative function in \overline{Ω} × (0, T_{0}) and unbounded in Ω at some T_{0} < +∞. From Lemma 2.2, (\underline{u}, \underline{v}) is a lower solution of (4.10) in \overline{Ω} × [0, T_{0}) if

w_{t} − d_{1}Δw ≤ w(−b_{1}δ_{1}w + c_{1}δ_{2}w),  R^{N} × (0, T_{0}),  (4.14)
w_{t} − d_{2}Δw ≤ w(b_{2}δ_{2}w − c_{2}δ_{2}w),  R^{N} × (0, T_{0}),  (4.15)
δ_{1}w(x, 0) ≤ u(x, 0), δ_{2}w(x, 0) ≤ v(x, 0), x ∈ R^{N}.  (4.16)

Since b_{2}c_{1} > b_{1}c_{2}, choose δ_{1}, δ_{2} as in [20] such that c_{2}/b_{2} < δ_{1}/δ_{2} < c_{1}/b_{1} and set

d = max{d_{1}^{−1}, d_{2}^{−1}},
b = min{(c_{1}δ_{2} − b_{1}δ_{1})/d_{1}, (b_{2}δ_{1} − c_{2}δ_{2})/d_{2}}.

Then d, b > 0 and (4.14), (4.15) hold if

d_{1}^{−1}w_{t} − Δw ≤ bw^{2},
d_{2}^{−1}w_{t} − Δw ≤ bw^{2}.

By choosing w as the solution of the scalar problem

dw_{t} − Δw = bw^{2},  (4.17)

(4.14), (4.15) hold provided that w_{t} ≥ 0.

Now for arbitrary nontrivial nonnegative solution (u, v) of (4.10), by Lemma 4.1, the solution is positive for t > 0. Without loss of generality, we may assume that u(x, 0) > 0 and v(x, 0) > 0 for x ∈ R^{N}, otherwise replace the initial function (u(x, 0), v(x, 0)) by (u(x, t_{1}), v(x, t_{1})) for t_{1} > 0. Since the initial data is positive, there exists a radially symmetric, radially nondecreasing function ψ(x) such that

δ_{1}ψ(x) ≤ u(x, 0), δ_{2}ψ(x) ≤ v(x, 0), x ∈ R^{N},
Δψ(x) + bψ^{2}(x) ≥ 0,  x ∈ R^{N}

and define w^{∗} be the solution of (4.17) when w(x, 0) = ψ(x). By Lemma 4.2, w^{∗} is monotone nondecreasing in t. Moreover, w^{∗} is radially symmetric, radially nondecreasing and therefore satisfies the growth condition. It follows from comparison principle Lemma 2.2 that u(x, t) ≥ δ_{1}w^{∗}(x, t) and v(x, t) ≥ δ_{2}w^{∗}(x, t) in R^{N} × [0, T_{0}). Hence (\underline{u}, \underline{v}) = (δ_{1}w^{∗}, δ_{2}w^{∗}) is a lower solution of (4.10).

On the other hand, Lemma 4.3 ensures the existence of a finite T_{0} such that the solution w^{∗} exists in R^{N} × [0, T_{0}) and is unbounded in R^{N} as t → T_{0} if N ≤ 2. Thus the solution of (4.10) cannot exist beyond T_{0} and is nonglobal.

Theorem 4.3 If b_{2}c_{1} > b_{1}c_{2} and N = 1, then any nontrivial nonnegative solution of

\{ \begin{matrix}u_{t} − d_{1}Δu = u(−b_{1}u + c_{1}v), & x ∈ H_{c}, t > 0, \\ v_{t} − d_{2}Δv = v(b_{2}u − c_{2}v), & x ∈ H_{c}, t > 0, \\ u(x, t) = 0, v(x, t) = 0, & x_{1} = −c, t > 0, \\ u(x, 0) ≥ 0, v(x, 0) ≥ 0, & x ∈ H_{c}, \\ u(x, 0), v(x, 0) ∈ L^{∞}(H_{c}) \end{matrix}  (4.18)

is nonglobal.

Proof: The proof of Theorem 4.3 is similar to that of Theorem 4.2. The only difference is that in the proof of Theorem 4.2 the related scalar problem (4.12) is nonglobal if N ≤ 2 and in the proof of Theorem 4.3, the related scalar problem (4.13) is nonglobal if N = 1, see Lemma 4.3.

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