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Blow-up for an evolution p-laplace system with nonlocal sources and inner absorptions

Yan Zhang1, Dengming Liu2*, Chunlai Mu2 and Pan Zheng2

Author affiliations

1 School of Mathematics and Computer Engineering, Xihua University, Chengdu, Sichuan 610039, PR China

2 College of Mathematics and Statistics, Chongqing University, Chongqing 410031, PR China

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Citation and License

Boundary Value Problems 2011, 2011:29  doi:10.1186/1687-2770-2011-29

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/29


Received:10 June 2011
Accepted:6 October 2011
Published:6 October 2011

© 2011 Zhang et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the blow-up properties of positive solutions to the following system of evolution p-Laplace equations with nonlocal sources and inner absorptions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M1">View MathML</a>

with homogeneous Dirichlet boundary conditions in a smooth bounded domain Ω ∈ RN(N ≥ 1), where p, q > 2, m, n, r, s ≥ 1, α, β > 0. Under appropriate hypotheses, the authors discuss the global existence and blow-up of positive weak solutions by using a comparison principle.

2010 Mathematics Subject Classification: 35B35; 35K60; 35K65; 35K57.

Keywords:
evolution p-Laplace system; global existence; blow-up; nonlocal sources; absorptions

1 Introduction

In this paper, we deal with the blow-up properties of positive solutions to an evolution p-Laplace system of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M2">View MathML</a>

(1.1)

where p, q > 2, m, n, r, s ≥ 1, α, β > 0, Ω is a bounded domain in RN(N ≥ 1) with a smooth boundary ∂Ω, the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M6">View MathML</a>, where v denotes the unit outer normal vector on ∂Ω.

System (1.1) is the classical reaction-diffusion system of Fujita-type for p = q = 2. If p ≠ 2, q ≠ 2, (1.1) appears in the theory of non-Newtonian fluids [1,2] and in nonlinear filtration theory [3]. In the non-Newtonian fluids theory, the pair (p, q) is a characteristic quantity of the medium. Media with (p, q) > (2, 2) are called dilatant fluids and those with (p, q) < (2, 2) are called pseudoplastics. If (p, q) = (2, 2), they are Newtonian fluids.

System (1.1) has been studied by many authors. For p = q = 2, Escobedo and Herrero [4] considered the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M7">View MathML</a>

(1.2)

where p, q > 0. Their main results read as follows. (i) If pq ≤ 1, every solution of (1.2) is global in time. (ii) If pq > 1, some solutions are global while some others blow up in finite time.

In the last three decades, many authors studied the following degenerate parabolic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M8">View MathML</a>

(1.3)

under different conditions (see [5,6] for nonlinear boundary conditions; see [7-10] for local nonlinear reaction terms; see [11] for nonlocal nonlinear reaction terms). In [12], the existence, uniqueness, and regularity of solutions were obtained. When f(u) = -uq, q > 0 or f(u) ≡ 0 extinction phenomenon of the solution may appear [13-15]; However, if f(u) = uq, q > 1 the solution may blow up in finite time [7-10,14].

Especially, in [11], Li and Xie dealt with the following p-Laplace equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M9">View MathML</a>

(1.4)

Under appropriate hypotheses, they established the local existence and uniqueness of its solution. Furthermore, they obtained that the solution u exists globally if q < p - 1; u blows up in finite time if q > p - 1 and u0(x) is large enough.

Recently, in [16], Li generalized (1.4) to system and studied the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M10">View MathML</a>

(1.5)

Similar to [11], he proved that whether the solution blows up in finite time depends on the initial data, constants α, β, and the relations between mn and (p - 1)(q - 1).

For other works on parabolic system like (1.1), we refer readers to [17-30] and the references therein.

When p = q, m = n, r = s, α = β, u0(x) = v0(x), system (1.1) is then reduced to a single p-Laplace equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M11">View MathML</a>

(1.6)

However, to the authors' best knowledge, there is little literature on the study of the global existence and blow-up properties for problems (1.1) and (1.6). Motivated by the above works, in this paper, we investigate the blow-up properties of solutions of the problem (1.1) and extend the results of [4,11,16,19] to more generalized cases.

In order to state our results, we introduce some useful symbols. Throughout this paper, we let φ(x), ψ(x) be the unique solution of the following elliptic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M12">View MathML</a>

(1.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M13">View MathML</a>

(1.8)

respectively. For convenience, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M14">View MathML</a>

Before starting the main results, we introduce a pair of parameters (μ, γ) solving the following characteristic algebraic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M15">View MathML</a>

namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M16">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M17">View MathML</a>

It is obvious that 1/τ and 1/θ share the same signs. We claim that the critical exponent of problem (1.1) should be (1/τ, 1/θ) = (0, 0), described by the following theorems.

Theorem 1.1. Assume that (1/τ, 1/θ) < (0, 0), then there exist solutions of (1.1) being globally bounded.

Theorem 1.2. Assume that (1/τ, 1/θ) > (0, 0), then the nonnegative solution of (1.1) blows up in finite time for sufficiently large initial values and exists globally for sufficiently small initial values.

Theorem 1.3. Assume that (1/τ, 1/θ) = (0, 0), φ(x) and ψ(x) are defined in (1.7) and (1.8), respectively.

(i) Suppose that r > p - 1 and s > q - 1. If αnβr ≥ |Ω|n+r, then the solutions are globally bounded for small initial data; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M19">View MathML</a>, then the solutions blow up in finite time for large data.

(ii) Suppose that p - 1 > r and q - 1 > s. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M20">View MathML</a>, then the solutions are globally bounded for small initial data; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M22">View MathML</a>then the solutions blow up in finite time for large data.

(iii) Suppose that p - 1 > r and s > q - 1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M23">View MathML</a>, then the solutions are globally bounded for small initial data; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M19">View MathML</a>, then the solutions blow up in finite time for large data.

(iv) Suppose that r > p - 1 and q - 1 > s. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M24">View MathML</a> , then the solutions are globally bounded for small initial data; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M18">View MathML</a>, then the solutions blow up in finite time for sufficiently large data.

The rest of this paper is organized as follows. In Section 2, we shall establish the comparison principle and local existence theorem for problem (1.1). Theorems 1.1 and 1.2 will be proved in Section 3 and Section 4, respectively. Finally, we will give the proof of Theorem 1.3 in Section 5.

2 Preliminaries

Since the equations in (1.1) are degenerate at points where ∇u = 0 or ∇v = 0, there is no classical solution in general, and we therefore consider its weak solutions. Let ΩT = Ω × (0, T), ST = ∂Ω × (0, T) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M26">View MathML</a>. We begin with the precise definition of a weak solution of problem (1.1).

Definition 2.1 A pair of functions (u(x, t), v(x, t)) is called a weak solution of problem (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27">View MathML</a> if and only if

(i) (u, v) is in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M28">View MathML</a> and (ut, vt) ∈ L2(0, T; L2(Ω)) × L2(0, T; L2(Ω)).

(ii) the following equalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M29">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M30">View MathML</a>

hold for all ϕ1, ϕ2, which belong to the class of test functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M31">View MathML</a>

(iii) u(x, t)|t = 0 = u0(x), v(x, t)|t = 0 = v0(x) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M32">View MathML</a>.

In a natural way, the notion of a weak subsolution for (1.1) is given as follows.

Definition 2.2 A pair of functions (

    u
(x, t),
    v
(x, t)) is called a weak subsolution of problem (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27">View MathML</a> if and only if

(i) (

    u
,
    v
) is in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M28">View MathML</a> and (
    u
t
,
    v
t
) ∈ L2(0, T; L2(Ω)) × L2(0, T; L2(Ω)).

(ii) the following inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M33">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M34">View MathML</a>

hold for any ϕ1, ϕ2, which belong to the class of test functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M35">View MathML</a>

(iii)

    u
(x, t)|t = 0 u0(x),
    v
(x, t)|t = 0 v0(x) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M32">View MathML</a>.

Similarly, a pair of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M36">View MathML</a> is a weak supersolution of (1.1) if the reversed inequalities hold in Definition 2.2. A weak solution of (1.1) is both a weak subsolution and a weak supersolution of (1.1).

We shall use the following comparison principle to prove our global and nonglobal existence results.

Proposition 2.3 Let (

    u
,
    v
) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M37">View MathML</a> be a nonnegative subsolution and supersolution of (1.1), respectively, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M38">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M32">View MathML</a>. Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M39">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27">View MathML</a>.

Proof. From the definitions of weak subsolution and supersolution, for any ϕ1, ϕ2 ∈ Θ2, we could obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M40">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M41">View MathML</a>

(2.2)

In addition, inequalities (2.1) and (2.2) remain true for any subcylinder of the form Ωτ = Ω × (0, τ) ⊂ ΩT and corresponding lateral boundary Sτ = ∂Ω × (0, τ) ⊂ ST. Taking a special test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M42">View MathML</a> in (2.1), where χ[0, τ] is the characteristic function defined on [0, τ] and s+ = max{s, 0}, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M43">View MathML</a>

(2.3)

where |Ω| denotes the Lebesgue measure of Ω and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M44">View MathML</a>

Next, our task is to estimate the first term on the right-side of (2.3). In view of Cauchy's inequality, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M45">View MathML</a>

(2.4)

Furthermore, by Lemma 1.4.4 in [12], we know that there exists δ > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M46">View MathML</a>

(2.5)

Combining now (2.3)-(2.5), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M47">View MathML</a>

(2.6)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M49">View MathML</a>.

Likewise, taking test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M50">View MathML</a> in (2.2), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M51">View MathML</a>

(2.7)

where C3, C4 denote some positive constants. Moreover, there exists a large enough constant C, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M52">View MathML</a>

(2.8)

Now, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M53">View MathML</a>

then, (2.8) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M54">View MathML</a>

(2.9)

By Gronwall's inequality, we know that y(τ) = 0, for any τ ∈ [0, T]. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M55">View MathML</a>, this means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M57">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M58">View MathML</a> as desired. The proof of Proposition 2.3 is complete. □

With the above established comparison principle in hand, we are able to show the basic existence theorem of weak solutions. Here, we only state the local existence theorem, and its proof is standard [12, 16, for more details].

Theorem 2.1 Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M59">View MathML</a>, there is some T0 > 0 such that the problem (1.1) admits a nonnegative unique weak solution (u, v) for each t < T0, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M60">View MathML</a>. Furthermore, either T0 = ∞ or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M61">View MathML</a>

3 Proof of Theorem 1.1

Proof of Theorem 1.1. Notice that (1/τ, 1/θ) < (0, 0) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M62">View MathML</a>

We will prove Theorem 1.1 in four subcases.

(a) For μ = r, γ = s, we then have mn < rs. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M63">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65">View MathML</a> will be determined later. After a simple computation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M66">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M67">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M68">View MathML</a> is a time-independent supersolution of problem (1.1) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M69">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M70">View MathML</a>

(3.1)

(b) For μ = p - 1, γ = q - 1, we then have mn < (p - 1)(q - 1). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M71">View MathML</a>

where φ, ψ satisfying (1.7) and (1.8), respectively. Taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M72">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M73">View MathML</a>

then it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M68">View MathML</a> is a global supersolution for system (1.1).

(c) For μ = r, γ = q - 1, we then have mn < r(q - 1). Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M74">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M75">View MathML</a> with ψ defined by (1.8). By direct Computation, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M76">View MathML</a>

(3.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M77">View MathML</a>

(3.3)

(d) For μ = p - 1, γ = s, we then have mn < r(q - 1). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M78">View MathML</a> with φ defined by (1.7), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65">View MathML</a>. Then, (3.2) and (3.3) hold if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M79">View MathML</a>

The proof of Theorem 1.1 is complete. □

4 Proof of Theorem 1.2

Proof of Theorem 1.2. Observe that 1/τ, 1/θ > 0 implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M80">View MathML</a>

For μ = r, γ = s. Choosing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M81">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M63">View MathML</a> is a global supersolution for problem (1.1) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65">View MathML</a>.

For μ = p - 1, γ = q - 1. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M82">View MathML</a>, where φ and ψ satisfying (1.7) and (1.8), respectively. Choosing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M83">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M84">View MathML</a>

therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M68">View MathML</a> is a global supersolution for system (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65">View MathML</a>.

For other cases, the solutions of (1.1) should be global due to the above discussion.

Next, we begin to prove our blow-up conclusion under large enough initial data. Due to the requirement of the comparison principle, we will construct blow-up subsolutions in some subdomain of Ω in which u, v > 0. We use an idea from Souplet [31] and apply it to degenerate equations. Since problem (1.1) does not make sense for negative values of (u, v), we actually consider the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M85">View MathML</a>

(4.1)

where u+ = max{0, u}, v+ = max{0, v}. Let ϖ(x) be a nontrivial nonnegative continuous function and vanish on ∂Ω. Without loss of generality, we may assume that 0 ∈ Ω and ϖ(0) > 0. We shall construct a self-similar blow-up subsolution to complete our proof.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M86">View MathML</a>

(4.2)

here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M87">View MathML</a>

and li, σi > 0(i = 1, 2), 0 < T < 1 are to be determined later. Notice the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M88">View MathML</a>

(4.3)

for sufficiently small T > 0.

Calculating directly, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M89">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M90">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M91">View MathML</a>

On the other hand, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M92">View MathML</a>

(4.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M93">View MathML</a>

(4.5)

here Hx(

    u
), Hx(
    v
) denotes the Hessian matrix of
    u
(x, t),
    v
(x, t) respect to x, respectively. Use the notation d(Ω) = diam(Ω), then from (4.4) and (4.5), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M94">View MathML</a>

Further, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M95">View MathML</a>

(4.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M96">View MathML</a>

(4.7)

Since 1/τ, 1/θ < 0, we see that μγ < mn. In addition, it is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M97">View MathML</a>

(4.8)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M98">View MathML</a>, we choose l1 and l2 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M99">View MathML</a>

(4.9)

Recall that μ = max{p - 1, r} and γ = max{q - 1, s}, then (4.9) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M100">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M101">View MathML</a>

Next, we can choose positive constants σ1, σ2 sufficiently small such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M102">View MathML</a>

consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M103">View MathML</a>

(4.10)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M104">View MathML</a>, we fix l1 and l2 to satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M105">View MathML</a>

(4.11)

then we can also select σ1, σ2 small enough such that (4.10) holds.

From (4.6), (4.7) and (4.10), for sufficiently small T > 0, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M106">View MathML</a>

(4.12)

Since ϖ(0) > 0 and ϖ(x) are continuous, there exist two positive constants ρ and ε such that ϖ(x) ≥ ε for all x B(0, ρ) ⊂ Ω. Choose T small enough to insure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M107">View MathML</a>, hence

    u
≤ 0,
    v
≤ 0 on ST. From (4.1) and (4.2), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M109">View MathML</a> for sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M110">View MathML</a>. By comparison principle, we have (
    u
,
    v
) ≤ (u, v) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M112">View MathML</a>. It shows that (u, v) blows up in finite time. The proof of Theorem 1.2 is complete. □

5 Proof of Theorem 1.3

Proof of Theorem 1.3. In the critical case of (1/τ, 1/θ) = (0, 0), we have mn = μγ.

(i) For r > p - 1, s > q - 1, we know mn = rs. Thanks to αnβr ≥ |Ω|n+r, we can choose A and B sufficiently large such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M65">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M113">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M63">View MathML</a> is a supersolution of problem (1.1), then by comparison principle, the solution of (1.1) should be global.

Next, we begin to prove our blow-up conclusion. Since mn = rs, we can choose constants l1, l2 > 1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M114">View MathML</a>

(5.1)

According to Proposition 2.3, we only need to construct a suitable blow-up subsolution of problem (1.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27">View MathML</a>. Let y(t) be the solution of the following ordinary differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M115">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M116">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M19">View MathML</a>, we have c1 > 0. On the other hand, by virtue of (5.1), it is easy to see that δ1 > δ2. Then, it is obvious that there exists a constant 0 < T' < +∞ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M117">View MathML</a>

Construct

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M118">View MathML</a>

where φ, ψ satisfying (1.7) and (1.8), respectively. Moreover, by the assumptions on initial data, we can take small enough constant y0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M119">View MathML</a>

(5.2)

Now, we begin to verify that (

    u
(x, t),
    v
(x, t)) is a blow-up subsolution of the problem (1.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27">View MathML</a>, T < T'. In fact, ∀(x, t) ∈ ΩT × (0, T), a series of computations show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M120">View MathML</a>

(5.3)

Similarly, we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M121">View MathML</a>

(5.4)

On the other hand, ∀t ∈ [0, T], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M122">View MathML</a>

(5.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M123">View MathML</a>

(5.6)

Combining now (5.2)-(5.6), we see that (

    u
,
    v
) is a subsolution of (1.1) and (
    u
,
    v
) < (u, v) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27">View MathML</a> by comparison principle, thus (u, v) must blow up in finite time since (
    u
,
    v
) does.

(ii) For p - 1 > r, q - 1 > s, we know mn = (p - 1)(q - 1). Under the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M124">View MathML</a>, we can choose A, B such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M125">View MathML</a>

Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M126">View MathML</a> is a global supersolution of (1.1).

Since mn = (p - 1)(q - 1), we can choose constants l1, l2 > 1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M127">View MathML</a>

(5.7)

Next, we consider the following ordinary differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M128">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M129">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M25">View MathML</a>, we have c1 > 0. On the other hand, in light of (5.7), it is easy to show that δ1 > δ2. Then, it is clear that y(t) will become infinite in a finite time T' < +∞.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M130">View MathML</a>

where φ(x), ψ(x) satisfies (1.7) and (1.8), respectively. Similar to the arguments for the case r > p - 1, s > q - 1, we can prove that (

    u
(x, t),
    v
(x, t)) is a blow-up subsolution of the problem (1.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M27">View MathML</a>, T < T'. Then, the solution (u, v) of (1.1) blows up in finite time.

(iii) For p - 1> r, s > q - 1, we know mn = s(p - 1). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M23">View MathML</a>, we can choose A, B such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M131">View MathML</a>

We can check <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M132">View MathML</a> is a global supersolution of (1.1).

Thanks to mn = s(p - 1), we can choose constants l1, l2 > 1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M133">View MathML</a>

(5.8)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M134">View MathML</a>

where φ(x), ψ(x) are defined in (1.7) and (1.8), respectively, and y(t) satisfies the following Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M135">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/29/mathml/M136">View MathML</a>

Then, the left arguments are the same as those for the case r > p - 1, s > q - 1, so we omit them.

(iv) The proof of this case is parallel to (iii). The proof of Theorem 1.3 is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The authors are very grateful to the anonymous referees and the editor for their careful reading and useful suggestions, which greatly improved the presentation of the paper. Dengming Liu is supported by the Fundamental Research Funds for the Central Universities (Project No. CDJXS 11 10 00 19). Chunlai Mu is supported in part by NSF of China (Project No. 10771226) and in part by Natural Science Foundation Project of CQ CSTC (Project No. 2007BB0124).

References

  1. Astrita, G, Marrucci, G: Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, New York, NY (1974)

  2. Martinson, LK, Pavlov, KB: Unsteady shear flows of a conducting fluid with a rheological power law. Magnitnaya Gidrodinamika. 7, 50–58 (1971)

  3. Esteban, JR, Vázquez, JL: On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Anal. 10, 1303–1325 (1986). Publisher Full Text OpenURL

  4. Escobedo, M, Herrero, MA: A semilinear parabolic system in a bounded domain. Ann Mat Pura Appl. IV CLXV, 315–336 (1993)

  5. Galaktionov, VA, Levine, HA: On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary. Israel J Math. 94, 125–146 (1996). Publisher Full Text OpenURL

  6. Zhou, J, Mu, CL: On critical Fujita exponent for degenerate parabolic system coupled via nonlinear boundary flux. Proc Edinb Math Soc. 51, 785–805 (2008). Publisher Full Text OpenURL

  7. Ishii, H: Asymptotic stability and blowing up of solutions of some nonlinear equations. J Differ Equ. 26, 291–319 (1997)

  8. Levine, HA, Payne, LE: Nonexistence theorems for the heat equation with nonlinear boundary conditions for the porous medium equation backward in time. J Differ Equ. 16, 319–334 (1974). Publisher Full Text OpenURL

  9. Tsutsumi, M: Existence and nonexistence of global solutions for nonlinear parabolic equations. Publ Res Inst Math Sci. 8, 221–229 (1972)

  10. Zhao, JN: Existence and nonexistence of solutions for ut - div(|∇u|p-2u) = f(∇u, u, x, t). J Math Anal Appl. 172, 130–146 (1993). Publisher Full Text OpenURL

  11. Li, FC, Xie, HC: Global and blow-up of solutions to a p-Laplace equation with nonlocal source. Comput Math Appl. 46, 1525–1533 (2003). Publisher Full Text OpenURL

  12. Dibenedetto, E: Degenerate Parabolic Equations. Springer, Berlin (1993)

  13. Tsutsumi, M: On solutions of some doubly nonlinear degenerate parabolic equations with absorption. J Math Anal Appl. 132, 187–212 (1988). Publisher Full Text OpenURL

  14. Yin, JX, Jin, CH: Critical extinction and blow-up exponents for fast diffusion p-Laplace with sources. Math Methods Appl Sci. 30, 1147–1167 (2007). Publisher Full Text OpenURL

  15. Yuan, HJ: Extinction and positivity of the evolution p-Laplacian equation. J Math Anal Appl. 196, 754–763 (1995). Publisher Full Text OpenURL

  16. Li, FC: Global existence and blow-up of solutions to a nonlocal quasilinear degenerate parabolic system. Nonlinear Anal. 67, 1387–1402 (2007). Publisher Full Text OpenURL

  17. Bedjaoui, N, Souplet, P: Critical blow-up exponents for a system of reaction-diffusion equations with absorption. Z Angew Math Phys. 53, 197–210 (2002). Publisher Full Text OpenURL

  18. Chen, YP: Blow-up for a system of heat equations with nonlocal sources and absorptions. Comput Math Appl. 48, 361–372 (2004). Publisher Full Text OpenURL

  19. Cui, ZJ, Yang, ZD: Global existence and blow-up solutions and blow-up estimates for some evolution systems with p-Laplacian with nonlocal sources. Int J Math Math Sci. 2007, 17 (2007) (Article ID 34301)

  20. Galaktionov, VA, Kurdyumov, SP, Samarskii, AA: A parabolic system of quasilinear equations I. Differ Equ. 19, 1558–1571 (1983)

  21. Galaktionov, VA, Kurdyumov, SP, Samarskii, AA: A parabolic system of quasilinear equations II. Differ Equ. 21, 1049–1062 (1985)

  22. Li, FC, Huang, SX, Xie, HC: Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete Contin Dyn Syst. 9, 1519–1532 (2003)

  23. Wu, XS, Gao, WJ: Global existence and blow-up of solutions to an evolution p-Laplace system coupled via nonlocal sources. J Math Anal Appl. 358, 229–237 (2009). Publisher Full Text OpenURL

  24. Yang, ZD, Lu, QS: Blow-up estimates for a quasilinear reaction-diffusion system. Math Method Appl Sci. 26, 1005–1023 (2003). Publisher Full Text OpenURL

  25. Zhang, R, Yang, ZD: Global existence and blow-up solutions and blow-up estimates for a non-local quasilinear degenerate parabolic system. Appl Math Comput. 200, 267–282 (2008). Publisher Full Text OpenURL

  26. Zheng, SN: Global existence and global non-existence of solution to a reaction-diffusion system. Nonlinear Anal. 39, 327–340 (2000). Publisher Full Text OpenURL

  27. Zheng, SN, Su, H: A quasilinear reaction-diffusion system coupled via nonlocal sources. Appl Math Comput. 180, 295–308 (2006). Publisher Full Text OpenURL

  28. Zhou, J, Mu, CL: Blow-up for a non-Newton polytropic filtration system with nonlinear nonlocal source. Commun Korean Math Soc. 23, 529–540 (2008). Publisher Full Text OpenURL

  29. Zhou, J, Mu, CL: Global existence and blow-up for non-Newton polytropic filtration system with nonlocal source. Glasgow Math J. 51, 39–47 (2009). Publisher Full Text OpenURL

  30. Zhou, J, Mu, CL: Global existence and blow-up for non-Newton polytropic filtration system with nonlocal source. ANZIAM J. 50, 13–29 (2008). Publisher Full Text OpenURL

  31. Souplet, P: Blow-up in nonlocal reaction-diffusion equations. SIAM J Math Anal. 29, 1301–1334 (1998). Publisher Full Text OpenURL