Open Access

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

Boundary Value Problems20112011:29

DOI: 10.1186/1687-2770-2011-29

Received: 10 June 2011

Accepted: 6 October 2011

Published: 6 October 2011

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

{ u t div ( | u | p 2 u ) = Ω v m d x α u r , x Ω , t > 0, v t div ( | v | q 2 v ) = Ω u n d x β v s , x Ω , t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equa_HTML.gif

with homogeneous Dirichlet boundary conditions in a smooth bounded domain Ω R N (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
{ u t div ( | u | p 2 u ) = Ω v m d x α u r , x Ω , t > 0, v t div ( | v | q 2 v ) = Ω u n d x β v s , x Ω , t > 0, u ( x , t ) = v ( x , t ) = 0, x Ω , t > 0, u ( x ,0 ) = u 0 ( x ), v ( x ,0 ) = v 0 ( x ), x Ω ¯ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ1_HTML.gif
(1.1)

where p, q > 2, m, n, r, s ≥ 1, α, β > 0, Ω is a bounded domain in R N (N ≥ 1) with a smooth boundary ∂Ω, the initial data u 0 ( x ) C ( Ω ¯ ) W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq1_HTML.gif, v 0 ( x ) C ( Ω ¯ ) W 0 1 , q ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq2_HTML.gif and u 0 ( x ) ν < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq3_HTML.gif, v 0 ( x ) ν < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq4_HTML.gif, 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
u t = Δ u + v p , v t = Δ v + u q , x Ω , t > 0 , u ( x , t ) = v ( x , t ) = 0 , x Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x Ω ¯ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ2_HTML.gif
(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
{ u t div ( | u | p 2 u ) = f ( u ), x Ω , t > 0, u ( x , t ) = 0, x Ω , t > 0 u ( x ,0 ) = u 0 ( x ), x Ω ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ3_HTML.gif
(1.3)

under different conditions (see [5, 6] for nonlinear boundary conditions; see [710] 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) = -u q , q > 0 or f(u) ≡ 0 extinction phenomenon of the solution may appear [1315]; However, if f(u) = u q , q > 1 the solution may blow up in finite time [710, 14].

Especially, in [11], Li and Xie dealt with the following p-Laplace equation
{ u t div ( | u | p 2 u ) = Ω u q ( x , t ) d x , x Ω , t > 0, u ( x , t ) = 0, x Ω , t > 0 u ( x ,0 ) = u 0 ( x ), x Ω ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ4_HTML.gif
(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
{ u t div ( | u | p 2 u ) = α Ω v m d x , x Ω , t > 0, v t div ( | v | q 2 v ) = β Ω u n d x , x Ω , t > 0, u ( x , t ) = v ( x , t ) = 0, x Ω , t > 0, u ( x ,0 ) = u 0 ( x ), v ( x ,0 ) = v 0 ( x ), x Ω ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ5_HTML.gif
(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 [1730] 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
u t div ( | u | p 2 u ) = Ω u m d x α u r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ6_HTML.gif
(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
div ( | φ | p 2 φ ) = 1, x Ω ; φ ( x ) = 0, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ7_HTML.gif
(1.7)
and
div ( | ψ | q 2 ψ ) = 1, x Ω ; ψ ( x ) = 0, x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ8_HTML.gif
(1.8)
respectively. For convenience, we denote
m 1 = min Ω ¯ φ ( x ) , M 1 = max Ω ¯ φ ( x ) , m 2 = min Ω ¯ ψ ( x ) , M 2 = max Ω ¯ ψ ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equb_HTML.gif
Before starting the main results, we introduce a pair of parameters (μ, γ) solving the following characteristic algebraic system
( - μ m n - γ ) ( τ θ ) = ( 1 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equc_HTML.gif
namely,
τ = m + γ m n - μ γ , θ = n + μ m n - μ γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equd_HTML.gif
with
μ = max { p - 1 , r } , γ = max { q - 1 , s } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Eque_HTML.gif

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.
  1. (i)

    Suppose that r > p - 1 and s > q - 1. If α n β r ≥ |Ω|n+r, then the solutions are globally bounded for small initial data; if Ω ψ m d x > α φ r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq5_HTML.gif, Ω φ n d x > β ψ s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq6_HTML.gif, then the solutions blow up in finite time for large data.

     
  2. (ii)

    Suppose that p - 1 > r and q - 1 > s. If ( Ω φ n d x ) 1 q - 1 ( Ω ψ m d x ) 1 m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq7_HTML.gif, then the solutions are globally bounded for small initial data; if Ω ψ m d x > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq8_HTML.gif, Ω φ n d x > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq9_HTML.gif then the solutions blow up in finite time for large data.

     
  3. (iii)

    Suppose that p - 1 > r and s > q - 1. If Ω φ n d x Ω - 1 m β 1 s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq10_HTML.gif, then the solutions are globally bounded for small initial data; if Ω ψ m d x > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq8_HTML.gif, Ω φ n d x > β ψ s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq6_HTML.gif, then the solutions blow up in finite time for large data.

     
  4. (iv)

    Suppose that r > p - 1 and q - 1 > s. If Ω ψ m d x Ω - 1 n α 1 r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq11_HTML.gif , then the solutions are globally bounded for small initial data; if Ω φ n d x > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq12_HTML.gif, Ω ψ m d x > α φ r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq5_HTML.gif, 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 Ω T = Ω × (0, T), S T = ∂Ω × (0, T) and Ω ¯ T = Ω ¯ × [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq13_HTML.gif. 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 Ω ¯ T × Ω ¯ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq14_HTML.gif if and only if
  1. (i)

    (u, v) is in the space ( C ( 0 , T ; L ( Ω ) ) L p ( 0 , T ; W 0 1 , p ( Ω ) ) ) × ( C ( 0 , T ; L ( Ω ) ) L q ( 0 , T ; W 0 1 , q ( Ω ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq15_HTML.gif and (u t , v t ) L 2(0, T; L 2(Ω)) × L 2(0, T; L 2(Ω)).

     
  2. (ii)
    the following equalities
    Ω T u t ϕ 1 d x d t + Ω T u p - 2 u ϕ 1 d x d t = Ω T ϕ 1 ( Ω v m d x - α u r ) d x d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equf_HTML.gif
     
and
Ω T v t ϕ 2 d x d t + Ω T v q - 2 v ϕ 2 d x d t = Ω T ϕ 2 ( Ω u n d x - β v s ) d x d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equg_HTML.gif
hold for all ϕ1, ϕ2, which belong to the class of test functions
Θ 1 Ψ C 1 , 1 ( Ω ¯ T ) ; Ψ ( x , T ) = 0 ; Ψ ( x , t ) = 0 o n S T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equh_HTML.gif
  1. (iii)

    u(x, t)|t = 0= u 0(x), v(x, t)|t = 0= v 0(x) for all x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq16_HTML.gif.

     

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 Ω ¯ T × Ω ¯ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq14_HTML.gif if and only if
  1. (i)

    ( u , v ) is in the space ( C ( 0 , T ; L ( Ω ) ) L p ( 0 , T ; W 0 1 , p ( Ω ) ) ) × ( C ( 0 , T ; L ( Ω ) ) L q ( 0 , T ; W 0 1 , q ( Ω ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq15_HTML.gif and (u t , v t ) L 2(0, T; L 2(Ω)) × L 2(0, T; L 2(Ω)).

     
  2. (ii)
    the following inequalities
    Ω T u t ϕ 1 d x d t + Ω T u p - 2 u ϕ 1 d x d t Ω T ϕ 1 ( Ω v m d x - α u r ) d x d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equi_HTML.gif
     
and
Ω T v t ϕ 2 d x d t + Ω T v q - 2 v ϕ 2 d x d t Ω T ϕ 2 ( Ω u n d x - β v s ) d x d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equj_HTML.gif
hold for any ϕ1, ϕ2, which belong to the class of test functions
Θ 2 { Ψ C 1 , 1 ( Ω ¯ T ) ; Ψ ( x , t ) 0 ; Ψ ( x , T ) = 0 ; Ψ ( x , t ) = 0 o n S T } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equk_HTML.gif
  1. (iii)

    u (x, t)|t = 0u 0(x), v (x, t)|t = 0v 0(x) for all x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq16_HTML.gif.

     

Similarly, a pair of functions ( u ¯ ( x , t ) , v ¯ ( x , t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq17_HTML.gif 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 ( u ¯ , v ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq18_HTML.gif be a nonnegative subsolution and supersolution of (1.1), respectively, with ( u ( x , 0 ) , v ( x , 0 ) ) ( u ¯ ( x , 0 ) , v ¯ ( x , 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq19_HTML.gif for all x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq16_HTML.gif. Then, ( u , v ) ( u ¯ , v ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq20_HTML.gif a.e. in Ω ¯ T × Ω ¯ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq14_HTML.gif.

Proof. From the definitions of weak subsolution and supersolution, for any ϕ1, ϕ2 Θ2, we could obtain that
Ω T ( u t - u ¯ t ) ϕ 1 d x d t + Ω T ( u p - 2 u - u ¯ p - 2 u ¯ ) ϕ 1 d x d t Ω T ϕ 1 Ω ( v m - v ¯ m ) d x - α ( u r - u ¯ r ) d x d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ9_HTML.gif
(2.1)
and
Ω T ( v t - v ¯ t ) ϕ 2 d x d t + Ω T ( v q - 2 v - v ¯ q - 2 v ¯ ) ϕ 2 d x d t Ω T ϕ 2 Ω ( u n - u ¯ n ) d x - β ( v s - v ¯ s ) d x d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ10_HTML.gif
(2.2)
In addition, inequalities (2.1) and (2.2) remain true for any subcylinder of the form Ω τ = Ω × (0, τ) Ω T and corresponding lateral boundary S τ = ∂Ω × (0, τ) S T . Taking a special test function ϕ 1 = χ [ 0 , τ ] ( u - u ¯ ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq21_HTML.gif in (2.1), where χ[0, τ]is the characteristic function defined on [0, τ] and s+ = max{s, 0}, we find that
Ω τ ( u ¯ t u ¯ t ) ( u ¯ u ¯ ) + d x d t + Ω τ ( | u ¯ | p 2 u ¯ | u ¯ | p 2 u ¯ ) ( u ¯ u ¯ ) + d x d t m | Ω | M ^ m 1 Ω τ ( v ¯ v ¯ ) + ( u ¯ u ¯ ) + d x d t + α r M ^ r 1 Ω τ ( u ¯ u ¯ ) + 2 d x d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ11_HTML.gif
(2.3)
where |Ω| denotes the Lebesgue measure of Ω and
M ^ = max u L ( Ω T ) , u ¯ L ( Ω T ) , v L ( Ω T ) , v ¯ L ( Ω T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equl_HTML.gif
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
m Ω M ^ m - 1 Ω τ ( v - v ¯ ) + ( u - u ¯ ) + d x d t 1 2 m Ω M ^ m - 1 Ω τ ( v - v ¯ ) + 2 d x d t + Ω τ ( u - u ¯ ) + 2 d x d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ12_HTML.gif
(2.4)
Furthermore, by Lemma 1.4.4 in [12], we know that there exists δ > 0 such that
( u p - 2 u - u ¯ p - 2 u ¯ ) χ [ 0 , τ ] ( u - u ¯ ) min 0 , δ ( u - u ¯ ) + p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ13_HTML.gif
(2.5)
Combining now (2.3)-(2.5), we deduce that
Ω ( u - u ¯ ) + 2 d x C 1 Ω τ ( u - u ¯ ) + 2 d x d t + C 2 Ω τ ( v - v ¯ ) + 2 d x d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ14_HTML.gif
(2.6)

here C 1 = 1 2 m Ω M ^ m - 1 + α r M ^ r - 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq22_HTML.gif, C 2 = 1 2 m Ω M ^ m - 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq23_HTML.gif.

Likewise, taking test function ϕ 2 = χ [ 0 , τ ] ( v - v ¯ ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq24_HTML.gif in (2.2), we have that
Ω ( v - v ¯ ) + 2 d x C 3 Ω τ ( u - u ¯ ) + 2 d x d t + C 4 Ω τ ( v - v ¯ ) + 2 d x d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ15_HTML.gif
(2.7)
where C3, C4 denote some positive constants. Moreover, there exists a large enough constant C, such that
Ω ( u - u ¯ ) + 2 + ( v - v ¯ ) + 2 d x C Ω τ ( u - u ¯ ) + 2 + ( v - v ¯ ) + 2 d x d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ16_HTML.gif
(2.8)
Now, we write
y ( τ ) = ( u - u ¯ ) + 2 + ( v - v ¯ ) + 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equm_HTML.gif
then, (2.8) implies that
y ( τ ) C 0 τ y ( t ) d t f o r a . e . 0 τ T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ17_HTML.gif
(2.9)

By Gronwall's inequality, we know that y(τ) = 0, for any τ [0, T]. Thus, ( u - u ¯ ) + = ( v - v ¯ ) + = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq25_HTML.gif, this means that u u ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq26_HTML.gif, v v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq27_HTML.gif in Ω ¯ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq28_HTML.gif 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 ( 0 , 0 ) ( u 0 , v 0 ) ( C ( Ω ¯ ) W 0 1 , p ) × ( C ( Ω ¯ ) W 0 1 , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq29_HTML.gif, there is some T0 > 0 such that the problem (1.1) admits a nonnegative unique weak solution (u, v) for each t < T0, and ( u , v ) ( C ( 0 , T 0 ; L ( Ω ) ) L p ( 0 , T ; W 0 1 , p ( Ω ) ) ) × ( C ( 0 , T ; L ( Ω ) ) L q ( 0 , T 0 ; W 0 1 , q ( Ω ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq30_HTML.gif. Furthermore, either T0 = ∞ or
lim t T 0 - sup ( u ( x , t ) | | + v ( x , t ) | | ) = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equn_HTML.gif

3 Proof of Theorem 1.1

Proof of Theorem 1.1. Notice that (1/τ, 1/θ) < (0, 0) implies
m n < μ γ = max { p - 1 , r } max { q - 1 , s } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equo_HTML.gif
We will prove Theorem 1.1 in four subcases.
  1. (a)
    For μ = r, γ = s, we then have mn < rs. Let ( u ¯ , v ¯ ) = ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq31_HTML.gif, where A max x Ω ¯ u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq32_HTML.gif, B max x Ω ¯ v 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq33_HTML.gif will be determined later. After a simple computation, we have
    u ¯ t div ( | u ¯ | p 2 u ¯ ) Ω v ¯ m d x + α u ¯ r = α A r | Ω | B m , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equp_HTML.gif
     
and
v ¯ t div ( | v ¯ | p 2 v ¯ ) Ω u ¯ n d x + β v ¯ s = β B s | Ω | A n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equq_HTML.gif
So, ( u ¯ , v ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq34_HTML.gif is a time-independent supersolution of problem (1.1) if
α A r | Ω | B m and β B s | Ω | A n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equr_HTML.gif
i.e.,
B m r ( | Ω | α ) 1 r A B s n ( | Ω | β ) 1 n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ18_HTML.gif
(3.1)
  1. (b)
    For μ = p - 1, γ = q - 1, we then have mn < (p - 1)(q - 1). Let
    ( u ¯ , v ¯ ) = ( A ( φ + 1 ) , B ( ψ + 1 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equs_HTML.gif
     
where φ, ψ satisfying (1.7) and (1.8), respectively. Taking
A max { max Ω ¯ u 0 ( x ), ( ( m 1 + 1 ) m n q 1 ( M 2 + 1 ) m | Ω | m + q 1 q 1 ) q 1 ( p 1 ) ( q 1 ) m n } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equt_HTML.gif
and
B max { max Ω ¯ v 0 ( x ), ( ( m 1 + 1 ) n ( M 2 + 1 ) m n p 1 | Ω | n + p 1 q 1 ) p 1 ( p 1 ) ( q 1 ) m n } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equu_HTML.gif
then it is easy to verify that ( u ¯ , v ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq34_HTML.gif is a global supersolution for system (1.1).
  1. (c)
    For μ = r, γ = q - 1, we then have mn < r(q - 1). Choose A max x Ω ¯ u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq32_HTML.gif and B max x Ω ¯ v 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq33_HTML.gif satisfy
    ( | Ω | A n ) 1 q - 1 B ( α | Ω | A r ( M 2 + 1 ) - m ) 1 m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equv_HTML.gif
     
Let ( u ¯ , v ¯ ) = ( A , B ( ψ + 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq35_HTML.gif with ψ defined by (1.8). By direct Computation, we arrive at
u ¯ t div ( | u ¯ | p 2 u ¯ ) Ω v ¯ m d x + α u ¯ r 0, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ19_HTML.gif
(3.2)
and
v ¯ t div ( | v ¯ | p 2 v ¯ ) Ω u ¯ n d x + β v ¯ s 0. https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ20_HTML.gif
(3.3)
  1. (d)
    For μ = p - 1, γ = s, we then have mn < r(q - 1). Let ( u ¯ , v ¯ ) = ( A ( φ + 1 ) , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq36_HTML.gif with φ defined by (1.7), where A max x Ω ¯ u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq32_HTML.gif and B max x Ω ¯ v 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq33_HTML.gif. Then, (3.2) and (3.3) hold if
    ( | Ω | B m ) 1 p - 1 A ( β | Ω | B s ( M 1 + 1 ) - n ) 1 n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equw_HTML.gif
     

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
p q > μ γ = max { p - 1 , r } max { q - 1 , s } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equx_HTML.gif
For μ = r, γ = s. Choosing
B = α n β r | Ω | n + r 1 m n - r s a n d A = 1 2 | Ω | α 1 r B m r + β | Ω | 1 n B s n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equy_HTML.gif

then ( u ¯ , v ¯ ) = ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq31_HTML.gif is a global supersolution for problem (1.1) provided that A max x Ω ¯ u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq32_HTML.gif and B max x Ω ¯ v 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq33_HTML.gif.

For μ = p - 1, γ = q - 1. Let ( u ¯ , v ¯ ) = ( A ( φ + 1 ) , B ( ψ + 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq37_HTML.gif, where φ and ψ satisfying (1.7) and (1.8), respectively. Choosing
A = 1 2 ( | Ω | 1 p - 1 ( M 2 + 1 ) m p - 1 B m p - 1 + 1 m 1 + 1 | Ω | - 1 n B q - 1 n ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equz_HTML.gif
and
B = ( | Ω | n + p 1 ( m 1 + 1 ) n ( p 1 ) ( M 2 + 1 ) m n ) 1 m n ( p 1 ) ( q 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equaa_HTML.gif

therefore, ( u ¯ , v ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq34_HTML.gif is a global supersolution for system (1.1) if A max x Ω ¯ u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq32_HTML.gif and B max x Ω ¯ v 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq33_HTML.gif.

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
{ P u ( x , t ) u t div ( | u | p 2 u ) Ω v + m d x + α u + r = 0, x Ω , t > 0, Q v ( x , t ) v t div ( | v | q 2 v ) Ω u + n d x + β v + s = 0, x Ω , t > 0, u ( x , t ) = v ( x , t ) = 0, x Ω , t > 0, u ( x ,0 ) = u 0 ( x ), v ( x ,0 ) = v 0 ( x ), x Ω ¯ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ21_HTML.gif
(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
u ( x , t ) = W ( y 1 ) ( T - t ) l 1 , v ( x , t ) = W ( y 2 ) ( T - t ) l 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ22_HTML.gif
(4.2)
here
y i = | x | ( T - t ) σ i 0 , W ( y i ) = 1 - y i 2 , i = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equab_HTML.gif
and l i , σ i > 0(i = 1, 2), 0 < T < 1 are to be determined later. Notice the fact that
supp u ¯ ( x , t ) + = B ( 0, ( T t ) σ 1 ) ¯ B ( 0, T σ 1 ) ¯ Ω , supp v ¯ ( x , t ) + = B ( 0, ( T t ) σ 2 ) ¯ B ( 0, T σ 2 ) ¯ Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ23_HTML.gif
(4.3)

for sufficiently small T > 0.

Calculating directly, we obtain
u t = l 1 W ( y 1 ) + σ 1 y 1 W ( y 1 ) ( T - t ) l 1 + 1 , - Δ u = 2 N ( T - t ) l 1 + 2 σ 1 , v t = l 2 W ( y 2 ) + σ 2 y 2 W ( y 2 ) ( T - t ) l 2 + 1 , - Δ v = 2 N ( T - t ) l 2 + 2 σ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equac_HTML.gif
and
Ω v ¯ + m d x = 1 ( T t ) m l 2 B ( 0, ( T t ) σ 2 ) W m ( | x | ( T t ) σ 2 ) d x S 1 ( T t ) m l 2 N σ 2 , Ω u ¯ + n d x = 1 ( T t ) n l 1 B ( 0, ( T t ) σ 1 ) W n ( | x | ( T t ) σ 1 ) d x S 2 ( T t ) n l 1 N σ 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equad_HTML.gif
where
S 1 = B ( 0 , 1 ) W m ( | ξ | ) d ξ , S 2 = B ( 0 , 1 ) W n ( | ξ | ) d ξ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equae_HTML.gif
On the other hand, we know
div ( | u ¯ | p 2 u ¯ ) = | u ¯ | p 2 Δ u ¯ + ( p 2 ) | u ¯ | p 4 ( u ¯ ) ( H x ( u ¯ ) ) u ¯ = | u ¯ | p 2 Δ u ¯ + ( p 2 ) | u ¯ | p 4 j = 1 N i = 1 N u ¯ x i 2 u ¯ x i x j u ¯ x j , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ24_HTML.gif
(4.4)
div ( | v ¯ | q 2 v ¯ ) = | v ¯ | q 2 Δ v ¯ + ( q 2 ) | v ¯ | q 4 ( v ¯ ) ( H x ( v ¯ ) ) v ¯ = | v ¯ | q 2 Δ v ¯ + ( q 2 ) | v ¯ | q 4 j = 1 N i = 1 N v ¯ x i 2 v ¯ x i x j v ¯ x j , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ25_HTML.gif
(4.5)
here H x (u), H x (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
| div ( | u ¯ | p 2 u ¯ ) | 2 N ( T t ) l 1 + 2 σ 1 ( d ( Ω ) ( T t ) l 1 + 2 σ 1 ) p 2 + 2 N ( p 2 ) ( T t ) l 1 + 2 σ 1 ( d ( Ω ) ( T t ) l 1 + 2 σ 1 ) p 4 ( d ( Ω ) ( T t ) l 1 + 2 σ 1 ) 2 = 2 N ( p 1 ) d ( Ω ) p 2 ( T t ) ( l 1 + 2 σ 1 ) ( p 1 ) , | div ( | v ¯ | q 2 v ¯ ) | 2 N ( T t ) l 2 + 2 σ 2 ( d ( Ω ) ( T t ) l 2 + 2 σ 2 ) q 2 + 2 N ( q 2 ) ( T t ) l 2 + 2 σ 2 ( d ( Ω ) ( T t ) l 2 + 2 σ 2 ) q 4 ( d ( Ω ) ( T t ) l 2 + 2 σ 2 ) 2 = 2 N ( q 1 ) d ( Ω ) q 2 ( T t ) l 2 + 2 σ 2 ( q 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equaf_HTML.gif
Further, we have
P u ( x , t ) l 1 ( T - t ) l 1 + 1 + 2 N ( p - 1 ) d ( Ω ) p - 2 ( T - t ) ( l 1 + 2 σ 1 ) ( p - 1 ) + α ( T - t ) r l 1 - S 1 ( T - t ) m l 2 - N σ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ26_HTML.gif
(4.6)
and
Q v ( x , t ) l 2 ( T - t ) l 2 + 1 + 2 N ( q - 1 ) d ( Ω ) q - 2 ( T - t ) ( l 2 + 2 σ 2 ) ( q - 1 ) + β ( T - t ) s l 2 - S 2 ( T - t ) n l 1 - N σ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ27_HTML.gif
(4.7)
Since 1/τ, 1/θ < 0, we see that μγ < mn. In addition, it is clear that
μ m < n + 1 m + 1 o r γ n < m + 1 n + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ28_HTML.gif
(4.8)
For μ m < n + 1 m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq38_HTML.gif, we choose l1 and l2 such that
μ m < l 2 l 1 < min n + 1 m + 1 , n γ a n d μ < 1 + l 1 l 1 < m l 2 l 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ29_HTML.gif
(4.9)
Recall that μ = max{p - 1, r} and γ = max{q - 1, s}, then (4.9) implies
m l 2 > r l 1 , m l 2 > l 1 ( p - 1 ) , m l 2 > l 1 + 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equag_HTML.gif
and
n l 1 > s l 2 , n l 1 > l 2 ( q - 1 ) , n l 1 > l 2 + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equah_HTML.gif
Next, we can choose positive constants σ1, σ2 sufficiently small such that
σ 1 = σ 2 < min m l 2 - ( l 1 + 1 ) N , m l 2 - r l 1 N , m l 2 - l 1 ( p - 1 ) N + 2 ( p - 1 ) , n l 1 - ( l 2 + 1 ) N , n l 1 - s l 2 N , n l 1 - l 2 ( q - 1 ) N + 2 ( q - 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equai_HTML.gif
consequently, we have
m l 2 - N σ 2 > max l 1 + 1 , ( l 1 + 2 σ 1 ) ( p - 1 ) , r l 1 , n l 1 - N σ 1 > max l 2 + 1 , ( l 2 + 2 σ 2 ) ( q - 1 ) , s l 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ30_HTML.gif
(4.10)
For γ n < m + 1 n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq39_HTML.gif, we fix l1 and l2 to satisfy
γ n < l 1 l 2 < min m + 1 n + 1 , m μ a n d γ < 1 + l 2 l 2 < n l 1 l 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ31_HTML.gif
(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
P u ( x , t ) 0 , Q v ( x , t ) 0 i n Ω ¯ T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ32_HTML.gif
(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 B ( 0 , T σ 1 ) B ( 0 , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq40_HTML.gif, hence u≤ 0, v≤ 0 on S T . From (4.1) and (4.2), it follows that u ( x , 0 ) M ¯ ϖ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq41_HTML.gif, v ( x , 0 ) M ¯ ϖ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq42_HTML.gif for sufficiently large M ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq43_HTML.gif. By comparison principle, we have (u, v) ≤ (u, v) provided that u 0 ( x ) M ¯ ϖ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq44_HTML.gif and v 0 ( x ) M ¯ ϖ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq45_HTML.gif. 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.
  1. (i)
    For r > p - 1, s > q - 1, we know mn = rs. Thanks to α n β r ≥ |Ω|n+r, we can choose A and B sufficiently large such that A max x Ω ¯ u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq32_HTML.gif, B max x Ω ¯ v 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq33_HTML.gif and
    B m r ( | Ω | α ) 1 r A B s n ( | Ω | β ) 1 n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equaj_HTML.gif
     

Clearly, ( u ¯ , v ¯ ) = ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq31_HTML.gif 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
q - 2 r - 1 < s n = l 1 l 2 = m r < s - 1 p - 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ33_HTML.gif
(5.1)
According to Proposition 2.3, we only need to construct a suitable blow-up subsolution of problem (1.1) on Ω ¯ T × Ω ¯ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq14_HTML.gif. Let y(t) be the solution of the following ordinary differential equation
y ( t ) = c 1 y δ 1 - c 2 y δ 2 , t > 0 , y ( 0 ) = y 0 > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equak_HTML.gif
where
c 1 = min Ω ψ m d x - α φ r l 1 φ , Ω φ n d x - β ψ s l 2 ψ , c 2 = max 1 l 1 φ , 1 l 2 ψ , δ 1 = min ( r - 1 ) l 1 + 1 , ( s - 1 ) l 2 + 1 , δ 2 = max ( p - 2 ) l 1 + 1 , ( q - 2 ) l 2 + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equal_HTML.gif
Since Ω ψ m d x > α φ r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq5_HTML.gif and Ω φ n d x > β ψ s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq6_HTML.gif, 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
lim t T y ( t ) = + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equam_HTML.gif
Construct
( u ( x , t ) , v ( x , t ) ) = ( y l 1 ( t ) φ ( x ) , y l 2 ( t ) ψ ( x ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equan_HTML.gif
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
u 0 ( x ) y 0 l 1 M 1 a n d v 0 ( x ) y 0 l 2 M 2 f o r a l l x Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ34_HTML.gif
(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 Ω ¯ T × Ω ¯ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq14_HTML.gif, T < T'. In fact, (x, t) Ω T × (0, T), a series of computations show
P u ¯ ( x , t ) u ¯ t div ( | u ¯ | p 2 u ¯ ) Ω v ¯ m d x + α u ¯ r = l 1 φ y l 1 1 y ( t ) + y l 1 ( p 1 ) y m l 2 Ω ψ m d x + α y r l 1 φ r = l 1 φ y l 1 1 ( y ( t ) + 1 l 1 φ y ( p 2 ) l 1 + 1 Ω ψ m d x α φ r l 1 φ y l 1 ( r 1 ) + 1 ) 0. https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ35_HTML.gif
(5.3)
Similarly, we also have
Q v ¯ ( x , t ) v ¯ t div ( | v ¯ | q 2 v ¯ ) Ω u ¯ n d x + β v ¯ s = l 2 ψ y l 2 1 y ( t ) + y l 2 ( q 1 ) y n l 1 Ω φ n d x + β y s l 2 ψ s = l 2 ψ y l 2 1 ( y ( t ) + 1 l 2 ψ y ( q 2 ) l 2 + 1 Ω φ n d x β ψ s l 2 ψ y l 2 ( s 1 ) + 1 ) 0. https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ36_HTML.gif
(5.4)
On the other hand, t [0, T], we have
u ( x , t ) | x Ω = y l 1 ( t ) φ ( x ) | x Ω = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ37_HTML.gif
(5.5)
and
v ( x , t ) | x Ω = y l 2 ( t ) ψ ( x ) | x Ω = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ38_HTML.gif
(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 Ω ¯ T × Ω ¯ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq14_HTML.gif by comparison principle, thus (u, v) must blow up in finite time since (u, v) does.
  1. (ii)
    For p - 1 > r, q - 1 > s, we know mn = (p - 1)(q - 1). Under the assumption ( Ω φ n d x ) 1 q - 1 ( Ω ψ m d x ) 1 m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq46_HTML.gif, we can choose A, B such that
    A n q - 1 Ω φ n d x 1 q - 1 B A p - 1 m Ω ψ m d x - 1 m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equao_HTML.gif
     

Then, ( u ¯ , v ¯ ) = ( A φ , B ψ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq47_HTML.gif is a global supersolution of (1.1).

Since mn = (p - 1)(q - 1), we can choose constants l1, l2 > 1 such that
s - 1 p - 2 < q - 1 n = l 1 l 2 = m p - 1 < q - 2 r - 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ39_HTML.gif
(5.7)
Next, we consider the following ordinary differential equation
y ( t ) = c 1 y δ 1 - c 2 y δ 2 , t > 0 , y ( 0 ) = y 0 > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equap_HTML.gif
where
c 1 = min Ω ψ m d x - 1 , Ω φ n d x - 1 , c 2 = max α φ r - 1 l 1 , β ψ s - 1 l 2 , δ 1 = min ( p - 2 ) l 1 + 1 , ( q - 2 ) l 2 + 1 , δ 2 = max ( r - 1 ) l 1 + 1 , ( s - 1 ) l 2 + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equaq_HTML.gif

Since Ω ψ m d x > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq8_HTML.gif, Ω φ n d x > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq12_HTML.gif, 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
( u ( x , t ) , v ( x , t ) ) = ( y l 1 ( t ) φ ( x ) , y l 2 ( t ) ψ ( x ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equar_HTML.gif
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 Ω ¯ T × Ω ¯ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq14_HTML.gif, T < T'. Then, the solution (u, v) of (1.1) blows up in finite time.
  1. (iii)
    For p - 1> r, s > q - 1, we know mn = s(p - 1). Since Ω φ n d x Ω - 1 m β 1 s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq10_HTML.gif, we can choose A, B such that
    β - 1 s A n s Ω φ n d x B | Ω | - 1 m A p - 1 m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equas_HTML.gif
     

We can check ( u ¯ , v ¯ ) = ( A φ , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_IEq48_HTML.gif is a global supersolution of (1.1).

Thanks to mn = s(p - 1), we can choose constants l1, l2 > 1 such that
q - 1 n < s n = l 1 l 2 = m p - 1 < m r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equ40_HTML.gif
(5.8)
Let
( u ( x , t ) , v ( x , t ) ) = ( y l 1 ( t ) φ ( x ) , y l 2 ( t ) ψ ( x ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equat_HTML.gif
where φ(x), ψ(x) are defined in (1.7) and (1.8), respectively, and y(t) satisfies the following Cauchy problem
y ( t ) = c 1 y δ 1 - c 2 y δ 2 , t > 0 , y ( 0 ) = y 0 > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equau_HTML.gif
where
c 1 = min Ω ψ m d x - 1 , Ω φ n d x - β ψ s l 2 ψ , c 2 = max α φ r - 1 l 1 , 1 l 2 ψ , δ 1 = min ( p - 2 ) l 1 + 1 , ( s - 1 ) l 2 + 1 , δ 2 = max ( r - 1 ) l 1 + 1 , ( q - 2 ) l 2 + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-29/MediaObjects/13661_2011_Article_75_Equav_HTML.gif
Then, the left arguments are the same as those for the case r > p - 1, s > q - 1, so we omit them.
  1. (iv)

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

     

Declarations

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).

Authors’ Affiliations

(1)
School of Mathematics and Computer Engineering, Xihua University
(2)
College of Mathematics and Statistics, Chongqing University

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 1971, 7: 50-58.
  3. Esteban JR, Vázquez JL: On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Anal 1986, 10: 1303-1325. 10.1016/0362-546X(86)90068-4View ArticleMathSciNet
  4. Escobedo M, Herrero MA: A semilinear parabolic system in a bounded domain. Ann Mat Pura Appl 1993, IV CLXV: 315-336.View ArticleMathSciNet
  5. Galaktionov VA, Levine HA: On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary. Israel J Math 1996, 94: 125-146. 10.1007/BF02762700View ArticleMathSciNet
  6. Zhou J, Mu CL: On critical Fujita exponent for degenerate parabolic system coupled via nonlinear boundary flux. Proc Edinb Math Soc 2008, 51: 785-805. 10.1017/S0013091505001537View ArticleMathSciNet
  7. Ishii H: Asymptotic stability and blowing up of solutions of some nonlinear equations. J Differ Equ 1997, 26: 291-319.View Article
  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 1974, 16: 319-334. 10.1016/0022-0396(74)90018-7View ArticleMathSciNet
  9. Tsutsumi M: Existence and nonexistence of global solutions for nonlinear parabolic equations. Publ Res Inst Math Sci 1972, 8: 221-229.View ArticleMathSciNet
  10. Zhao JN: Existence and nonexistence of solutions for u t - div( u p-2 u ) = f ( u , u , x , t ). J Math Anal Appl 1993, 172: 130-146. 10.1006/jmaa.1993.1012View ArticleMathSciNet
  11. Li FC, Xie HC: Global and blow-up of solutions to a p -Laplace equation with nonlocal source. Comput Math Appl 2003, 46: 1525-1533. 10.1016/S0898-1221(03)90188-XView ArticleMathSciNet
  12. Dibenedetto E: Degenerate Parabolic Equations. Springer, Berlin; 1993.View Article
  13. Tsutsumi M: On solutions of some doubly nonlinear degenerate parabolic equations with absorption. J Math Anal Appl 1988, 132: 187-212. 10.1016/0022-247X(88)90053-4View ArticleMathSciNet
  14. Yin JX, Jin CH: Critical extinction and blow-up exponents for fast diffusion p -Laplace with sources. Math Methods Appl Sci 2007, 30: 1147-1167. 10.1002/mma.833View ArticleMathSciNet
  15. Yuan HJ: Extinction and positivity of the evolution p -Laplacian equation. J Math Anal Appl 1995, 196: 754-763. 10.1006/jmaa.1995.1439View ArticleMathSciNet
  16. Li FC: Global existence and blow-up of solutions to a nonlocal quasilinear degenerate parabolic system. Nonlinear Anal 2007, 67: 1387-1402. 10.1016/j.na.2006.07.024View ArticleMathSciNet
  17. Bedjaoui N, Souplet P: Critical blow-up exponents for a system of reaction-diffusion equations with absorption. Z Angew Math Phys 2002, 53: 197-210. 10.1007/s00033-002-8152-9View ArticleMathSciNet
  18. Chen YP: Blow-up for a system of heat equations with nonlocal sources and absorptions. Comput Math Appl 2004, 48: 361-372. 10.1016/j.camwa.2004.05.002View ArticleMathSciNet
  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, 2007: 17. (Article ID 34301)View ArticleMathSciNet
  20. Galaktionov VA, Kurdyumov SP, Samarskii AA: A parabolic system of quasilinear equations I. Differ Equ 1983, 19: 1558-1571.MathSciNet
  21. Galaktionov VA, Kurdyumov SP, Samarskii AA: A parabolic system of quasilinear equations II. Differ Equ 1985, 21: 1049-1062.
  22. Li FC, Huang SX, Xie HC: Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete Contin Dyn Syst 2003, 9: 1519-1532.View ArticleMathSciNet
  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 2009, 358: 229-237. 10.1016/j.jmaa.2009.04.059View ArticleMathSciNet
  24. Yang ZD, Lu QS: Blow-up estimates for a quasilinear reaction-diffusion system. Math Method Appl Sci 2003, 26: 1005-1023. 10.1002/mma.409View ArticleMathSciNet
  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 2008, 200: 267-282. 10.1016/j.amc.2007.11.012View ArticleMathSciNet
  26. Zheng SN: Global existence and global non-existence of solution to a reaction-diffusion system. Nonlinear Anal 2000, 39: 327-340. 10.1016/S0362-546X(98)00171-0View ArticleMathSciNet
  27. Zheng SN, Su H: A quasilinear reaction-diffusion system coupled via nonlocal sources. Appl Math Comput 2006, 180: 295-308. 10.1016/j.amc.2005.12.020View ArticleMathSciNet
  28. Zhou J, Mu CL: Blow-up for a non-Newton polytropic filtration system with nonlinear nonlocal source. Commun Korean Math Soc 2008, 23: 529-540. 10.4134/CKMS.2008.23.4.529View ArticleMathSciNet
  29. Zhou J, Mu CL: Global existence and blow-up for non-Newton polytropic filtration system with nonlocal source. Glasgow Math J 2009, 51: 39-47. 10.1017/S0017089508004515View ArticleMathSciNet
  30. Zhou J, Mu CL: Global existence and blow-up for non-Newton polytropic filtration system with nonlocal source. ANZIAM J 2008, 50: 13-29. 10.1017/S1446181108000242View ArticleMathSciNet
  31. Souplet P: Blow-up in nonlocal reaction-diffusion equations. SIAM J Math Anal 1998, 29: 1301-1334. 10.1137/S0036141097318900View ArticleMathSciNet

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