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Boundary Value Problems

, 2011:29 | Cite as

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

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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 Open image in new window

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 ¯ , Open image in new window
(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 ) Open image in new window, v 0 ( x ) C ( Ω ¯ ) W 0 1 , q ( Ω ) Open image in new window and u 0 ( x ) ν < 0 Open image in new window, v 0 ( x ) ν < 0 Open image in new window, 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 ¯ , Open image in new window
(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 ¯ . Open image in new window
(1.3)

under different conditions (see [5, 6] for nonlinear boundary conditions; see [7, 8, 9, 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) = -u q , q > 0 or f(u) ≡ 0 extinction phenomenon of the solution may appear [13, 14, 15]; However, if f(u) = u q , q > 1 the solution may blow up in finite time [7, 8, 9, 10, 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 ¯ . Open image in new window
(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 ¯ . Open image in new window
(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, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 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
u t div ( | u | p 2 u ) = Ω u m d x α u r . Open image in new window
(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 Open image in new window
(1.7)
and
div ( | ψ | q 2 ψ ) = 1, x Ω ; ψ ( x ) = 0, x Ω , Open image in new window
(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 ) . Open image in new window
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 ) , Open image in new window
namely,
τ = m + γ m n - μ γ , θ = n + μ m n - μ γ Open image in new window
with
μ = max { p - 1 , r } , γ = max { q - 1 , s } . Open image in new window

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 Open image in new window, Ω φ n d x > β ψ s Open image in new window, 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 Open image in new window, then the solutions are globally bounded for small initial data; if Ω ψ m d x > 1 Open image in new window, Ω φ n d x > 1 Open image in new window 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 Open image in new window, then the solutions are globally bounded for small initial data; if Ω ψ m d x > 1 Open image in new window, Ω φ n d x > β ψ s Open image in new window, 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 Open image in new window , then the solutions are globally bounded for small initial data; if Ω φ n d x > 1 Open image in new window, Ω ψ m d x > α φ r Open image in new window, 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 ) Open image in new window. 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 Open image in new window 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 ) ) ) Open image in new window 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 Open image in new window
     
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 Open image in new window
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 . Open image in new window
  1. (iii)

    u(x, t)|t = 0= u 0(x), v(x, t)|t = 0= v 0(x) for all x Ω ¯ Open image in new window.

     

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 Open image in new window 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 ) ) ) Open image in new window 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 Open image in new window
     
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 Open image in new window
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 } . Open image in new window
  1. (iii)

    u (x, t)|t = 0u 0(x), v (x, t)|t = 0v 0(x) for all x Ω ¯ Open image in new window.

     

Similarly, a pair of functions ( u ¯ ( x , t ) , v ¯ ( x , t ) ) Open image in new window 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 ¯ ) Open image in new window be a nonnegative subsolution and supersolution of (1.1), respectively, with ( u ( x , 0 ) , v ( x , 0 ) ) ( u ¯ ( x , 0 ) , v ¯ ( x , 0 ) ) Open image in new window for all x Ω ¯ Open image in new window. Then, ( u , v ) ( u ¯ , v ¯ ) Open image in new window a.e. in Ω ¯ T × Ω ¯ T Open image in new window.

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 , Open image in new window
(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 . Open image in new window
(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 ¯ ) + Open image in new window 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 , Open image in new window
(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 ) . Open image in new window
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 . Open image in new window
(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 . Open image in new window
(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 , Open image in new window
(2.6)

here C 1 = 1 2 m Ω M ^ m - 1 + α r M ^ r - 1 Open image in new window, C 2 = 1 2 m Ω M ^ m - 1 Open image in new window.

Likewise, taking test function ϕ 2 = χ [ 0 , τ ] ( v - v ¯ ) + Open image in new window 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 , Open image in new window
(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 . Open image in new window
(2.8)
Now, we write
y ( τ ) = ( u - u ¯ ) + 2 + ( v - v ¯ ) + 2 , Open image in new window
then, (2.8) implies that
y ( τ ) C 0 τ y ( t ) d t f o r a . e . 0 τ T . Open image in new window
(2.9)

By Gronwall's inequality, we know that y(τ) = 0, for any τ ∈ [0, T]. Thus, ( u - u ¯ ) + = ( v - v ¯ ) + = 0 Open image in new window, this means that u u ¯ Open image in new window, v v ¯ Open image in new window in Ω ¯ T Open image in new window 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 ) Open image in new window, 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 ) ) ) Open image in new window. Furthermore, either T0 = ∞ or
lim t T 0 - sup ( u ( x , t ) | | + v ( x , t ) | | ) = . Open image in new window

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 } . Open image in new window
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 ) Open image in new window, where A max x Ω ¯ u 0 ( x ) Open image in new window, B max x Ω ¯ v 0 ( x ) Open image in new window 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 , Open image in new window
     
and
v ¯ t div ( | v ¯ | p 2 v ¯ ) Ω u ¯ n d x + β v ¯ s = β B s | Ω | A n . Open image in new window
So, ( u ¯ , v ¯ ) Open image in new window is a time-independent supersolution of problem (1.1) if
α A r | Ω | B m and β B s | Ω | A n , Open image in new window
i.e.,
B m r ( | Ω | α ) 1 r A B s n ( | Ω | β ) 1 n . Open image in new window
(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 ) ) , Open image in new window
     
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 } , Open image in new window
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 } , Open image in new window
then it is easy to verify that ( u ¯ , v ¯ ) Open image in new window 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 ) Open image in new window and B max x Ω ¯ v 0 ( x ) Open image in new window satisfy
    ( | Ω | A n ) 1 q - 1 B ( α | Ω | A r ( M 2 + 1 ) - m ) 1 m . Open image in new window
     
Let ( u ¯ , v ¯ ) = ( A , B ( ψ + 1 ) ) Open image in new window 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, Open image in new window
(3.2)
and
v ¯ t div ( | v ¯ | p 2 v ¯ ) Ω u ¯ n d x + β v ¯ s 0. Open image in new window
(3.3)
  1. (d)
    For μ = p - 1, γ = s, we then have mn < r(q - 1). Let ( u ¯ , v ¯ ) = ( A ( φ + 1 ) , B ) Open image in new window with φ defined by (1.7), where A max x Ω ¯ u 0 ( x ) Open image in new window and B max x Ω ¯ v 0 ( x ) Open image in new window. 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 . Open image in new window
     

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 } . Open image in new window
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 , Open image in new window

then ( u ¯ , v ¯ ) = ( A , B ) Open image in new window is a global supersolution for problem (1.1) provided that A max x Ω ¯ u 0 ( x ) Open image in new window and B max x Ω ¯ v 0 ( x ) Open image in new window.

For μ = p - 1, γ = q - 1. Let ( u ¯ , v ¯ ) = ( A ( φ + 1 ) , B ( ψ + 1 ) ) Open image in new window, 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 ) , Open image in new window
and
B = ( | Ω | n + p 1 ( m 1 + 1 ) n ( p 1 ) ( M 2 + 1 ) m n ) 1 m n ( p 1 ) ( q 1 ) , Open image in new window

therefore, ( u ¯ , v ¯ ) Open image in new window is a global supersolution for system (1.1) if A max x Ω ¯ u 0 ( x ) Open image in new window and B max x Ω ¯ v 0 ( x ) Open image in new window.

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 ¯ , Open image in new window
(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 , Open image in new window
(4.2)
here
y i = | x | ( T - t ) σ i 0 , W ( y i ) = 1 - y i 2 , i = 1 , 2 , Open image in new window
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 Open image in new window
(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 , Open image in new window
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 , Open image in new window
where
S 1 = B ( 0 , 1 ) W m ( | ξ | ) d ξ , S 2 = B ( 0 , 1 ) W n ( | ξ | ) d ξ . Open image in new window
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 , Open image in new window
(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 , Open image in new window
(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 ) . Open image in new window
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 , Open image in new window
(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 . Open image in new window
(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 . Open image in new window
(4.8)
For μ m < n + 1 m + 1 Open image in new window, 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 . Open image in new window
(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 , Open image in new window
and
n l 1 > s l 2 , n l 1 > l 2 ( q - 1 ) , n l 1 > l 2 + 1 . Open image in new window
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 ) , Open image in new window
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 . Open image in new window
(4.10)
For γ n < m + 1 n + 1 Open image in new window, 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 , Open image in new window
(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 . Open image in new window
(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 xB(0, ρ) ⊂ Ω. Choose T small enough to insure B ( 0 , T σ 1 ) B ( 0 , ρ ) Open image in new window, hence u≤ 0, v≤ 0 on S T . From (4.1) and (4.2), it follows that u ( x , 0 ) M ¯ ϖ ( x ) Open image in new window, v ( x , 0 ) M ¯ ϖ ( x ) Open image in new window for sufficiently large M ¯ Open image in new window. By comparison principle, we have (u, v) ≤ (u, v) provided that u 0 ( x ) M ¯ ϖ ( x ) Open image in new window and v 0 ( x ) M ¯ ϖ ( x ) Open image in new window. 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 ) Open image in new window, B max x Ω ¯ v 0 ( x ) Open image in new window and
    B m r ( | Ω | α ) 1 r A B s n ( | Ω | β ) 1 n . Open image in new window
     

Clearly, ( u ¯ , v ¯ ) = ( A , B ) Open image in new window 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 . Open image in new window
(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 Open image in new window. 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 , Open image in new window
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 . Open image in new window
Since Ω ψ m d x > α φ r Open image in new window and Ω φ n d x > β ψ s Open image in new window, 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 ) = + . Open image in new window
Construct
( u ( x , t ) , v ( x , t ) ) = ( y l 1 ( t ) φ ( x ) , y l 2 ( t ) ψ ( x ) ) , Open image in new window
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 . Open image in new window
(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 Open image in new window, 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. Open image in new window
(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. Open image in new window
(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 , Open image in new window
(5.5)
and
v ( x , t ) | x Ω = y l 2 ( t ) ψ ( x ) | x Ω = 0 . Open image in new window
(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 Open image in new window 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 Open image in new window, 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 . Open image in new window
     

Then, ( u ¯ , v ¯ ) = ( A φ , B ψ ) Open image in new window 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 . Open image in new window
(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 , Open image in new window
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 . Open image in new window

Since Ω ψ m d x > 1 Open image in new window, Ω φ n d x > 1 Open image in new window, 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 ) ) , Open image in new window
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 Open image in new window, 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 Open image in new window, 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 . Open image in new window
     

We can check ( u ¯ , v ¯ ) = ( A φ , B ) Open image in new window 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 . Open image in new window
(5.8)
Let
( u ( x , t ) , v ( x , t ) ) = ( y l 1 ( t ) φ ( x ) , y l 2 ( t ) ψ ( x ) ) , Open image in new window
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 , Open image in new window
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 . Open image in new window
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. □

     

Notes

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).

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Copyright information

© Zhang et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • Yan Zhang
    • 1
  • Dengming Liu
    • 2
  • Chunlai Mu
    • 2
  • Pan Zheng
    • 2
  1. 1.School of Mathematics and Computer EngineeringXihua UniversityChengduPR China
  2. 2.College of Mathematics and StatisticsChongqing UniversityChongqingPR China

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