Open Access

Critical parameter equations for degenerate parabolic equations coupled via nonlinear boundary flux

Boundary Value Problems20112011:15

DOI: 10.1186/1687-2770-2011-15

Received: 1 May 2011

Accepted: 19 August 2011

Published: 19 August 2011

Abstract

This paper deals with the critical parameter equations for a degenerate parabolic system coupled via nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence parameter equation. The critical Fujita type is conjectured with the aid of some new results.

Mathematics Subject Classification (2000). 35K55; 35K57.

Keywords

degenerate parabolic system global existence blow-up

1 Introduction

In this paper, we consider the following degenerate parabolic equations
u i t = ( u i p i ) x x , ( i = 1 , 2 , , k ) , x > 0 , 0 < t < T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ1_HTML.gif
(1.1)
coupled via nonlinear boundary flux
- ( u i p i ) x ( 0 , t ) = u i + 1 q i + 1 ( 0 , t ) , ( i = 1 , 2 , , k ) , u k + 1 : = u 1 , q k + 1 : = q 1 0 < t < T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ2_HTML.gif
(1.2)
with continuous, nonnegative initial data
u i ( x , 0 ) = u 0 i ( x ) , ( i = 1 , 2 , , k ) , x > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ3_HTML.gif
(1.3)

compactly supported in +, where p i > 1, q i > 0, (i = 1, 2, ..., k) are parameters.

Parabolic systems like (1.1)-(1.3) appear in several branches of applied mathematics. They have been used to models, for example, chemical reactions, heat transfer, or population dynamics (see [1] and the references therein).

As we shall see, under certain conditions the solutions of this problem can become unbounded in a finite time. This phenomenon is known as blow-up, and has been observed for several scalar equations since the pioneering work of Fujita [2]. For further references, see the review by Leivine [3]. Blow-up may also happen for systems (see [47]). Our main interest here will be to determine under which conditions there are solutions of (1.1)-(1.3) that blow up and, in the blow-up case, the speed at which blowup takes place, and the localization of blow-up points in terms of the parameters p i , q i , (i = 1, 2, ..., k).

As a precedent, we have the work of Galaktionov and Levine [8], where they studied the single equation
u 1 t = ( u 1 p 1 ) x x , x > 0 , 0 < t < T , - ( u 1 p 1 ) x ( 0 , t ) = u 1 q 2 ( 0 , t ) , 0 < t < T , u 1 ( x , 0 ) = u 0 1 ( x ) , x > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ4_HTML.gif
(1.4)

It was shown if 0 < q2q0 = (p1 +1)/2, then all nonnegative solutions of (1.4) are global in time, while for q2 > q0 there are solutions with finite time blow-up. That is, q0 is the critical global existence exponent. Moreover, it was shown that q c := p1 + 1 is a critical exponent of Fujita type. Precisely, q c has the following properties: if q0 < q2q c , the all nontrivial nonnegative solutions blow up in a finite time, while global nontrivial nonnegative solutions exist if q2 > q c .

We remark that there are some related works on the critical exponents for (1.1)-(1.3) in special cases.

In [911], the authors consider the case for p i = 1, (i = 1, 2, ..., k).

In [12], the authors consider the case for k = 2.

For the system (1.1)-(1.3), instead of critical exponents there are critical parameter equations, one for global existence and another of Fujita type. This is the content of our first theorem.

To state our results, we introduce some useful symbols. Denote by
A = 1 + p 1 - 2 q 2 0 0 0 0 0 0 1 + p 2 - 2 q 3 0 0 0 0 0 0 0 0 0 1 + p k - 1 - 2 q k - 2 q 1 0 0 0 0 0 1 + p k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equa_HTML.gif
A series of standard computations yield
det A = l = 1 k ( 1 + p l ) - l = 1 k 2 q l . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equb_HTML.gif
We shall see that det A = 0 is the critical global existence parameter equation. Let (α1, α2, ..., α k ) T be the solution of the following linear algebraic system
A ( α 1 , α 2 , , α k - 1 , α k ) T = ( 1 , 1 , , 1 , 1 ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equc_HTML.gif
that is
α i = l = 1 k ( 1 + p l ) 2 q i [ l = 1 k ( 1 + p l ) - l = 1 k 2 q l ] m = i k + i - 1 j = i m 2 q j 1 + p j , q k + i = q i , p k + i = p i ( i = 1 , 2 , , k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ5_HTML.gif
(1.5)
We define
β i = 1 + ( p i - 1 ) α i 2 , ( i = 1 , 2 , , k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ6_HTML.gif
(1.6)
Theorem 1.1.
  1. (I)

    If l = 1 k ( 1 + p l ) l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq1_HTML.gif (i.e. det A ≥ 0), every nonnegative solution of (1.1)-(1.3) is global in time.

     
  2. (II)

    If l = 1 k ( 1 + p l ) < l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq2_HTML.gif (i.e. det A < 0) and there exists j (1 ≤ jk) such that α j + β j ≤ 0, then every nonnegative, nontrivial solution blows up in finite time.

     
  3. (III)

    If l = 1 k ( 1 + p l ) < l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq3_HTML.gif (i.e. det A < 0), with α i + β i > 0 (i = 1, 2, ...,k), there exist nonnegative solutions with blow-up and nonnegative solutions that are global.

     
Therefore, the critical global existence parameter equation is
l = 1 k ( 1 + p l ) = l = 1 k 2 q l ( i . e . det A = 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equd_HTML.gif
and the critical Fujita type parameter equation is
min { α 1 + β 1 , α 2 + β 2 , , α k + β k } = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Eque_HTML.gif

The values of α i , β i (i = 1, 2, ..., k) are the exponents of self-similar solutions to problem (1.1)-(1.2). Such self-similar solutions are studied in Section 2, and play an important role in the proof of Theorem 1.1.

Let us observe that if we take k = 2, the critical parameter equations coincide with those found in [12].

The rest of this paper is organized as follows. In the next section, we study the existence of self-similar solutions of different type. In Section 3 we give some results concerning existence, comparison, monotonicity and uniqueness. In Section 4 we find the critical parameter equations (Theorem 1.1).

2 Self-similar solutions

In this section, we consider different kinds of self-similar solutions of problem (1.1)-(1.2). We have the following results.

Theorem 2.1. Let
u i ( x , t ) = ( T - t ) α i f i ( ξ i ) , ξ i = x ( T - t ) - β i , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ7_HTML.gif
(2.1)
If
l = 1 k ( 1 + p l ) < l = 1 k 2 q l , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ8_HTML.gif
(2.2)

there is a self-similar solution of problem (1.1)-(1.2) blowing up in a finite time T > 0, of form (2.1). Moreover, the support of f i is +if β i > 0, and a compact set if β i ≤ 0 (i = 1, 2, ..., k).

Theorem 2.2. Let
u i ( x , t ) = t α i f i ( ξ i ) , ξ i = x t - β i , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ9_HTML.gif
(2.3)
  1. (a)
    If
    l = 1 k ( 1 + p l ) > l = 1 k 2 q l , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ10_HTML.gif
    (2.4)
     

then there exist functions f i positive in +, such that u i given in (2.3) is a self-similar solution of problem (1.1)-(1.2) global in time. These solutions have α i > 0 and thus their initial data are identically zero. Then β i < 0 (i = 1, 2, ...,k).

(b)If
α i + β i > 0 , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equf_HTML.gif

then there exist functions f i , compactly supported in +, such that u i given in (2.3) is a self-similar solution of problem (1.1)-(1.2) global in time. These solutions have α i < 0 and thus they decay to zero as t → ∞. Then β i > 0, and hence their supports expand as time increases.

Remark 2.2. If there exists j (1 ≤ jk) such that α j + β j ≤ 0, there are no profiles f i L1(+) such that u i (i = 1, 2, ..., k,) given by (2.3) is a solution. Indeed
0 u j ( x , t ) d x = t α j + β j 0 f j ( ξ j ) d ξ j . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equg_HTML.gif

Then, if α j + β j ≤ 0, the mass of u j would not increase, a contradiction.

Theorem 2.3. Let
u i ( x , t ) = e α i t f i ( ξ i ) , ξ i = x e - β i t , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ11_HTML.gif
(2.5)
If
l = 1 k ( 1 + p l ) = l = 1 k 2 q l , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ12_HTML.gif
(2.6)
for any α1 > 0, there is a self-similar solution of problem (1.1)-(1.2) global in time of form (2.5) where
α i = α 1 j = 2 i 1 + p j - 1 2 q j ( i = 2 , , k ) , β i = ( p i - 1 ) α i 2 ( i = 1 , 2 , , k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ13_HTML.gif
(2.7)

Moreover, the supports of f i (i = 1, 2, ..., k) are compact.

Remark 2.3. The solutions are in principle weak. However, if they are positive everywhere, they are also classical.

In order to prove these theorems, we will use the following results of Gilding and Peletier (see [1315]):

Theorem 2.4. Let a, b, V and U ≥ 0. For fixed a and b, let S A denote the set of values of (U, V) such that there exists a weak, nonnegative, compactly supported solution f1of
( f 1 p 1 ) ( η ) + a η f 1 ( η ) = b f 1 ( η ) , 0 < η < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ14_HTML.gif
(2.8)
f 1 ( 0 ) = U , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ15_HTML.gif
(2.9)
( f 1 p 1 ) ( 0 ) = V , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ16_HTML.gif
(2.10)
and let S B denote the set of values (U, V) for which there exists a bounded, positive, classical solution f1of (2.8)-(2.10).
  1. (a)

    If b < 0 and 2a + b < 0, then S A = {(0, 0)} and S B = Ø.

     
  2. (b)

    If b < 0 and 2a + b = 0, then S A = {(0, V): 0 ≤ V < ∞} and S B = Ø.

     
  3. (c)

    If b ≤ 0 and 2a + b > 0, then there exists a unique V * such that S A = { ( U , U ( p 1 + 1 ) 2 V * ) : 0 U < 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq4_HTML.gif and S B = {(U, V): 0 ≤ U < ∞, U ( p 1 + 1 ) 2 V * < V < } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq5_HTML.gif, where V * > 0 if a + b < 0, V * = 0 if a + b = 0, and V * < 0 if a + b > 0.

     
  4. (d)

    If b > 0 and a ≥ 0, then there exists a unique V * < 0 such that S A = { ( U , U ( p 1 + 1 ) 2 V * ) : 0 U < 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq4_HTML.gif and S B = Ø.

     
  5. (e)

    If b > 0 and a < 0, or b = 0 and a ≤ 0, then S A = {(0, 0)} and there exists a unique V * such that S B = {(U, U (p 1+1)/2 V *): 0 ≤ U < ∞}, where V * < 0 if b > 0 and V * = 0 if b = 0.

     

Moreover, for each (U, V) S A S B there exists at most one weak solution of (2.8)-(2.10).

Remark 2.4. In the case where a = ((p1 - 1)/2)b > 0, we have V* = -1. This is a consequence of the existence for a self-similar solution of exponential form for the scalar problem (1.4) with q2 = (p1 + 1)/2 (see [8]).

Proof of Theorem 2.1. We consider solutions of form (2.1). Imposing that the porous equations (1.1) are fulfilled, we get the following relations for the parameters:
α i - 1 = α i p i - 2 β i , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ17_HTML.gif
(2.11)
On the other hand, the boundary conditions (1.2) imply that
α i p i - β i = α i + 1 q i + 1 , i = 1 , 2 , , k , α k + 1 = α 1 , q k + 1 = q 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ18_HTML.gif
(2.12)
Solving the linear systems (2.11)-(2.12), we get that α i , β i (i = 1, 2, ..., k) are given by (1.5) and (1.6). Therefore, α i < 0 (i = 1, 2, ..., k) if and only if l = 1 k ( 1 + p l ) < l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq6_HTML.gif. On the other hand, the profiles must satisfy
( f i p i ) ( ξ i ) - β i ξ i f i ( ξ i ) = - α i f i ( ξ i ) , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ19_HTML.gif
(2.13)
plus the boundary conditions
- ( f i p i ) ( 0 ) = f i + 1 q i + 1 ( 0 ) , i = 1 , 2 , , k , q k + 1 = q 1 , f k + 1 = f 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ20_HTML.gif
(2.14)
Then f i satisfy (2.8) with coefficients a i = -β I , b i = -α i (i = 1, 2, ..., k). Thus, Theorem 2.4 parts (d) and (e) says that there is an one-parameter family (parameter U i ) of (2.8) satisfying
f i ( 0 ) = U i , ( f i p i ) ( 0 ) = U i ( p i + 1 ) 2 V * i , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equh_HTML.gif
where V*i< 0 (i = 1, 2, ..., k) are constants. The profile f i has compact support if β i ≤ 0 and is positive in + if β i > 0. We choose U i such that the boundary conditions (2.14) are fulfilled, that is
- U i ( p i + 1 ) 2 V * i = U i + 1 q i + 1 , i = 1 , 2 , , k , U k + 1 = U 1 , q k + 1 = q 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equi_HTML.gif
Taking logarithms, this is equivalent to
A ( ln U 1 , ln U 2 , , ln U k - 1 , ln U k ) T = - 2 ( ln | V * 1 | , ln | V * 2 | , , ln | V * k - 1 | , ln | V * k | ) T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ21_HTML.gif
(2.15)

As l = 1 k ( 1 + p l ) l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq7_HTML.gif (i.e. det A ≠ 0), the above system has a unique solution. □

Proof of Theorem 2.2. We are considering solutions of the form (2.3). Imposing that the equations (1.1) and that boundary conditions (1.2) are fulfilled, we get that the exponents should satisfy the relations (2.11)-(2.12). Hence they are given by (1.5)-(1.6). Moreover, the boundary conditions for the profiles are given by (2.14). However, the equations for the profiles are now different:
( f i p i ) ( ξ i ) + β i ξ i f i ( ξ i ) = α i f i ( ξ i ) , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ22_HTML.gif
(2.16)
Thus, f i satisfy (2.8) with coefficients a i = β i , b i = α i (i = 1, 2, ..., k).
  1. (I)

    If α i > 0, that is, if (2.4) holds, then β i < 0 (i = 1, 2, ..., k). Therefore, applying Theorem 2.4 part (d) as in the proof of Theorem 2.1, and taking the solutions of (2.15) as values for parameters, we obtain that there exist positive profiles f i (i = 1, 2, ..., k) solving (2.16) and satisfying (2.14).

     
  2. (II)

    If α i < 0 and α i + β i > 0 (i = 1, 2, ..., k), we can apply Theorem 2.4 part (c) as in the proof of Theorem 2.1 and taking the solutions of (2.15) as the parameters, we obtain that there exist compactly supports profiles f i (i = 1, 2, ..., k) solving (2.16) and satisfying the boundary conditions (2.14).

     
Proof of Theorem 2.3. We are considering solutions of the form (2.5). Though the boundary conditions (1.2) impose (2.12) again, now equations (1.1) impose different relations for the exponents. Namely
α i = α i p i - 2 β i , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ23_HTML.gif
(2.17)
Thus,
A ( α 1 , α 2 , , α k - 1 , α k ) T = ( 0 , 0 , , 0 , 0 ) T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ24_HTML.gif
(2.18)

There are nontrivial solutions of (2.18) if and only if l = 1 k ( 1 + p l ) = l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq8_HTML.gif (i.e. det A = 0). In this case, β1, α I , β i (i = 2, ...,k) are related to α1 by (2.7).

The boundary conditions for the profiles are again given by (2.14), while the equations for the profiles are given by (2.16). If α1 > 0, then β1, α i , β i > 0 (i = 2, ..., k) and β i = ((p i - 1)/2)α i (i = 1, ..., k). Hence, using Remark 2.4, we have solutions of (2.16) with V*i= -1 (i = 1, 2, ...,k). Choosing one of the solutions of (2.15) with right-hand side zero (again we are using l = 1 k ( 1 + p l ) = l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq8_HTML.gif (i.e. det A = 0)), we obtain that there exist compactly supported profiles f i (i = 1, 2, ..., k) solving (2.16) and satisfying (2.14).

3 Existence and uniqueness

First, we state a theorem that guarantees the existence of a solution. It can be obtained using a standard monotonicity argument following ideas from [16].

Theorem 3.1. Given continuous, compactly supported initial data u0i(x) (i = 2, ..., k), there exists a local in time continuous weak solution of (1.1)-(1.3). Moreover, if the initial data are smooth and compatible in sense that
- ( u 0 i p i ) x ( 0 ) = u 0 i + 1 q i + 1 ( 0 ) , i = 2 , , k , u 0 k + 1 ( x ) = u 0 1 ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equj_HTML.gif

then the solution has continuous time derivatives down to t = 0.

Proof. Let us consider the Neumann problem
w t = ( w r ) x x , x > 0 , 0 < t < τ , - ( w r ) x ( 0 , t ) = h ( t ) , 0 < t < τ , w ( x , 0 ) = w 0 ( x ) , x > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ25_HTML.gif
(3.1)

with r > 1. We define the operator M q i + 1 : C ( [ 0 , τ ] ) C ( [ 0 , τ ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq9_HTML.gif as M q i + 1 ( h ) ( t ) = w q i + 1 ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq10_HTML.gif, where w(x, t) is the unique solution of (3.1) with r = p i and initial condition w0(x) = u0i(x) i = 1 , 2 , , k , M q k + 1 = M q 1 , w q k + 1 = w q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq11_HTML.gif.

It has been proved in [17] that M q i ( i = 1 , 2 , . . . , k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq12_HTML.gif is continuous and compact. Moreover, they are order preserving.

Now let A ( h ) = M q k M q k - 1 M q 2 M q 1 ( h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq13_HTML.gif. Using the method of monotone iterations, one can prove that there exist τ > 0 such that A has a fixed point in C([0, τ]). This fixed point provides us with a continuous weak solution of (1.1)-(1.3) up to time τ.

In order to obtain the regularity of the solution with compatible initial data, we only have to observe that the solution of (3.1) is regular if - ( w 0 r ) x = h ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq14_HTML.gif (see [18]).

Remark 3.1. If the initial data are compactly support, the solution u i (i = 1, 2, ..., k) also has compact support as long as it exists.

Remark 3.2. If the initial data are nontrivial, we can assume that they satisfy u0i(x) > 0 (i = 1, 2, ..., k). If not, u i (0, t) (i = 1, 2, ..., k) eventually become positive (compare with a Barenblatt solution of the corresponding equation).

Next, we define what called a subsolution and a supersolution for (1.1)-(1.2).

Definition 3.1. ( u 1 , u 2 , , u k - 1 , u k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq15_HTML.gifis a subsolution of (1.1)-(1.2) if it satisfies
u i t ( u i p i ) x x x > 0 , 0 < t < T , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ26_HTML.gif
(3.2)
- ( u i p i ) x ( 0 , t ) u i + 1 q i + 1 ( 0 , t ) , u k + 1 q k + 1 = u 1 q 1 , 0 < t < T , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ27_HTML.gif
(3.3)

Definition 3.2. We call ( ū 1 , ū 2 , , ū k - 1 , ū k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq16_HTML.gif a supersolution of (1.1)-(1.2) of it satisfies (3.2)-(3.3) with the opposite inequalities.

With these definitions of super and subsolutions, we can state a comparison lemma.

Lemma 3.1 Let ( ū 1 , ū 2 , , ū k - 1 , ū k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq16_HTML.gif be a supersolution and ( u 1 , u 2 , , u k - 1 , u k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq15_HTML.gif be a subsolution. If
u i ( x , 0 ) ū i ( x , 0 ) , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equk_HTML.gif
with
u i ( 0 , 0 ) ū i ( 0 , 0 ) , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equl_HTML.gif
then
u i ( x , t ) ū i ( x , t ) , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equm_HTML.gif

as long as both super and subsolutions exist.

Proof. It is standard, therefore we omit the details. Assume that the result is false. Let t0 be the maximum time such that
u i ( x , t ) ū i ( x , t ) , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equn_HTML.gif

up to t0. This time t0 must be positive, by continuity. At that time, we must have u j ( 0 , t 0 ) = ū j ( 0 , t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq17_HTML.gif for some j (1 ≤ jk). Let us assume that u 1 ( 0 , t 0 ) = ū 1 ( 0 , t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq18_HTML.gif. Now the result follows by an application of Hopf's lemma. Indeed, ū 1 - u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq19_HTML.gif satisfies a uniformly parabolic equation in a neighborhood of x = 0, attains a minimum at (0, t0), and the corresponding flux is greater or equal than zero, a contradiction.

Now we state a lemma that guarantees that, for certain initial data, the solution of (1.1)-(1.3) increases in time.

Lemma 3.2 Let u0i(x) be the initial data for (1.1) -(1.3) such that u0i(x) are smooth, satisfy the compatibility condition at the boundary and ( u 0 i p i ) x x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq20_HTML.gif . Then u i (x, t) increases in time, i.e., u it (x, t) ≥ 0 (i = 1, 2, ...,k).

Proof. Let w i = u it . Then, as the solutions are smooth (Theorem 3.1), we can differentiate to obtain the (w1, ..., w k ) is a solution of
w i t = ( p i u i p i - 1 w i ) x x , i = 1 , 2 , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ28_HTML.gif
(3.4)
- ( p i u i p i - 1 w i ) x ( 0 , t ) = q i + 1 u i + 1 q i + 1 - 1 w i + 1 ( 0 , t ) , q k + 1 = q 1 , u k + 1 = u 1 , w k + 1 = w 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ29_HTML.gif
(3.5)
with initial data satisfying
w i ( x , 0 ) 0 , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equo_HTML.gif

To conclude the proof we apply the maximum principle. Due to the degeneration of the equations this cannot be done directly. A standard regularization procedure is needed (see [8] for details).

Next, we deal with the problem of uniqueness versus non-uniqueness for (1.1)-(1.3) on the case of vanishing initial data (u0i(x) = 0, i = 1, 2, ..., k).

Theorem 3.2
  1. (a)

    Let l = 1 k ( 1 + p l ) > l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq21_HTML.gif. Then there exists a nontrivial solution with zero initial data that becomes positive at × = 0 instantaneously. Then there is no uniqueness for problem (1.1)-(1.3) with zero initial data.

     
  2. (b)

    Let l = 1 k ( 1 + p l ) l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq22_HTML.gif. Then the solution of (1.1)-(1.3) with zero initial data is unique.

     
Proof.
  1. (a)

    The self-similar solutions constructed in Theorem 2.2 become positive at x = 0 instantaneously.

     
  2. (b)
    We can construct small supersolution with the aid of the self-similar ones of exponential form that we found in Theorem 2.3. First, choose q ̃ 1 q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq23_HTML.gif such that 2 q ̃ 1 l = 2 k 2 q l = l = 1 k ( 1 + p l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq24_HTML.gif.
    ū i ( x , t ) = e α i ( t + τ ) f i ( x e - β i ( t + τ ) ) , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equp_HTML.gif
     
where α1 > 0 is arbitrary and β1, α i , β i , (i = 2, ..., k) are given by (2.7). Now we observe that ( ū 1 , ū 2 , , ū k - 1 , ū k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq25_HTML.gif be a supersolution is a supersolution of (1.1)-(1.3) as long as u ¯ 1 ( 0 , t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq26_HTML.gif. By the comparison Lemma 3.1, we obtain that every solution has initial data identically zero satisfies
ū i ( x , t ) u i ( x , t ) , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equq_HTML.gif

As ū i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq27_HTML.gif can be chosen as small as we want (using τ negative and large enough) we conclude that ū i 0 ( i = 1 , 2 , , k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq28_HTML.gif.

4 Blow-up versus global existence

We devote this section to prove Theorem 1.1. We borrow ideas from [8]. However, the fact that we are dealing with a system instead of a single equation forces us to develop a significantly different proof. We will organize the proof in several lemmas.

Our first lemma proves part (I) of Theorem 1.1.

Lemma 4.1. If l = 1 k ( 1 + p l ) l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq29_HTML.gif (i.e. det A ≥ 0), every nonnegative solution of (1.1)-(1.3) is global in time.

Proof. It is enough to construct global supersolutions with initial data as large as needed. We achieve this with the aid of the self-similar solutions of exponential form that we found in Theorem 2.3.

First we choose q ̃ 1 q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq30_HTML.gif such that 2 q ̃ 1 l = 2 k 2 q l = l = 1 k ( 1 + p l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq31_HTML.gif and we let
ū i ( x , t ) = e α i ( t + τ ) f i ( x e - β i ( t + τ ) ) , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equr_HTML.gif

where α1 > 0 is arbitrary and β1, α i , β i , (i = 2, ..., k) are given by (2.7). Now we observe that ( ū 1 , ū 2 , , ū k - 1 , ū k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq32_HTML.gif is a supersolution of (1.1)-(1.3) as long as ū 1 ( 0 , t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq33_HTML.gif. This can be done by choosing τ large enough. This also allows to assume ū i ( x , 0 ) u 0 i ( x ) ( i = 1 , 2 , , k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq34_HTML.gif. Then, by the comparison Lemma 3.1, we obtain that every solution is global.

Now we construct subsolutions with finite time blow-up.

Lemma 4.2. Let l = 1 k ( 1 + p l ) < l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq35_HTML.gif (i.e. det A < 0), then there exist compactly supported functions g i (i = 1, 2, ..., k), such that
u i ( x , t ) = ( T - t ) α i g i ( ξ i ) , ξ i = x ( T - t ) - β i , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equs_HTML.gif

is a subsolution of (1.1)-(1.2).

Proof. To satisfy (3.2) and (3.3), we need that
( g i p i ) ( ξ i ) - α i g i ( ξ i ) + β i ξ i g i ( ξ i ) , i = 1 , 2 , , k , - ( g i p i ) ( 0 ) g i + 1 q i + 1 ( 0 ) , i = 1 , 2 , , k , q k + 1 = q 1 , g k + 1 = g 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equt_HTML.gif
We choose
g i ( ξ i ) = A i ( a i - ξ i ) + 1 ( p i - 1 ) , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equu_HTML.gif
Inserting this in the equation, we get
p i ( p i - 1 ) 2 A i p i - 1 - α i ( a i - ξ i ) + - β i p i - 1 ξ i f o r 0 ξ i a i , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equv_HTML.gif
Hence, it is enough to impose
p i ( p i - 1 ) 2 A i p i - 1 - α i a i + | β i | p i - 1 a i , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equw_HTML.gif
that is
C i A i p i - 1 a i , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ30_HTML.gif
(4.1)
The boundary conditions impose
p i p i - 1 A i p i a i 1 ( p i - 1 ) A i + 1 q i + 1 a i + 1 q i + 1 ( p i + 1 - 1 ) , i = 1 , 2 , , k , A k + 1 = A 1 , q k + 1 = q 1 , a k + 1 = a 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ31_HTML.gif
(4.2)
Let
b i = A i a i 1 ( p i - 1 ) , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equx_HTML.gif
Then conditions (4.2) become
p i p i - 1 A i p i - 1 b i b i + 1 q i + 1 , i = 1 , 2 , , k , b k + 1 q k + 1 = b 1 q 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ32_HTML.gif
(4.3)

We fix b i = 1 (i = 1, 2, , k) and then A i large enough (and thus a i small) to satisfy (4.1) and (4.3).

Corollary 4.1 Let l = 1 k ( 1 + p l ) < l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq36_HTML.gif (i.e. det A < 0). Then there exist solutions of (1.1)-(1.3) that blow up in a finite time.

Proof. We only have to apply Lemma 3.1, to obtain that every solution (u1, , u k ) that begins above the subsolutions provided by Lemma 4.2 has finite time blow-up.

Lemma 4.3 Let l = 1 k ( 1 + p l ) < l = 1 k 2 q l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq36_HTML.gif(i.e. det A < 0). If there exists j ( 1 ≤ jk) such that α j + β j ≤ 0, then every nontrivial solution of (1.1)-(1.3) blows up in finite time.

Proof. Without loss of generality, we consider the case α1 + β1 ≤ 0.

Assume that there exists a global nonnegative nontrivial solution of (1.1)-(1.3), we make the following change of variables
φ i ( ξ i , τ ) = ( 1 + t ) - α i u i ( ξ i ( 1 + t ) β i , t ) , τ = log ( 1 + t ) , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ33_HTML.gif
(4.4)
These functions satisfy
φ i τ = ( φ i p i ) ξ i ξ i + β i ξ i φ i ξ i - α i φ i , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ34_HTML.gif
(4.5)
- ( φ i p i ) ξ i ( 0 , τ ) = φ i + 1 q i + 1 ( 0 , τ ) , i = 1 , 2 , , k , φ k + 1 q k + 1 = φ 1 q 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ35_HTML.gif
(4.6)

As u i (x, t) (i = 1, 2, ..., k) are by hypothesis global, the same is true for φ i (i = 1, 2, ..., k,). We will construct a solution ( φ ^ 1 , , φ ^ k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq37_HTML.gif to system (4.5)-(4.6) increasing with time, with initial data ( φ ^ 0 1 , , φ ^ 0 k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq38_HTML.gif such that φ ^ 0 i ( ξ i ) u i ( ξ i , 0 ) ( i = 1 , 2 , , k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq39_HTML.gif. We will prove that ( φ ^ 1 , , φ ^ k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq37_HTML.gif cannot exists globally, thus contradicting the global existence of (u1, , u k ). In order to achieve our goal, we use an adaptation for systems of the general monotonicity for single quasilinear equation described in [19].

We take initial data ( φ ^ 0 1 , , φ ^ 0 k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq38_HTML.gif satisfying
( φ ^ 0 i p i ) ξ i ξ i + β i ξ i ( φ ^ 0 i ) ξ i - α i φ ^ 0 i 0 , i = 1 , 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equy_HTML.gif
and the compatibility conditions
- ( φ ^ 0 i p i ) ξ i ( 0 ) = φ ^ 0 i + 1 q i + 1 ( 0 ) , i = 1 , 2 , , k , φ ^ 0 k + 1 q k + 1 = φ ^ 0 1 q 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equz_HTML.gif

Hence, arguing as in Lemma 3.2, we have that φ ^ i τ 0 ( i = 1 , 2 , , k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq40_HTML.gif.

Following an idea for scalar equation from [8], we set
φ ^ 0 1 ( ξ 1 ) = h ( ξ 1 + b ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equaa_HTML.gif
where h is the Barenblatt profile
h ( ξ 1 ) = a p 1 ( c - ξ 1 2 ) + 1 ( p 1 - 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equab_HTML.gif
Then we have
( φ ^ 0 1 p 1 ) ξ 1 ξ 1 + β 1 ξ 1 ( φ ^ 0 1 ) ξ 1 - α 1 φ ^ 0 1 = - 1 p 1 + 1 b h ξ 1 ( ξ 1 + b ) + ( β 1 - 1 p 1 + 1 ) ξ 1 h ξ 1 ( ξ 1 + b ) + ( - α 1 - 1 p 1 + 1 ) h ( ξ 1 + b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equac_HTML.gif

The last expression is nonnegative if β1 - 1/(p1 + 1) ≤ 0 and -α1 - 1/(p1 + 1) ≥ 0. But these two conditions are equivalent α1 + β1 ≤ 0.

Now we take α ̃ i , β ̃ i > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq41_HTML.gif such that α ̃ i α i , β ̃ i β i ( i = 1 , 2 , , k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq42_HTML.gif. We take as φ ^ 0 i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq43_HTML.gif a solution to
( φ ^ 0 i p i ) = - β ̃ i ξ i φ ^ 0 i + α ̃ i φ ^ 0 i , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equad_HTML.gif
There is one-parameter family of solution to this equation (see Theorem 2.4), with φ ^ 0 i 0 , φ ^ 0 i 0 ( i = 2 , , k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq44_HTML.gif. Hence,
( φ ^ 0 i p i ) - β i ξ i φ ^ 0 i + α i φ ^ 0 i , i = 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equae_HTML.gif
Moreover,
φ ^ 0 i ( 0 ) = U i , ( φ ^ 0 i p i ) ( 0 ) = U i ( p i + 1 ) 2 V * i , i = 2 , , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equaf_HTML.gif

where V*i< 0 is a constant and U i is the free parameter.

We still have to control the boundary conditions. In order to do this, we choose the constants c, b and U i (i = 2, ...,k) conveniently. They have to satisfy
2 p 1 a p 1 p 1 - 1 b ( c - b 2 ) 1 ( p 1 - 1 ) = U 2 q 2 , b ( 0 , c 1 2 ) , - V * i U i ( p i + 1 ) 2 = a p 1 q i + 1 ( c - b 2 ) q i + 1 ( p 1 - 1 ) , i = 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equag_HTML.gif
Thus, we choose
U 2 = c 2 b 2 q 3 ( 2 q 2 q 3 - p 2 - 1 ) , U i = c i b 2 ( p 2 - 1 ) q i + 1 ( ( p i + 1 ) ( 2 q 2 q 3 - p 2 - 1 ) ) , ( i = 3 , , k , q k + 1 = q 1 ) , c = b 2 + γ b ( p 1 - 1 ) ( p 2 - 1 ) ( 2 q 2 q 3 - p 2 - 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equah_HTML.gif

where c i (i = 2, ..., k) and γ are positive constants. Taking b small enough, the initial data ( φ ^ 0 1 , , φ ^ 0 k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq45_HTML.gif is below (u1(ξ1,0), ...,u k (ξ k , 0)). This can be done as u0i(i = 1, 2, ... k) can be assumed to be positive at the origin.

To conclude the proof, we will show that ( φ ^ 1 , , φ ^ k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq46_HTML.gif converge to a self-similar profile that does not exist in this range of parameters.

Lemma 4.4. There exists j ( 1 ≤ jk) such that
lim τ φ ^ j ( ξ j , τ ) = φ ̃ j ( ξ j ) < , ξ j > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ36_HTML.gif
(4.7)
Proof. It is clear that φ ^ i ξ i 0 ( i = 1 , 2 , , k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq47_HTML.gif. Let us suppose that
φ ^ i ( ξ i , τ ) u n i f o r m l y i n ( 0 , ξ i 0 ) , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equai_HTML.gif
In the original variables ( u ^ 1 , , u ^ k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq48_HTML.gif, we have that for any M > 0 there is a value such that
( 1 + t 0 ) α i M u ^ i ( x , t 0 ) f o r 0 < x ( 1 + t 0 ) - β i < ξ i 0 , i = 1 , 2 , , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ37_HTML.gif
(4.8)
Now we will check that, under these conditions, we can put one of the blowing up subsolutions constructed in Lemma 4.2 below these data. This would lead to a contradiction, as ( u ^ 1 , , u ^ k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq48_HTML.gif is global. In order to do this, we need
( 1 + t 0 ) a 1 M A 1 a 1 1 ( p 1 - 1 ) T α 1 , ξ 1 0 ( 1 + t 0 ) β 1 a 1 T β 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ38_HTML.gif
(4.9)
The first equation says that the height at x = 0 of u ^ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq49_HTML.gif is bigger than that of u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq50_HTML.gif, and the second says that the support of u ^ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq49_HTML.gif is bigger than the support of u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq50_HTML.gif. Imposing analogous conditions for u ^ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq51_HTML.gif and u i ( i = 2 , , k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq52_HTML.gif we get
( 1 + t 0 ) a i M A i a i 1 ( p i - 1 ) T α i , ξ i 0 ( 1 + t 0 ) β i a i T β i . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ39_HTML.gif
(4.10)

Taking T = 1 + t0, then a i small enough and A i large enough (i = 1, 2, ..., k), and then M large, then the 2k conditions (4.9)-(4.10) are fulfilled.

Let us remark this parametric evolution comparison method to prove global non-existence for arbitrary data first introduced in [20], for scalar quasilinear heat equation.

End of the proof of Lemma 4.3. Let us assume that (4.7) holds. Using standard arguments, see [8], we may pass to the limit to obtain that
( φ ̃ 1 p 1 ) ξ 1 ξ 1 + β 1 ξ 1 φ ̃ 1 ξ 1 - α 1 φ ̃ 1 = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ40_HTML.gif
(4.11)
Let z = φ ̃ 1 p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq53_HTML.gif, then
z ξ 1 ξ 1 + β 1 p 1 ξ 1 z ( 1 - p 1 ) p 1 z ξ 1 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equaj_HTML.gif
Hence, in (0, ξ10), zc > 0,
z ξ 1 ξ 1 C z ξ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equak_HTML.gif
We conclude that z and therefore φ ̃ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq54_HTML.gif cannot be unbounded at ξ1 = 0. In particular, 0 < φ ̃ 1 ( 0 ) C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq55_HTML.gif Then, considering the regularity of φ ̃ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq54_HTML.gif in the region where φ ̃ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq56_HTML.gif, we can pass to the limit in the boundary condition for ( φ ^ 1 p 1 ) ξ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq57_HTML.gif to obtain that
- ( φ ̃ 1 p 1 ) ξ 1 ( 0 ) = φ ̃ 2 q 2 ( 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equ41_HTML.gif
(4.12)

However, as α1 + β1 ≤ 0, problem (4.11)-(4.12) does not have a nontrivial solution, see Theorem 2.4.

If (4.7) holds for some j > 1, we can proceed as before to obtain that φ ̃ j ( 0 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq58_HTML.gif. Thus, we can pass to the limit in the boundary condition for φ ^ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq59_HTML.gif, obtaining
- ( φ ̃ j p j ) ξ j ( 0 ) = φ ̃ j + 1 q j + 1 ( 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_Equal_HTML.gif

As φ ̃ j + 1 ( 0 ) φ ̃ j + 1 ( ξ j + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq60_HTML.gif, this implies that φ ̃ j + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-15/MediaObjects/13661_2011_Article_16_IEq61_HTML.gif is finite for every ξj+1≥ 0. We get the same contradiction as before.

Declarations

Acknowledgements

We would like to thank Professor Dimitru Motreanu, Christopher Rualizo and the referees for their valuable comments and suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, Jiangxi Vocational College of Finance and Economics

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© Xu and Song; licensee Springer. 2011

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