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 ,

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 ,

with continuous, nonnegative initial data

u i ( x , 0 ) = u 0 i ( x ) , ( i = 1 , 2 , … , k ) , x > 0 ,

compactly supported in ℝ₊, 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 4 5 6 7 ). 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 .

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

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

In 9 10 11 , 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

A series of standard computations yield

det A = ∏ l = 1 k ( 1 + p l ) - ∏ l = 1 k 2 q l .

We shall see that det A = 0 is the critical global existence parameter equation. Let (α ₁, α ₂, ..., 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 ,

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 ) .

We define

β i = 1 + ( p i - 1 ) α i 2 , ( i = 1 , 2 , ⋅ ⋅ ⋅ , k ) .

Theorem 1.1.

(I) If ∏ l = 1 k ( 1 + p l ) ≥ ∏ l = 1 k 2 q l (i.e. det A ≥ 0), every nonnegative solution of (1.1)-(1.3) is global in time.

(II) If ∏ l = 1 k ( 1 + p l ) < ∏ l = 1 k 2 q l (i.e. det A < 0) and there exists j (1 ≤ j ≤ k) such that α_j + β_j ≤ 0, then every nonnegative, nontrivial solution blows up in finite time.

(III) If ∏ l = 1 k ( 1 + p l ) < ∏ l = 1 k 2 q l (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 )

and the critical Fujita type parameter equation is

min { α 1 + β 1 , α 2 + β 2 , … , α k + β k } = 0 .

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).

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 .

If

∏ l = 1 k ( 1 + p l ) < ∏ l = 1 k 2 q l ,

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 ℝ₊ 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 .

(a) If

∏ l = 1 k ( 1 + p l ) > ∏ l = 1 k 2 q l ,

then there exist functions f_i positive in ℝ₊, 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 ,

then there exist functions f_i, compactly supported in ℝ₊, 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 ≤ j ≤ k) such that α _j + β_j ≤ 0, there are no profiles f_i ∈ L ¹(ℝ₊) 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 .

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 .

If

∏ l = 1 k ( 1 + p l ) = ∏ l = 1 k 2 q l ,

for any α ₁ > 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 ) .

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 13 14 15 ):

Theorem 2.4. Let a, b, V ∈ ℝ 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 f ₁ of

( f 1 p 1 ) ″ ( η ) + a η f 1 ′ ( η ) = b f 1 ( η ) , 0 < η < ∞ ,

f 1 ( 0 ) = U ,

( f 1 p 1 ) ′ ( 0 ) = V ,

and let S _B denote the set of values (U, V) for which there exists a bounded, positive, classical solution f ₁ of (2.8)-(2.10).

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

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

(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 } and S _B = {(U, V): 0 ≤ U < ∞, U ( p 1 + 1 ) ∕ 2 V * < V < ∞ } , where V _* > 0 if a + b < 0, V _* = 0 if a + b = 0, and V _* < 0 if a + b > 0.

(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 } and S _B = Ø.

(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 ^(p1+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 = ((p ₁ - 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 q ₂ = (p ₁ + 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 .

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 .

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 . 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 ,

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 .

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 ,

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 ℝ₊ 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 .

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 .

As ∏ l = 1 k ( 1 + p l ) ≠ ∏ l = 1 k 2 q l (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 .

Thus, f_i satisfy (2.8) with coefficients a_i = β_i , b_i = α_i (i = 1, 2, ..., k).

(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).

(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 .

Thus,

A ( α 1 , α 2 , … , α k - 1 , α k ) T = ( 0 , 0 , … , 0 , 0 ) T .

There are nontrivial solutions of (2.18) if and only if ∏ l = 1 k ( 1 + p l ) = ∏ l = 1 k 2 q l (i.e. det A = 0). In this case, β ₁, α_I , β_i (i = 2, ...,k) are related to α ₁ 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 α ₁ > 0, then β ₁, 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 (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).

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 u _0i (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 ) ,

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 ,

with r > 1. We define the operator M q i + 1 : C ( [ 0 , τ ] ) → C ( [ 0 , τ ] ) as M q i + 1 ( h ) ( t ) = w q i + 1 ( 0 , t ) , where w(x, t) is the unique solution of (3.1) with r = p_i and initial condition w ₀(x) = u _0i (x) i = 1 , 2 , … , k , M q k + 1 = M q 1 , w q k + 1 = w q 1 .

It has been proved in 17 that M q i ( i = 1 , 2 , . . . , k ) 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 ) . 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 ) (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 u _0i (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 ) is 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 ,

- ( 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 .

Definition 3.2. We call ( ū 1 , ū 2 , … , ū k - 1 , ū k ) 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 ) be a supersolution and ( u 1 , u 2 , … , u k - 1 , u k ) be a subsolution. If

u i ( x , 0 ) ≤ ū i ( x , 0 ) , i = 1 , 2 , … , k ,

with

u i ( 0 , 0 ) ≤ ū i ( 0 , 0 ) , i = 1 , 2 , … , k ,

then

u i ( x , t ) ≤ ū i ( x , t ) , i = 1 , 2 , … , k ,

as long as both super and subsolutions exist.

Proof. It is standard, therefore we omit the details. Assume that the result is false. Let t ₀ be the maximum time such that

u i ( x , t ) ≤ ū i ( x , t ) , i = 1 , 2 , … , k ,

up to t ₀. This time t ₀ must be positive, by continuity. At that time, we must have u j ( 0 , t 0 ) = ū j ( 0 , t 0 ) for some j (1 ≤ j ≤ k). Let us assume that u 1 ( 0 , t 0 ) = ū 1 ( 0 , t 0 ) . Now the result follows by an application of Hopf's lemma. Indeed, ū 1 - u 1 satisfies a uniformly parabolic equation in a neighborhood of x = 0, attains a minimum at (0, t ₀), 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 u _0i (x) be the initial data for (1.1) -(1.3) such that u _0i (x) are smooth, satisfy the compatibility condition at the boundary and ( u 0 i p i ) x x ≥ 0 . 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 (w ₁, ..., w_k ) is a solution of

w i t = ( p i u i p i - 1 w i ) x x , i = 1 , 2 , … k ,

- ( 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 ,

with initial data satisfying

w i ( x , 0 ) ≥ 0 , i = 1 , 2 , … , k .

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 (u _0i (x) = 0, i = 1, 2, ..., k).

Theorem 3.2

(a) Let ∏ l = 1 k ( 1 + p l ) > ∏ l = 1 k 2 q l . 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.

(b) Let ∏ l = 1 k ( 1 + p l ) ≤ ∏ l = 1 k 2 q l . Then the solution of (1.1)-(1.3) with zero initial data is unique.

Proof.

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

(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 such that 2 q ̃ 1 ∏ l = 2 k 2 q l = ∏ l = 1 k ( 1 + p l ) .

ū i ( x , t ) = e α i ( t + τ ) f i ( x e - β i ( t + τ ) ) , i = 1 , 2 , … , k ,

where α ₁ > 0 is arbitrary and β ₁, 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 ) be a supersolution is a supersolution of (1.1)-(1.3) as long as u ¯ 1 ( 0 , t ) ≤ 1 . 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 .

As ū i can be chosen as small as we want (using τ negative and large enough) we conclude that ū i ≡ 0 ( i = 1 , 2 , … , k ) .

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 (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 such that 2 q ̃ 1 ∏ l = 2 k 2 q l = ∏ l = 1 k ( 1 + p l ) and we let

ū i ( x , t ) = e α i ( t + τ ) f i ( x e - β i ( t + τ ) ) , i = 1 , 2 , … , k ,

where α ₁ > 0 is arbitrary and β ₁, 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 ) is a supersolution of (1.1)-(1.3) as long as ū 1 ( 0 , t ) ≥ 1 . 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 ) . 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 (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 ,

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 .

We choose

g i ( ξ i ) = A i ( a i - ξ i ) + 1 ∕ ( p i - 1 ) , i = 1 , 2 , … , k .

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 .

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 ,

that is

C i A i p i - 1 ≥ a i , i = 1 , 2 , … , k .

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 .

Let

b i = A i a i 1 ∕ ( p i - 1 ) , i = 1 , 2 , … , k .

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 .

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 (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 (u ₁, ⋯, 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 (i.e. det A < 0). If there exists j (1 ≤ j ≤ k) 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 α ₁ + β ₁ ≤ 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 .

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 ,

- ( 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 .

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 ) to system (4.5)-(4.6) increasing with time, with initial data ( φ ^ 0 1 , … , φ ^ 0 k ) such that φ ^ 0 i ( ξ i ) ≤ u i ( ξ i , 0 ) ( i = 1 , 2 , … , k ) . We will prove that ( φ ^ 1 , … , φ ^ k ) cannot exists globally, thus contradicting the global existence of (u ₁, ⋯, 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 ) 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 ,

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 .

Hence, arguing as in Lemma 3.2, we have that φ ^ i τ ≥ 0 ( i = 1 , 2 , … , k ) .

Following an idea for scalar equation from 8 , we set

φ ^ 0 1 ( ξ 1 ) = h ( ξ 1 + b ) ,

where h is the Barenblatt profile

h ( ξ 1 ) = a p 1 ( c - ξ 1 2 ) + 1 ∕ ( p 1 - 1 ) .

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 ) .

The last expression is nonnegative if β ₁ - 1/(p ₁ + 1) ≤ 0 and -α ₁ - 1/(p ₁ + 1) ≥ 0. But these two conditions are equivalent α ₁ + β ₁ ≤ 0.

Now we take α ̃ i , β ̃ i > 0 such that α ̃ i ≥ α i , β ̃ i ≥ β i ( i = 1 , 2 , … , k ) . We take as φ ^ 0 i a solution to

( φ ^ 0 i p i ) ″ = - β ̃ i ξ i φ ^ 0 i ′ + α ̃ i φ ^ 0 i , i = 1 , 2 , … , k .

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 ) . Hence,

( φ ^ 0 i p i ) ″ ≥ - β i ξ i φ ^ 0 i ′ + α i φ ^ 0 i , i = 2 , … , k .

Moreover,

φ ^ 0 i ( 0 ) = U i , ( φ ^ 0 i p i ) ′ ( 0 ) = U i ( p i + 1 ) ∕ 2 V * i , i = 2 , … , k ,

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 .