Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix

Nabila Barrouk; Karima Abdelmalek

doi:10.1515/dema-2023-0122

Article Open Access

Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix

Nabila Barrouk and Karima Abdelmalek

Published/Copyright: October 29, 2024

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

The aim of this work is to study the global existence in time of solutions for the tridiagonal system of reaction-diffusion by order m . Our techniques of proof are based on compact semigroup methods and some L 1 -estimates. We show that global solutions exist. Our investigation can be applied for a wide class of nonlinear terms of reaction.

Keywords: global solution; semigroups; local solution; reaction-diffusion systems

MSC 2010: 35K57; 35K40; 35K45; 37L05; 35K55

1 Introduction

Reaction-diffusion systems are systems involving constituents locally transformed into each other by chemical reactions and transported in space by diffusion. They arise, in many applications, in chemistry, chemical engineering and biology. They have been the subject of countless studies in the past few decades. One of the most important aspects of this broad field is proving the global existence of solutions under certain assumptions and restrictions.

The aim of this work is to study the global existence in time of solutions of an m -component reaction-diffusion system with tridiagonal symmetric Toeplitz diffusion matrix and homogeneous boundary conditions of the form

(1) ∂ U ∂ t − A m Δ U = F ( U ) on ] 0 , + ∞ [ × Ω , ∂ η U = 0 or U = 0 on ] 0 , + ∞ [ × ∂ Ω , U ( 0 , x ) = U 0 ( x ) on Ω .

Our techniques of proof are based on compact semigroup methods and some L 1 -estimates, and we show that global solutions exist.

We consider the m -equations of reaction-diffusion system (1), with m ≥ 2 , where Ω is an open bounded domain of class C 1 in R n , ( n ≥ 1 ), the vectors U , F , U 0 and the matrix A m are defined as follows:

U = ( u 1 , … , u m ) T = ( ( u i ) i = 1 m ) T , F = ( F 1 , … , F m ) T = ( ( F i ) i = 1 m ) T , U 0 = ( u 1 0 , … , u m 0 ) T = ( ( u i 0 ) i = 1 m ) T .

(2) A m = a 1 b 1 0 ⋯ 0 b 1 a 2 b 2 ⋱ ⋮ 0 b 2 a 3 ⋱ 0 ⋮ ⋱ ⋱ ⋱ b m − 1 0 ⋯ 0 b m − 1 a m .

The constants ( a i ) i = 1 m , ( b i ) i = 1 m − 1 are supposed to be strictly positive and satisfy the condition

(3) 4 b i 2 cos 2 π m + 1 < a i a i + 1 ,

which reflects the parabolicity of the system and implies at the same time that the diffusion matrix A m is positive definite. That means the eigenvalues ( λ s ) s = 1 m , ( λ 1 > λ 2 > … > λ m ) of A m are positive.

In 2 × 2 diagonal case, Alikakos [1] established global existence and L ∞ -bounds of solutions for positive initial data for

f ( u , v ) = − g ( u , v ) = − u v σ , and 1 < σ < n + 2 n .

Masuda [2] showed that solutions to this system exist globally for every σ > 1 and converge to a constant vector as t → + ∞ .

Haraux and Youkana [3] have generalized the method of Masuda to nonlinearities f ( u , v ) = g ( u , v ) = − u Ψ ( v ) satisfying

lim v → + ∞ [ log ( 1 + Ψ ( v ) ) ] v = 0 .

In the same direction, Kouachi [4] has proved the global existence of solutions for two-component reaction-diffusion systems with a general full matrix of diffusion coefficients, nonhomogeneous boundary conditions and polynomial growth conditions on the nonlinear terms and he obtained in [5] the global existence of solutions for the same system with homogeneous Neumann boundary conditions.

In [6,7], we find new developed methods based on truncation functions, fixed point theorems and compactness, etc. to prove the existence of global solutions.

In [8–12], m -equations of reaction-diffusion system (1) with the homogeneous and nonhomogeneous boundary conditions were treated with nonlinearities of exponential growth via a Lyapunov functional.

In this present study, we shall generalize the results obtained in [13–16]. In [13–16], the authors obtained a global existence of solutions for the coupled reaction-diffusion semilinear system with diagonal by order 2, and m, triangular, and full matrix of diffusion coefficients. By combining the compact semigroup methods and some L ¹ estimates, we show that global solutions exist.

We prove the existence of global solutions of m × m tridiagonal symmetric Toeplitz diffusion matrix and homogeneous Neumann or Dirichlet conditions. The reaction terms are assumed to be of polynomial growth.

The initial data are assumed to be in the regions:

(4) ∑ S,Z = U 0 ∈ R m : w z 0 = ⟨ V z , U 0 ⟩ ≤ 0 if z ∈ Z w s 0 = ⟨ V s , U 0 ⟩ ≥ 0 if s ∈ S ,

where

S ∩ Z = ϕ , S ∪ Z = { 1 , 2 , … , m } .

The notation ⟨ . , . ⟩ denotes the inner product in R m and V s = ( v s 1 , … , v s m ) T the eigenvector of the diffusion matrix A m associated with the eigenvalue ( λ s ) s = 1 m . Hence, we can see that there are 2 m regions.

2 Statement of the main result

2.1 Assumptions

The following assumptions are also made on the function Ψ defined by

Ψ = ( ( Ψ s ) s = 1 m ) T , Ψ s = ⟨ ( − 1 ) i s V s , F ⟩ , i s = 1 , 2 .

Ψ s are continuously differentiable on R + m and Ψ s , s = 1 , m ¯ , are quasi-positive functions, which means that, for s = 1 , m ¯
[ w 1 ≥ 0 , … , w s − 1 ≥ 0 , w s + 1 ≥ 0 , … , w m ≥ 0 ] ,
which implies
(5) Ψ s ( w 1 , … , w s − 1 , 0 , w s + 1 , … , w m ) ≥ 0 .
These conditions on Ψ guarantee local existence of unique, nonnegative classical solutions on a maximal time interval [ 0 , T max ) , see Hollis and Morgan [17].
The inequality
(6) ⟨ S , Ψ ( W ) ⟩ ≤ C 1 ( 1 + ⟨ W , 1 ⟩ ) ,
such that
W = ( w 1 , … , w m ) , S = ( d 1 , d 2 , … , d m − 1 , 1 ) ,
for all w s ≥ 0 , s = 1 , … , m and all constants d s satisfy d s ≥ d ¯ s , s = 1 , … , m − 1 , where C 1 ≥ 0 and d ¯ s are positive constants sufficiently large.
W s 0 , i = 1 , m ¯ are nonnegative functions in L 1 ( Ω ) .
There exists a positive constant C 2 independent of w s , s = 1 , m ¯ such that
(7) ∑ s = 1 m Ψ s ( w 1 , … , w m ) ≤ C 2 ∑ s = 1 m w s , ∀ w s ≥ 0 , s = 1 , m ¯ .

3 Eigenvalues and eigenvectors of the diffusion matrix

The usual norms in the spaces L 1 ( Ω ) , L ∞ ( Ω ) and C ( Ω ¯ ) are, respectively, by

∥ u ∥ 1 = ∫ Ω ∣ u ( x ) ∣ d x ,

∥ u ∥ ∞ = ess sup x ∈ Ω ∣ u ( x ) ∣ and ∥ u ∥ C ( Ω ¯ ) = max x ∈ Ω ¯ ∣ u ( x ) ∣ .

For any initial data in C ( Ω ¯ ) or L ∞ ( Ω ) , local existence and uniqueness of solutions to the initial value problem (1) follow from the basic existence theory for abstract semilinear differential equations (Friedman [18], Henry [19], and Pazy [20]).

The aim in this section is to derive a three-term recurrence relation for the characteristic polynomial of the matrix A m (of dimension m × m ) in terms of matrices of dimensions ( m − 1 ) × ( m − 1 ) and ( m − 2 ) × ( m − 2 ) . Additionally, this recurrence relation will help in determining the eigenvectors of the matrix A_m . The eigenvalues of the matrix A m are the values of λ that satisfy the equation: det ( A m − λ I m ) = 0 . We denote the characteristic polynomial of A m , A m − 1 , A m − 2 by ϕ m ( λ ) , ϕ m − 1 ( λ ) , ϕ m − 2 ( λ ) , respectively.

Lemma 1

[21] Let A m be the tridiagonal matrix defined in (2). The eigenvalues of A m are distinct and interlace strictly with the eigenvalues of A m − 1 and A m − 2 , for m ≥ 2 .

(8) ϕ 0 ( λ ) = 1 , ϕ 1 ( λ ) = a 1 − λ , ϕ m ( λ ) = a m ϕ m − 1 ( λ ) − b m − 1 2 ϕ m − 2 ( λ ) .

Lemma 2

[22] Let A m be the real symmetric tridiagonal matrix defined in (2), with diagonal entries positive. If

4 b i 2 cos 2 π m + 1 < a i a i + 1 , i = 1 , … , m − 1 ,

then A m is positive definite.

Lemma 3

The real symmetric tridiagonal matrix A m , defined in (2), is positive definite if and only if its principal minors det A s , for s = 1 , … , m , are positive.

Proof

See Andelic and da Fonseca [22].□

Lemma 4

The eigenvector of the matrix A m given in (2) associated with eigenvalue λ s , for s = 1 , … , m is given by V s = ( v s 1 , … , v s m ) T , where v s ℓ ( ℓ = 1 , … , m ) are given by the following expressions:

(9) v s m = 1 , v s ( m − 1 ) = λ s − a m b m − 1 , v s ( ℓ − 1 ) = − b ℓ v s ( ℓ + 1 ) + ( a ℓ − λ s ) v s ℓ b ℓ − 1 , ℓ = 2 , … , m − 1 .

Proof

Recall that the diffusion matrix is positive definite, hence its eigenvalues are necessarily positive, the eigenvectors of the diffusion matrix associated with the eigenvalues λ ℓ are defined as V = ( v 1 , … , v m ) T . For an eigenpair ( λ , V ) , the components in A m V = λ V are

a 1 v 1 + b 1 v 2 = λ v 1 , b ℓ − 1 v ℓ − 1 + a ℓ v ℓ + b ℓ v ℓ + 1 = λ v ℓ , 2 ≤ ℓ ≤ m − 1 , b m − 1 v m − 1 + a m v m = λ v m .

If v m = 0 , then under the assumption b ℓ ≠ 0 for all ℓ = 1 , … , m − 1 we have that all v ℓ are zero. We can therefore take v m = 1 and V = ( v 1 , … , v m − 1 ) is a solution of upper triangular system

b ℓ − 1 v ℓ − 1 + ( a ℓ − λ ) v ℓ + b ℓ v ℓ + 1 = 0 , 2 ≤ ℓ ≤ m − 2 , b m − 2 v m − 2 + ( a m − 1 − λ ) v m − 1 = − b m − 1 , b m − 1 v m − 1 = λ − a m .

The solution of this system is given by

□ v m − 1 = λ − a m b m − 1 , v ℓ − 1 = − b ℓ v ℓ + 1 + ( a ℓ − λ ) v ℓ b ℓ − 1 , ℓ = 2 , … , m − 1 .

3.1 Formulation of the result

Since the initial conditions are in ∑ S,Z , under assumptions (A1) and (A2), the next proposition says that the classical solution of system (1) remains in ∑ S,Z for all t in [ 0 , T max ) .

Proposition 1

Suppose that assumptions (A1)–(A2) are satisfied. Then, for any U 0 in ∑ S,Z , the classical solution U of system (1) on [ 0 , T max ) × Ω remains in ∑ S,Z for all t in [ 0 , T max ) .

Usually, to construct an invariant regions for systems such as (1) we make a linear change of variables u i to obtain a new equivalent system with diagonal diffusion matrix for which standard techniques can be applied to deduce global existence [8–12].

Let V s = ( v s 1 , … , v s m ) T be an eigenvector of the matrix A m associated with its eigenvalue ( λ s ) s = 1 m where λ 1 > λ 2 > … > λ m . Multiplying the kth equation of (1) by ( − 1 ) i s V s k , i s = 1 , 2 and k = 1 , … , m , and adding the resulting equations, we obtain

(10) ∂ W ∂ t − diag ( λ 1 , λ 2 , … , λ m ) Δ W = Ψ ( W ) on ] 0 , + ∞ [ × Ω , ∂ η W = 0 or W = 0 on ] 0 , + ∞ [ × ∂ Ω , W ( 0 , x ) = W 0 ( x ) on Ω ,

where

W = ( ( w s ) s = 1 m ) T , w s = ⟨ ( − 1 ) i s V s , U ⟩ , Ψ = ( ( Ψ s ) s = 1 m ) T , Ψ s = ⟨ ( − 1 ) i s V s , F ⟩ , W 0 = ( ( w s 0 ) s = 1 m ) T , w s 0 = ⟨ ( − 1 ) i s V s , U 0 ⟩ .

Proposition 2

System (10) admits a unique classical solution W on [ 0 , T max ) × Ω , where

T max ( ∥ w 1 0 ∥ ∞ , ∥ w 2 0 ∥ ∞ , … , ∥ w m 0 ∥ ∞ )

denotes the eventual blow-up time. Furthermore, if T max < + ∞ , then

lim t → T max ∑ s = 1 m ∥ w s ( t , . ) ∥ ∞ = + ∞ .

Therefore, if there exists a positive constant C such that

∑ s = 1 m ∥ w s ( t , . ) ∥ ∞ ≤ C for a l l t ∈ [ 0 , T max ) ,

then, T max = + ∞ .

3.2 Compactness of the solution

Lemma 5

We consider the generator A of a semigroup of contractions S t and a function F : X → X which is Lipschitz continuous on bounded sets. We consider the semilinear evolution equation

(5.1) d u d t − A u = F u t , u 0 = u 0 .

in classical form with u ∈ C 0 , T , D A ∩ C 1 0 , T , X , and u 0 ∈ X as well as the associated formulation

u t = S t u 0 + ∫ 0 t S t − s F u s d s .

(11) u ∈ C ( [ 0 , T ] , D ( A ) ) ∩ C 1 ( [ 0 , T ] , X ) , d u d t − A u = F ( u ( t ) ) , u ( 0 ) = u 0 ,

admits a unique solution u verifying

(12) u ( t ) = S ( t ) u 0 + ∫ 0 t S ( t − s ) F ( u ( s ) ) d s , ∀ t ∈ [ 0 , T max [ .

Proof

See Pazy [20].□

In this section, we will give a compactness result of operator L defining the solution of problem (11) in the case where the initial value equals zero [ u ( 0 ) = 0 ] , i.e.,

(13) L ( F ) ( t ) = u ( t ) = ∫ 0 t S ( t − s ) F ( u ( s ) ) d s , ∀ t ∈ [ 0 , T ] .

Theorem 1

[13–16,23] If for all t > 0 , the operators S ( t ) are compact, then L is compact of L 1 ( [ 0 , T ] , X ) in L 1 ( [ 0 , T ] , X ) .

Proof

Step 1. To show that S ( λ ) L : F → S ( λ ) L ( F ) is compact in L 1 ( [ 0 , T ] , X ) , it suffices to prove that the set { S ( λ ) L ( F ) ( t ) ; ∥ F ∥ 1 ≤ 1 } is relatively compact in L 1 ( [ 0 , T ] , X ) , ∀ t ∈ [ 0 , T ] .

Since S ( t ) is compact, the application t → S ( t ) is continuous of ] 0 , + ∞ [ in £ ( X ) ; therefore,

∀ ε > 0 , ∀ δ > 0 , ∃ η > 0 , ∀ 0 ≤ h ≤ η , ∀ t ≥ δ , ∥ S ( t + h ) − S ( t ) ∥ £ ( X ) ≤ ε ,

By choosing λ = δ , we have for 0 ≤ t ≤ T − h

S ( λ ) u ( t + h ) − S ( λ ) u ( t ) = ∫ 0 t + h S ( λ + t + h − s ) F ( u ( s ) ) d s − ∫ 0 t S ( λ + t − s ) F ( u ( s ) ) d s = ∫ t t + h S ( λ + t + h − s ) F ( u ( s ) ) d s + ∫ 0 t ( S ( λ + t + h − s ) − S ( λ + t − s ) ) F ( u ( s ) ) d s ,

where from

∥ S ( λ ) u ( t + h ) − S ( λ ) u ( t ) ∥ X ≤ ∫ t t + h ∥ F ( u ( s ) ) ∥ X d s + ε ∫ 0 t ∥ F ( u ( s ) ) ∥ X d s

we define v ( t ) by

v ( t ) = u ( t ) if 0 ≤ t ≤ T 0 otherwise ,

therefore,

∥ S ( λ ) v ( t + h ) − S ( λ ) v ( t ) ∥ 1 ≤ ( h + ε T ) ∥ F ( u ( s ) ) ∥ 1 ,

which implies that { S ( λ ) v , ∥ F ∥ 1 ≤ 1 } is equi-integrable, then { S ( λ ) L ( F ) ( t ) , ∥ F ∥ 1 ≤ 1 } is relatively compact in L 1 ( [ 0 , T ] , X ) , which means that S ( λ ) L is compact.

Step 2. We prove that S ( λ ) L converges to L when λ tends to 0 in L 1 ( [ 0 , T ] , X ) . We have

S ( λ ) u ( t ) − u ( t ) = ∫ 0 t S ( λ + t − s ) F ( u ( s ) ) d s − ∫ 0 t S ( t − s ) F ( u ( s ) ) d s .

So, for t ≥ δ , we have

∥ S ( λ ) u ( t ) − u ( t ) ∥ ≤ ∫ δ t ∥ S ( λ + s ) − S ( s ) ∥ £ ( X ) ∥ F ( u ( s ) ) ∥ d s + 2 ∫ t − δ t ∥ F ( u ( s ) ) ∥ d s .

We choose 0 < λ < η , then

∥ S ( λ ) u ( t ) − u ( t ) ∥ ≤ ε ∫ δ t ∥ F ( u ( s ) ) ∥ d s + 2 ∫ t − δ t ∥ F ( u ( s ) ) ∥ d s

and for 0 ≤ t < δ , we have

∥ S ( λ ) u ( t ) − u ( t ) ∥ ≤ 2 ∫ 0 t ∥ F ( u ( s ) ) ∥ d s .

Since F ∈ L 1 ( 0 , T , X ) , we obtain

∥ S ( λ ) u ( t ) − u ( t ) ∥ ≤ ( ε T + 2 δ ) ∥ F ( u ( s ) ) ∥ 1 .

So, if λ → 0 , then S ( λ ) u → u in L 1 ( [ 0 , T ] , X ) .

The operator L is a uniform limit with compact linear operator between two Banach spaces, then L is compact in L 1 ( [ 0 , T ] , X ) .□

Remark 1

The semigroup S ( t ) generated by the operator Δ is compact in L 1 ( Ω ) .

4 Approximating problem

For all n > 0 , we define the functions w s 0 n , s = 1 , m ¯ , by:

w s 0 n = min ( w s 0 , n ) ,

from which it is clear that w s 0 n verifies (A3), i.e.,

w s 0 n ∈ L 1 ( Ω ) , w s 0 n ≥ 0 , ∀ s = 1 , m ¯ .

Let us consider the following system:

(14) ∂ w 1 n ∂ t − λ 1 Δ w 1 n = Ψ 1 ( w 1 n , … , w m n ) in ] 0 , + ∞ [ × Ω , ⋮ ∂ w m n ∂ t − λ m Δ w m n = Ψ m ( w 1 n , … , w m n ) in ] 0 , + ∞ [ × Ω , ∂ w s n ∂ η = 0 , or w s n = 0 , s = 1 , m ¯ on ] 0 , + ∞ [ × ∂ Ω , w s n ( 0 , x ) = w s 0 n ( x ) ≥ 0 , s = 1 , m ¯ in Ω .

4.1 Existence of a local solution and its positivity of the solution of problem (14)

We convert problem (14) to an abstract first-order system in the Banach space X = ( L 1 ( Ω ) ) m of the form

(15) ∂ ω n ∂ t = A ω n + Ψ ( ω n ) in [ 0 , T ] × Ω , ∂ ω n ∂ η = 0 or ω n = 0 in [ 0 , T ] × ∂ Ω , ω n ( 0 , . ) = ω 0 n ( . ) ∈ X in Ω .

Here ω n = ( w s n ) s = 1 m , the operator A is defined as A = diag ( λ 1 Δ , λ 2 Δ , … , λ m Δ ) , where D ( A ) ≔ { ω n ∈ X : Δ ω n ∈ X } , the function Ψ is defined as Ψ = ( Ψ s ) i = 1 m , and ω 0 n = ( w s 0 n ) s = 1 m .

So system (15) can be returned to the shape of system (11), thus, if ( w 1 n , … , w m n ) is a solution of (15), it then verifies the integral equation

(16) w s n ( t ) = S s ( t ) w s 0 n + ∫ 0 t S s ( t − τ ) Ψ s ( w 1 n ( τ ) , … , w m n ( τ ) ) d τ , s = 1 , m ¯ ,

where S s ( t ) is the semigroup generated by the operator λ s Δ , s = 1 , m ¯ .

Theorem 2

There exists T M > 0 and ω n a local solution of (15) for all t ∈ [ 0 , T M ] .

Proof

We know that S s ( t ) are semigroups of contraction and as Ψ is locally Lipschitz in ω n in the space X , so we have ∃ T M > 0 and ω n is a local solution of (15) on [ 0 , T M ] . □

Lemma 6

Let ( w 1 n , … , w m n ) be the solution of problem (14) such that

w s 0 n ( x ) ≥ 0 , ∀ x ∈ Ω .

Then,

w s n ( t , x ) ≥ 0 , ∀ ( t , x ) ∈ ( 0 , T ) × Ω .

Proof

Let w ¯ s n = e − σ t w s n , σ > 0 , then

∂ w s n ∂ t = e σ t ∂ w ¯ s n ∂ t + σ w ¯ s n , for all 1 ≤ s ≤ m .

Consequently, by problem (15), we have w ¯ s n , 1 ≤ s ≤ m is a solution of the system

(17) ∂ w ¯ 1 n ∂ t + σ w ¯ 1 n − λ 1 Δ w ¯ 1 n = e − σ t Ψ 1 ( w ¯ 1 n , … , w ¯ m n ) in ] 0 , T [ × Ω , ⋮ ⋮ ∂ w ¯ m n ∂ t + σ w ¯ m n − λ m Δ w ¯ m n = e − σ t Ψ m ( w ¯ 1 n , … , w ¯ m n ) in ] 0 , T [ × Ω , ∂ w ¯ s n ∂ η = 0 or w ¯ s n = 0 , 1 ≤ s ≤ m on ] 0 , T [ × ∂ Ω , w ¯ s n ( 0 , x ) = w s 0 n ( x ) ≥ 0 , 1 ≤ s ≤ m in Ω .

Let W 0 = ( t 0 , x 0 ) be the minimum of w ¯ 1 n on ] 0 , T [ × Ω . We will show that w ¯ 1 n ( W 0 ) ≥ 0 , which will imply that w ¯ 1 n ≥ 0 on ] 0 , T [ × Ω and then w 1 n ≥ 0 on ] 0 , T [ × Ω .

Suppose the contrary, namely, w ¯ 1 n ( W 0 ) < 0 .

By the properties of the minimum, we can ensure that W 0 ∈ ] 0 , T ] × Ω and

∂ w ¯ 1 n ∂ t ( W 0 ) = 0 , Δ w ¯ 1 n ( W 0 ) ≥ 0 if 0 < t 0 < T , ∂ w ¯ 1 n ∂ t ( W 0 ) ≤ 0 , Δ w ¯ 1 n ( W 0 ) ≥ 0 if t 0 = T .

Hence, the first equation in (17) yields

σ w ¯ 1 n ( W 0 ) = − ∂ w ¯ 1 n ∂ t ( W 0 ) + λ 1 Δ w ¯ 1 n ( W 0 ) + e − σ t 0 Ψ ( w ¯ 1 n ( W 0 ) , … , w ¯ m n ( W 0 ) ) ≥ e − σ t 0 Ψ ( w ¯ 1 n ( W 0 ) , … , w ¯ m n ( W 0 ) ) .

Now, we use the structure of w ¯ 1 n ( W 0 ) and hypothesis (5) to write

Ψ ( w ¯ 1 n ( W 0 ) , … , w ¯ m n ( W 0 ) ) = Ψ ( 0 , … , w ¯ m n ( W 0 ) ) ≥ 0 .

This implies that w ¯ 1 n ( W 0 ) ≥ 0 , which is impossible by the hypotheses.

In the same way we find

w k n ( t , x ) ≥ 0 , k = 2 , m ¯ ,

then we obtain the positivity of w s n ( t , x ) ≥ 0 , ∀ ( t , x ) ∈ ( 0 , T ) × Ω and for all s = 1 , m ¯ .□

4.2 Global existence of the solution of problem (14)

To prove the global existence of the solution of problem (14) for all nonnegative t , it is enough to find an estimate of the solution for everything t ≥ 0 .

For this, the following lemma shows the existence of an estimate of the solution of (14) in L 1 ( Ω ) .

Lemma 7

Let ( w 1 n , … , w m n ) , the solution of problem (14), so there exists M ( t ) , which depends only on t, such that for all 0 ≤ t ≤ T M , we have

∑ s = 1 m w s n ( t ) L 1 ( Ω ) ≤ M ( t ) .

Proof

Let us add the equations of (14), we obtain

∂ ∂ t ∑ s = 1 m w s n − ∑ s = 1 m λ s Δ w s n = ∑ s = 1 m Ψ s ( w 1 n , … , w m n ) .

By (7) we get

∂ ∂ t ∑ s = 1 m w s n − ∑ s = 1 m λ s Δ w s n ≤ C 2 ∑ s = 1 m w s n .

Let us integrate on Ω and apply the formula of Green, we find that

∂ ∂ t ∫ Ω ∑ s = 1 m w s n d x ≤ C 2 ∫ Ω ∑ s = 1 m w s n d x ,

so,

∂ ∂ t ∫ Ω ∑ s = 1 m w s n d x ∫ Ω ∑ s = 1 m w s n d x ≤ C 2 ,

integrating on [ 0 , t ] , we find

ln ∫ Ω ∑ s = 1 m w s n d x 0 t ≤ C 2 t ,

thus

ln ∫ Ω ∑ s = 1 m w s n ( t ) d x ∫ Ω ∑ s = 1 m w s 0 n d x ≤ C 2 t ,

which implies that

∫ Ω ∑ s = 1 m w s n ( t ) d x ∫ Ω ∑ s = 1 m w s 0 n d x ≤ exp ( C 2 t )

and for w s 0 n ≤ w s 0 , we have

∫ Ω ∑ s = 1 m w s n ( t ) d x ≤ exp ( C 2 t ) ∫ Ω ∑ s = 1 m w s 0 d x .

Let us put

M ( t ) = exp ( C 2 t ) ∑ s = 1 m w s 0 L 1 ( Ω ) .

Since w s n are positives, we obtain

□ ∑ s = 1 m w s n ( t ) L 1 ( Ω ) ≤ M ( t ) , 0 ≤ t ≤ T M .

We can conclude from this estimate that the solution ( w 1 n , … , w m n ) given by the theory 2 is a global solution.

Now, we give the following lemma, which shows the existence of estimate of the solution ( w 1 n , … , w m n ) of problem (14) in L 1 ( Q ) .

Lemma 8

For any solution ( w 1 n , … , w m n ) of (14), there is a constant K ( t ) , which depends only on t, such that

∑ s = 1 m w s n ( t ) L 1 ( Q ) ≤ K ( t ) ∑ s = 1 m w s 0 L 1 ( Ω ) .

Proof

To prove this lemma, we use the following results given by Bonafede and Schmitt [24].

So, we introduce θ ∈ C 0 ∞ ( Q ) , θ ≥ 0 and Φ ∈ C 1 , 2 ( Q ) a nonnegative solution of the following system:

(18) − Φ t − d Δ Φ = θ on Q , ∂ Φ ∂ η = 0 or Φ ( t , . ) = 0 on [ 0 , T [ × ∂ Ω , Φ ( T , . ) = 0 on Ω .

Furthermore, for all q ∈ ] 1 , + ∞ [ , there exists a nonnegative constant c independent of θ , such that

∥ Φ ∥ L q ′ ( Q ) ≤ c ∥ θ ∥ L q ( Q ) .

According to Bonafede and Schmitt [24]:

∫ Q S s ( t ) w s 0 n ( x ) − ∂ Φ ∂ t − d Δ Φ d t d x = ∫ Ω w s 0 n ( x ) Φ ( 0 , x ) d x ,

and that

∫ Q ∫ 0 t S s ( t − τ ) Ψ s ( w 1 n , … , w m n ) d τ − ∂ Φ ∂ t − d Δ Φ d t d x = ∫ Q Ψ s ( w 1 n , … , w m n ) Φ ( τ , x ) d τ d x ,

where

(19) ∫ Q ( S s ( t ) w s 0 n ( x ) ) θ d t d x = ∫ Ω w s 0 n ( x ) Φ ( 0 , x ) d x

and

(20) ∫ Q ∫ 0 t S s ( t − τ ) Ψ s ( w 1 n , … , w m n ) d τ θ d t d x = ∫ Q Ψ s ( w 1 n , … , w m n ) Φ ( τ , x ) d τ d x .

Let us multiply equation (16) by θ , and let us integrate on Q , by using (19) and (20), we obtain

∫ Q w s n ( t ) θ d t d x = ∫ Q S s ( t ) w s 0 n ( x ) θ d t d x + ∫ Q ∫ 0 t S s ( t − τ ) Ψ s ( w 1 n , … , w m n ) d τ θ d t d x = ∫ Ω w s 0 n ( x ) Φ ( 0 , x ) d x + ∫ Q Ψ s ( w 1 n , … , w m n ) Φ ( τ , x ) d τ d x , s = 1 , m ¯ ,

therefore,

∫ Q ∑ s = 1 m w s n ( t ) θ d t d x = ∫ Ω ∑ s = 1 m w s 0 n ( x ) Φ ( 0 , x ) d x + ∫ Q ∑ s = 1 m Ψ s ( w 1 n , … , w m n ) Φ ( τ , x ) d τ d x ,

according to (7) and as w s 0 n ≤ w s 0 we have

∫ Q ∑ s = 1 m w s n ( t ) θ d t d x ≤ ∫ Ω ∑ s = 1 m w s 0 ( x ) Φ ( 0 , x ) d x + ∫ Q C 2 ∑ s = 1 m w s n ( t ) Φ ( τ , x ) d τ d x .

Using the Holder inequality, we deduce

∫ Q ∑ s = 1 m w s n ( t ) θ d t d x ≤ ∑ s = 1 m w s 0 L 1 ( Ω ) . ∥ Φ ( 0 , . ) ∥ L ∞ ( Q ) + C 2 ∑ s = 1 m w s n ( t ) L 1 ( Q ) . ∥ Φ ∥ L ∞ ( Q ) ≤ ∑ s = 1 m w s 0 L 1 ( Ω ) + C 2 ∑ s = 1 m w s n ( t ) L 1 ( Q ) . ∥ Φ ∥ L ∞ ( Q ) ≤ max ( 1 , C 2 ) ∑ s = 1 m w s 0 L 1 ( Ω ) + ∑ s = 1 m w s n ( t ) L 1 ( Q ) . ∥ Φ ∥ L ∞ ( Q ) ≤ k 1 ( t ) ∑ s = 1 m w s 0 L 1 ( Ω ) + ∑ s = 1 m w s n ( t ) L 1 ( Q ) . ∥ θ ∥ L ∞ ( Q ) ,

where k 1 ( t ) ≥ max ( c , c C 2 ) .

Since θ is arbitrary in C 0 ∞ ( Q ) , this implies

∑ s = 1 m w s n ( t ) L 1 ( Q ) ≤ k 1 ( t ) ∑ s = 1 m w s 0 L 1 ( Ω ) + ∑ s = 1 m w s n ( t ) L 1 ( Q ) .

Taking k ( t ) = k 1 ( t ) 1 − k 1 ( t ) we find that

□ ∑ s = 1 m w s n ( t ) L 1 ( Q ) ≤ k ( t ) ∑ s = 1 m w s 0 L 1 ( Ω ) .

Now, we will prove the main result of this work: The existence of global solutions for the system (1) is equivalent to the existence of w s , s = 1 , m ¯ , illustrated by the following main theorem:

Theorem 3

Suppose that the hypotheses (A1)–(A4) are satisfied, then system (1) has a nonnegative solution w s , s = 1 , m ¯ , in the sense:

(21) w s ∈ C ( [ 0 , + ∞ [ , L 1 ( Ω ) ) , s = 1 , m ¯ , Ψ s ( w 1 , … , w m ) ∈ L 1 ( Q ) where Q = ( 0 , T ) × Ω for all T > 0 , w s ( t ) = S s ( t ) w s 0 + ∫ 0 t S s ( t − τ ) Ψ s ( w 1 ( τ ) , … , w m ( τ ) ) d τ , s = 1 , m ¯ , ∀ t ∈ [ 0 , T [ , .

where S s ( t ) are the semigroups of contractions in L 1 ( Ω ) generated by λ s Δ , s = 1 , m ¯ .

Proof

Let us define the application L by

L : ( ω 0 , h ) → S ( t ) ω 0 + ∫ 0 t S ( t − τ ) h ( τ ) d τ ,

where S ( t ) is the semigroup of contraction generated by the operator d Δ , according to the previous result of Theorem 1 and as S ( t ) is compact, then the application L is adding two compact applications in L 1 ( Q ) .

This is how L is compact from ( L 1 ( Q T ) ) m in L 1 ( Q T ) .

Therefore, there is a subsequence ( w 1 n j , … , w m n j ) of ( w 1 n , … , w m n ) and ( w 1 , … , w m ) of ( L 1 ( Q ) ) m , such that:

( w 1 n j , … , w m n j ) converges toward ( w 1 , … , w m ) .

Let us now show that ( w 1 n j , … , w m n j ) is a solution of (16). We have

(22) w s n j ( t , x ) = S s ( t ) w s 0 j + ∫ 0 t S s ( t − τ ) Ψ s ( w 1 n j ( τ ) , … , w m n j ( τ ) ) d τ , s = 1 , m ¯ ,

so, it is enough to show that ( w 1 , … , w m ) verifies (21), and it is clear that if j → + ∞ , we have the following limits:

(23) Ψ s ( w 1 n j , … , w m n j ) → Ψ s ( w 1 , … , w m ) , a.e, s = 1 , m ¯

and

w s 0 j → w s 0 , s = 1 , m ¯ .

Thus, to show that ( w 1 , … , w m ) verifies (21), it remains to show that

Ψ s ( w 1 n j , … , w m n j ) → Ψ s ( w 1 , … , w m ) , s = 1 , m ¯ , in L 1 ( Q ) when j → + ∞ .

We integrate the equations of (14) on Q by taking into account that

− λ s ∫ Q Δ w s n j d t d x = 0 , s = 1 , m ¯ .

We have

∫ Ω w s n j ( t ) d x − ∫ Ω w s 0 j d x = ∫ Q Ψ s ( w 1 n j , … , w m n j ) d t d x ,

where

(24) − ∫ Q Ψ s ( w 1 n j , … , w m n j ) d t d x ≤ ∫ Ω w s 0 d x , s = 1 , m ¯ .

Let us put

ϒ s n = C 2 ∑ s = 1 m w s n j − Ψ s ( w 1 n j , … , w m n j ) , s = 1 , m ¯ ,

it is clear that ϒ s n is positive according to (7), from (24) we obtain

∫ Q ϒ s n d t d x ≤ C 2 ∫ Q ∑ s = 1 m w s n j d t d x + ∫ Ω w s 0 d x .

Lemma 8 gives us

∫ Q ϒ s n d t d x < + ∞ ,

which implies

∫ Q ∣ Ψ s ( w 1 n j , … , w m n j ) ∣ d t d x ≤ C 2 ∫ Q ∑ s = 1 m w s n j d t d x + ∫ Q ϒ s n d t d x < + ∞ .

Let

φ s n = ϒ s n + C 2 ∑ s = 1 m w s n j , s = 1 , m ¯ ,

φ s n are in L 1 ( Q ) and positives and furthermore

∣ Ψ s ( w 1 n j , … , w m n j ) ∣ ≤ φ s n a.e , s = 1 , m ¯ .

Let us combine this result with (23) and we apply the theorem of convergence dominated by Lebesgue, we obtain

Ψ s ( w 1 n j , … , w m n j ) → Ψ s ( w 1 , … , w m ) in L 1 ( Q ) .

By passing in the limit j → + ∞ of (22) in L 1 ( Q ) , we find

w s ( t ) = S s ( t ) w s 0 + ∫ 0 t S s ( t − τ ) Ψ s ( w 1 ( τ ) , … , w m ( τ ) ) d τ , s = 1 , m ¯ .

Then, ( w 1 , … , w m ) verifies (21), consequently ( w 1 , … , w m ) is the solution of (1).□

Acknowledgments

The authors would like to thank the anonymous referees for their useful comments and suggestions.

Funding information: Authors state no funding involved.
Author contributions: All authors have read and approved the manuscript submission.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: Not applicable.

References

[1] N. D. Alikakos, Lp-bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), no. 8, 827–868, DOI: https://doi.org/10.1080/03605307908820113. 10.1080/03605307908820113Search in Google Scholar

[2] K. Masuda, On the global existence and asymptotic behaviour of solution of reaction-diffusion equations, Hokkaido Math. J. 12 (1983), no. 3, 360–370, DOI: https://doi.org/10.14492/hokmj/1470081012. 10.14492/hokmj/1470081012Search in Google Scholar

[3] A. Haraux and A. Youkana, On a result of K. Masuda concerning reaction-diffusion equations, Tohoku Math. J. 40 (1988), no 1, 159–163, DOI: https://doi.org/10.2748/tmj/1178228084. 10.2748/tmj/1178228084Search in Google Scholar

[4] S. Kouachi, Invariant regions and global existence of solutions for reaction-diffusion systems with full matrix of diffusion coefficients and nonhomogeneous boundary conditions, Georgian Math. J. 11 (2004), 349–359, DOI: https://doi.org/10.1515/GMJ.2004.349. 10.1515/GMJ.2004.349Search in Google Scholar

[5] S. Kouachi and A. Youkana, Global existence for a class of reaction-diffusion systems, Bull. Pol. Acad. Sci. Math. 49 (2001), no. 3, 303–308. Search in Google Scholar

[6] N. Alaa and I. Mounir, Global existence for reaction diffusion systems with mass control and critical growth with respect to the gradient, J. Math. Anal. Appl. 253 (2001), no. 2, 532–557, DOI: https://doi.org/10.1006/jmaa.2000.7163. 10.1006/jmaa.2000.7163Search in Google Scholar

[7] W. Bouarifi, N. Alaa, and S. Mesbahi, Global existence of weak solutions for parabolic triangular reaction-diffusion systems applied to a climate model, An. Univ. Craiova Ser. Mat. Inform. 42 (2015), no. 1, 80–97.Search in Google Scholar

[8] S. Abdelmalek, Existence of global solutions via invariant regions for a generalized reaction-diffusion system with a tridiagonal Toeplitz matrix of diffusion coefficients, Funct. Anal. Theory Methods Appl. 2 (2016), 12–27, DOI: https://doi.org/10.48550/arXiv.1411.5727. Search in Google Scholar

[9] S. Abdelmalek, Invariant regions and global solutions for reaction-diffusion systems with a tridiagonal symmetric Toeplitz matrix of diffusion coefficients, Electron. J. Differential Equations 2014 (2014), no. 247, 1–14. Search in Google Scholar

[10] S. Abdelmalek, Invariant regions and global existence of solutions for reaction-diffusion systems with a tridiagonal matrix of diffusion coefficients and nonhomogeneous boundary conditions, J. Appl. Math. 2007 (2007), 1–15, DOI: https://doi.org/10.1155/2007/12375. 10.1155/2007/12375Search in Google Scholar

[11] S. Abdelmalek and S. Kouachi, Proof of existence of global solutions for m-component reaction-diffusion systems with mixed boundary conditions via the Lyapunov functional method, J. Phys. A 40 (2007), no. 41, 12335–12350. 10.1088/1751-8113/40/41/005Search in Google Scholar

[12] K. Abdelmalek, B. Rebiai, and S. Abdelmalek, Invariant regions and existence of global solutions to generalized m-component reaction-diffusion system with tridiagonal symmetric Toeplitz diffusion matrix, Adv. Pure Appl. Math. 12 (2021), no. 1, 1–15, DOI: https://doi.org/10.21494/ISTE.OP.2020.0579. 10.21494/ISTE.OP.2020.0579Search in Google Scholar

[13] A. Moumeni and N. Barrouk, Existence of global solutions for systems of reaction-diffusion with compact result, Int. J. Pure Appl. Math. 102 (2015), no. 2, 169–186, DOI: https://doi.org/10.12732/ijpam.v102i2.1. 10.12732/ijpam.v102i2.1Search in Google Scholar

[14] A. Moumeni and N. Barrouk, Triangular reaction-diffusion systems with compact result, Glob. J. Pure Appl. Math. 11 (2015), no. 6, 4729–4747. Search in Google Scholar

[15] A. Moumeni and M. Dehimi, Global existence’s solution of a system of reaction-diffusion, Int. J. Math. Arch. 4 (2013), no. 1, 122–129. Search in Google Scholar

[16] A. Moumeni and M. Mebarki, Global existence of solution for reaction-diffusion system with full matrix via the compactness, Glob. J. Pure Appl. Math. 12 (2016), no. 6, 4913–4928. Search in Google Scholar

[17] S. L. Hollis and J. J. Morgan, On the blow-up of solutions to some semilinear and quasilinear reaction-diffusion systems, Rocky Mountain J. Math, 24 (1994), no. 4, 1447–1465. 10.1216/rmjm/1181072348Search in Google Scholar

[18] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, New Jersey, 1964. Search in Google Scholar

[19] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, New York, 1984. Search in Google Scholar

[20] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. 10.1007/978-1-4612-5561-1Search in Google Scholar

[21] D. Kulkarni, D. Schmidh, and S. Tsui, Eigenvalues of tridiagonal pseudo-Toeplitz matrices, Linear Algebra Appl. 297 (1999), 63–80. 10.1016/S0024-3795(99)00114-7Search in Google Scholar

[22] M. Andelic and C. M. da Fonseca, Sufficient conditions for positive definiteness of tridiagonal matrices revisited, Positivity 15 (2011), no. 1, 155–159. 10.1007/s11117-010-0047-ySearch in Google Scholar

[23] P. Baras, J. C. Hasan, and L. Veron, Compacite de l’operateur definissant la solution d’une équation d’évolution non homogène, C. R. Acad. Sci. Paris 284 (1977), no. 14, 799–802. Search in Google Scholar

[24] S. Bonafede and D. Schmitt, Triangular reaction diffusion systems with integrable initial data, Nonlinear Anal. 33 (1998), no. 7, 785–801. 10.1016/S0362-546X(98)00042-XSearch in Google Scholar

Received: 2022-05-21

Revised: 2023-06-21

Accepted: 2023-10-13

Published Online: 2024-10-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2023-0122

Keywords for this article

global solution; semigroups; local solution; reaction-diffusion systems

Creative Commons

BY 4.0