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Invariant regions and global existence of solutions for reaction-diffusion systems with a general full matrix

Abstract

The aim of this study is to prove the global existence in time of solutions for reaction-diffusion systems. We make use of the appropriate techniques which are based on invariant regions and Lyapunov functional methods. We consider a full matrix of diffusion coefficients and we show the global existence of the solutions.

MSC:35K45, 35K57.

1 Introduction

We are mainly interested in the global existence in time of solutions to a reaction-diffusion system of the form

u t aΔubΔv=Πf(u,v)σuin ]0,+[×Ω,
(1.1)
v t cΔudΔv=f(u,v)σvin ]0,+[×Ω
(1.2)

with the following boundary conditions:

u η = v η =0in ]0,+[×Ω
(1.3)

and the initial data

u(0,x)= u 0 ,v(0,x)= v 0 in Ω,
(1.4)

where Ω is an open bounded domain in R n with boundary Ω of class C 1 , η denotes the outward normal derivative on Ω, Δ denotes the Laplacian operator with respect to the x variable, a, b, c, d, σ are positive constants satisfying the condition ( b + c ) 2 <4ad, which reflects the parabolicity of the system and implies at the same time that the matrix of diffusion is positive definite, Π0. The eigenvalues λ 1 and λ 2 ( λ 1 < λ 2 ) of the matrix are positive. We assume that

λ 1 <a<d< λ 2 <a+c,

and the initial data are assumed to be in the following region:

Σ= { ( u 0 , v 0 ) R 2  such that  a λ 2 c v 0 u 0 a λ 1 c v 0 } .
(1.5)

For more details, one may consult [1].

The function f is a nonnegative continuously differentiable function on Σ such that

f ( a λ 2 c η , η ) =0andf ( a λ 1 c η , η ) Π ( 1 + a λ 1 c ) for all η0.
(1.6)

In addition we suppose that

(ξ,η)Σ0f(ξ,η)φ(ξ) ( 1 + η ) β ,
(1.7)

where β1 and φ is a nonnegative function of class C(R) such that

lim ξ φ ( ξ ) ξ =0.
(1.8)

Melkemi et al. [2] established the existence of global solutions (eventually uniformly bounded in time) using a novel approach that involved the use of a Lyapunov function for system (1.1)-(1.4) when c=b=0. Along the same lines, Rebai [3] has proved the global existence of solutions for system (1.1)-(1.4), in the case b=0, c>0 (triangular matrix). The present investigation is a continuation of results obtained in [3]. Here, we follow the same reasoning as in [2], in the study of system (1.1)-(1.4), when b>0, c>0, that is, for a model that involves a general full matrix.

The components u(t,x) and v(t,x) represent either chemical concentrations or biological population densities and system (1.1)-(1.2) is a mathematical model describing various chemical and biological phenomena (see, e.g., Cussler [4]).

Remark 1 If a<d, then we have λ 1 <a<d< λ 2 . We note that the condition of parabolicity implies that det(A)=adbc>0, where A is the matrix of diffusion.

2 Local existence and invariant regions

Throughout the text we shall denote by p the norm in L p (Ω), and by the norm in L (Ω) or C( Ω ¯ ).

For any initial data in C( Ω ¯ ) or L p (Ω), p]1,+[, local existence and uniqueness of solutions to the initial value problem (1.1)-(1.4) follow from the basic existence theory for abstract semilinear differential equations (see Henry [5] and Pazy [6]). The solutions are classical on ]0; T [, where T denotes the eventual blowing-up time in L (Ω).

Furthermore, if T <+, then

lim t T ( u ( t ) + v ( t ) ) =+.

Therefore, if there exists a positive constant C such that

u ( t ) + v ( t ) C,t]0, T [,

then T =+.

Multiplying (1.2) first through by a λ 2 c and adding (1.1) and then by a λ 1 c and subtracting (1.1), we get

w t λ 2 Δw=Π ( 1 + a λ 2 c ) F(w,z)σwin ]0, T [×Ω,
(2.1)
z t λ 1 Δz=Π+ ( 1 + a λ 1 c ) F(w,z)σzin ]0, T [×Ω,
(2.2)

with the boundary conditions

w η = z η =0in ]0, T [×Ω,
(2.3)

and the initial data

w(0,x)= w 0 (x),z(0,x)= z 0 (x)in Ω,
(2.4)

where

w ( t , x ) = u ( t , x ) a λ 2 c v ( t , x ) , z ( t , x ) = u ( t , x ) + a λ 1 c v ( t , x )
(2.5)

for any (t,x) in ]0, T [×Ω and

F(w,z)=f(u,v)for all (u,v) in Σ.
(2.6)

To prove that Σ is an invariant region for system (1.1)-(1.4) it suffices to prove that the region

Σ 1 = { ( w 0 , z 0 ) R 2  such that  w 0 0 , z 0 0 }

is invariant for system (2.1)-(2.4).

Now, to prove that the region Σ 1 is invariant for system (2.1)-(2.4), it suffices to show that (Π(1+ a λ 2 c )F(0,z))0 for z0, and (Π+(1+ a λ 1 c )F(w,0))0, for w0, see [7].

From (1.6), its clear that the region Σ 1 is invariant for system (2.1)-(2.4) and from (2.5) we have

v ( t , x ) = c λ 2 λ 1 ( w ( t , x ) + z ( t , x ) ) , u ( t , x ) = a λ 1 λ 2 λ 1 w ( t , x ) + a λ 2 λ 2 λ 1 z ( t , x ) .
(2.7)

Remark 2 We note that if (ξ,η)Σ, then ξR and η0.

3 Existence of global solutions

A simple application of the comparison theorem [[7], Theorem 10.1] to system (2.1)-(2.4) implies that for any initial conditions w 0 0 and z 0 0, we have

0w(t,x)max ( w 0 , Π σ ) =K.
(3.1)

To prove the global existence of the solutions of problem (1.1)-(1.4), one needs to prove it for problem (2.1)-(2.4). As regards this subject, it is well known that it suffices to derive a uniform estimate of Π + ( 1 + a λ 1 c ) F ( w , z ) σ z p for some p> n 2 , i.e.

Π + ( 1 + a λ 1 c ) F ( w , z ) σ z p C,

where C is a nonnegative constant independent of t.

From the assumptions (1.7) and (1.8), we are led to establish the uniform boundedness of the z p on ]0, T [ in order to get that of z on ]0, T [.

For p2, we put

α= ( λ 2 λ 1 ) 2 4 λ 1 λ 2 ,α(p)= p α + 1 p 1 , M p =K+ Π σ α ( p ) .
(3.2)

We firstly introduce the following lemmas, which are useful in our main results.

Lemma 1 Let (w,z) be a solution of (2.1)-(2.4). Then

d d t Ω wdx+ ( 1 + a λ 2 c ) Ω F(w,z)dx+σ Ω wdx=Π|Ω|.
(3.3)

Proof We integrate both sides of (2.1), satisfied by w, which is positive and then we obtain

d d t Ω wdx=Π|Ω| ( 1 + a λ 2 c ) Ω F(w,z)dxσ Ω wdx.

 □

Lemma 2 Assume that p2 and let

G q (t)= Ω [ q w + exp ( p 1 p α + 1 ln ( α ( p ) ( M p w ) ) ) z p ] dt,
(3.4)

where (w,z) is the solution of (2.1)-(2.4) on ]0, T [. Then under the assumptions (1.7)-(1.8) there exist two positive constants q>0 and s>0 such that

d d t G q (t)(p1)σ G q +s.
(3.5)

Proof The proof is similar to that in Melkemi et al. [2].

Let

h(w)= p 1 p α + 1 ln ( α ( p ) ( M p w ) ) ,
(3.6)

then

G q (t)=q Ω wdx+N(t),
(3.7)

where

N(t)= Ω e h ( w ) z p dx.
(3.8)

Differentiating N(t) with respect to t and using the Green formula one obtains

d d t N=H+S,
(3.9)

where

H = λ 2 Ω ( ( h ( w ) ) 2 + h ( w ) ) e h ( w ) z p ( w ) 2 d x p ( λ 2 + λ 1 ) Ω h ( w ) e h ( w ) z p 1 w z d x λ 1 Ω p ( p 1 ) e h ( w ) z p 2 ( z ) 2 d x
(3.10)

and

S = Π Ω h ( w ) e h ( w ) z p d x + Ω [ p z p 1 ( 1 + a λ 1 c ) F ( w , z ) ( 1 + a λ 2 c ) h ( w ) z p F ( w , z ) ] e h ( w ) d x σ Ω h ( w ) w e h ( w ) z p d x p σ Ω e h ( w ) z p d x p Π Ω e h ( w ) z p 1 d x .
(3.11)

We observe that H is given by

H= Ω Q e h ( w ) dx,

where

Q = λ 2 ( ( h ( w ) ) 2 + h ( w ) ) z p ( w ) 2 + p ( λ 2 + λ 1 ) h ( w ) z p 1 w z + λ 1 p ( p 1 ) z p 2 ( z ) 2
(3.12)

is a quadratic form with respect to w and z, which is nonnegative if

( p ( λ 2 + λ 1 ) h ( w ) z p 1 ) 2 4 λ 1 λ 2 p(p1) ( ( h ( w ) ) 2 + h ( w ) ) z 2 p 2 0,
(3.13)

and we have chosen h(w) such that

h (w)= 1 α ( p ) ( M p w ) , h (w)= α ( p ) [ α ( p ) ( M p w ) ] 2 .
(3.14)

It is easy to see that the left-hand side of (3.13) can be written as

4 λ 1 λ 2 p z 2 p 2 { p [ α 1 ( α ( p ) ( M p w ) ) 2 α ( p ) ( α ( p ) ( M p w ) ) 2 ] + 1 + α ( p ) ( α ( p ) ( M p w ) ) 2 } = 0 ,
(3.15)

since

pαpα(p)+1+α(p)=0,

the inequality (3.13) holds, Q0, and consequently

H= Ω Q e h ( w ) dx0,
(3.16)

and the second term S can be estimated as

S Ω ( Π h ( w ) σ p ) e h ( w ) z p d x + Ω [ p z p 1 ( 1 + a λ 1 c ) F ( w , z ) h ( w ) z p ( 1 + a λ 2 c ) F ( w , z ) ] e h ( w ) d x ( p 1 ) σ Ω e h ( w ) z p d x + Ω [ ( 1 + a λ 1 c ) p z p 1 F ( w , z ) ( 1 + a λ 2 c ) h ( w ) z p F ( w , z ) ] e h ( w ) d x ,
(3.17)

since

h (w)= 1 α ( p ) ( M p w ) 1 α ( p ) ( M p K ) = σ Π .
(3.18)

On the other hand

h ( w ) = 1 α ( p ) ( M p w ) 1 α ( p ) M p , h ( w ) 1 α ( p ) ln Π σ .
(3.19)

Taking into account the fact that z0, and using (3.19), we observe that

p ( 1 + a λ 1 c ) z p 1 F ( w , z ) ( 1 + a λ 2 c ) h ( w ) z p F ( w , z ) ( p ( 1 + a λ 1 c ) z p 1 1 α ( p ) M p ( 1 + a λ 2 c ) z p ) F ( w , z ) .

Then for η 0 =p(1+ a λ 1 c )( 1 ( 1 + a λ 2 c ) +1)α(p) M p >0, and for 0ξK, η η 0 , we have

( p ( 1 + a λ 1 c ) η p 1 1 α ( p ) M p ( 1 + a λ 2 c ) η p ) F ( ξ , η ) = [ p ( 1 + a λ 1 c ) η ( 1 + a λ 2 c ) α ( p ) M p ] η p F ( ξ , η ) 0 .

On the other hand, we deduce that the function

(ξ,η)p ( 1 + a λ 1 c ) η p 1 1 α ( p ) M p ( 1 + a λ 2 c ) η p

is bounded on the compact interval [0, η 0 ]; then there exists c 1 >0 such that

p ( 1 + a λ 1 c ) z p 1 F(w,z) ( 1 + a λ 2 c ) h (w) z p F(w,z) c 1 F(w,z).
(3.20)

From (3.17) and (3.20), we deduce immediately the following inequality:

S(p1)σN+ c 1 Ω F(w,z) e h ( w ) dx(p1)σN+ c 1 e 1 α ( p ) ln Π σ Ω F(w,z)dx,

and putting

q= c 1 ( 1 + a λ 2 c ) e 1 α ( p ) ln Π σ ,
(3.21)

by (3.3), we have

S(p1)σN+qΠ|Ω|q d d t Ω w(t,x)dx,
(3.22)

and from (3.7), it follows that

S(p1)σ G q +q ( ( p 1 ) σ K + Π ) |Ω|q d d t Ω w(t,x)dx,
(3.23)

and from (3.7) and (3.9), we conclude that

d d t G q (p1)σ G q +s,
(3.24)

where

s=q ( ( p 1 ) σ K + Π ) |Ω|.
(3.25)

 □

Now we can establish the global existence and uniform boundedness of the solutions of (2.1)-(2.4).

Theorem 1 Under the assumptions (1.7) and (1.8), the solutions of (2.1)-(2.4) are global and uniformly bounded on [0,+[×Ω.

Proof Multiplying the inequality (3.24) by e ( p 1 ) σ t and then integrating, we deduce that there exists a positive constant C>0 independent of t, such that

G q (t)C.
(3.26)

From (3.6), we observe that

e h ( w ) e 1 α ( p ) ln α ( p ) M p ,
(3.27)

and it follows, for all p2, that

Ω z p dx e 1 α ( p ) ln ( K α ( p ) + Π σ ) G q (t) C 1 (p),
(3.28)

where

C 1 (p)=C e 1 α ( p ) ln ( K α ( p ) + Π σ ) ,
(3.29)

and as we select p> n 2 we can proceed to bound Π + ( 1 + a λ 1 c ) F ( w , z ) σ z p .

Let

A= max ξ 0 ξ K 1 φ(ξ),
(3.30)

where

K 1 = a λ 1 λ 2 λ 1 K,

and ξ 0 is such that

ξ ξ 0 φ(ξ)<|ξ|,
(3.31)

using (1.7), we deduce

F(w,z)=f(u,v)φ(u) ( 1 + v ) β ,

which implies

Ω F p ( w , z ) d x Ω ( φ ( u ) ) p ( 1 + v ) β p d x = u ξ 0 ( φ ( u ) ) p ( 1 + v ) β p d x + ξ 0 u ( φ ( u ) ) p ( 1 + v ) β p d x u ξ 0 | u | p ( 1 + v ) β p d x + A p ξ 0 u ( 1 + v ) β p d x .

From (2.7), we have

| u | p = | a λ 1 λ 2 λ 1 w ( t , x ) + a λ 2 λ 2 λ 1 z ( t , x ) | p ( a λ 1 λ 2 λ 1 w ( t , x ) + λ 2 a λ 2 λ 1 z ( t , x ) ) p ( λ 2 a λ 2 λ 1 ) p ( w ( t , x ) + z ( t , x ) ) p ,

then

Ω F p ( w , z ) d x u ξ 0 ( λ 2 a λ 2 λ 1 ) p ( w + z ) p ( 1 + c λ 2 λ 1 ( w + z ) ) β p d x Ω F p ( w , z ) d x + A p ξ 0 u ( 1 + c λ 2 λ 1 ( w + z ) ) β p d x Ω F p ( w , z ) d x max ( A p , ( λ 2 a λ 2 λ 1 ) p ) ( u ξ 0 ( w + z ) p ( 1 + c λ 2 λ 1 ( w + z ) ) β p d x Ω F p ( w , z ) d x + ξ 0 u ( 1 + c λ 2 λ 1 ( w + z ) ) β p d x ) Ω F p ( w , z ) d x max ( A p , ( λ 2 a λ 2 λ 1 ) p ) ( Ω ( w + z ) p ( 1 + c λ 2 λ 1 ( w + z ) ) β p d x Ω F p ( w , z ) d x + Ω ( 1 + c λ 2 λ 1 ( w + z ) ) β p d x ) , Ω ( w + z ) p ( 1 + c λ 2 λ 1 ( w + z ) ) β p d x 2 β p 1 ( Ω ( w + z ) p + ( c λ 2 λ 1 ) β p ( w + z ) ( β + 1 ) p d x ) 2 ( β + 1 ) p 2 ( K P | Ω | + C 1 ( p ) ) + 2 ( 2 β + 1 ) p 2 ( c λ 2 λ 1 ) β p ( K ( β + 1 ) p | Ω | + C 1 ( ( β + 1 ) p ) ) = C 2 ( β , p , K , Ω ) , Ω ( 1 + c λ 2 λ 1 ( w + z ) ) β p d x 2 β p 1 ( | Ω | + ( c λ 2 λ 1 ) β p × 2 β p 1 ( K β p | Ω | + C 1 ( β p ) ) ) = C 3 ( β , p , K , Ω ) .

Consequently

Ω F p (w,z)dx C 4 (A,β,p,K,Ω).

Finally

Π + a λ 1 c F ( w , z ) σ z p = a λ 1 c F ( w , z ) ( σ z + Π ) p a λ 1 c F ( w , z ) p + σ z p + Π | Ω | a λ 1 c C 4 ( A , β , p , K ) p + σ C 1 ( p ) p + Π | Ω | = C 5 ( A , β , p , K , Ω , σ ) .
(3.32)

Using the regularity results for the solutions of the parabolic equations in [5], we conclude that the solutions of problem (2.1)-(2.4) are uniformly bounded on [0,+[×Ω. □

By (2.7), it is easy to see that the solutions of problem (1.1)-(1.4) are also uniformly bounded on [0,+[×Ω.

Remark 3 Because 0w(t,x)K and z(t,x)0, we deduce that

u(t,x) a λ 1 λ 2 λ 1 K= K 1 .

Remark 4 We note that a λ 2 λ 2 λ 1 <0 and λ 2 a λ 2 λ 1 a λ 1 λ 2 λ 1 , because λ 2 + λ 1 =a+d and d>a.

We conclude by noting that the study of the global existence of strongly coupled systems has been a major development, and several articles are devoted to this subject. In our opinion, many other systems with non-constant diffusion matrix which are in the actual scope of the results previously given, should be taken in consideration and studied with more interest.

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Acknowledgements

The author would like to thank the anonymous referee for his/her valuable suggestions.

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Boukerrioua, K. Invariant regions and global existence of solutions for reaction-diffusion systems with a general full matrix. J Inequal Appl 2014, 24 (2014). https://doi.org/10.1186/1029-242X-2014-24

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Keywords

  • global existence
  • reaction-diffusion systems
  • Lyapunov functional