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# Global Existence of Solutions to a Class of Reaction–Diffusion Systems on Rn Written By Salah Badraoui Submitted: 02 October 2020 Reviewed: 18 December 2020 Published: 26 March 2021 DOI: 10.5772/intechopen.95543 IntechOpen Recent Developments in the Solution of Nonlinear Differential Equ... Edited by Bruno Carpentieri From the Edited Volume Recent Developments in the Solution of Nonlinear Differential Equations Edited by Bruno Carpentieri Book Details Order Print

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## Abstract

We prove in this work the existence of a unique global nonnegative classical solution to the class of reaction–diffusion systemsuttx=aΔutx−guvm,vttx=dΔvtx+λtxguvm,

### Keywords

• reaction–diffusion systems
• local existence
• positivity
• comparison principle
• global existence

## 1. Introduction

In the sequel, we use the notations.

R+=0,R+=0.

N=01 the set of natural numbers and N=N\0.

For pR:p the integer part of p.

For nN and x=x1xnRn:x2=j=1nxj2.

Z=1,0,1 the set of integers.

For x0Rn and ρR+,:

Bx0ρ=xRn:xx0ρ the closed ball of center x0 and radius ρ.

Sx0ρ=xRn:xx0=ρ the boundary of Bx0ρ.

Let QRnnN a subset. Q denote the boundary of Q.

ln: the natural logaritm function.

ωnρ=2πn/2ρn1Γn/2 the surface area of S0ρ, where Γx=0ettxdtxR+ is the Gamma function.

BUCRn the Banach space of bounded and uniformly continuous functions on Rn with the supremum norm u=supxRnux.

X=BUCRn×BUCRn which is a Banach space endowed with the norm uvX=u+v.

For uLpRn(p1), we denote by upp=Rnupdx.

For u,v:RnR two regular functions, u=ux1uxn and u.v=j=1nuxj.vxj.

Reaction-Diffuison equations are nonlinear parabolic partial differential equations arises in many fields of sciences like chemistry, physics, biology, ecology and even medicine. It appears usually as coupled systems.

The somewhat general form of these systems of two equations is

uttx=aΔutx+f1txuv,vttx=dΔvtx+f2txuv,

where t>0,xΩ with ΩRnnN is an open set, Δ is the Laplacian operator, a, d are two real positive constants called the coefficients of the diffusion. For a chemical reaction where two substances S1 and S2, u and v represent their concentrations at time t and position x respectively, and f1 and f2 represent the rate of production of these substances in the given order. For more details see [1, 2].

In this chapter, we are concerned with the existence of global solutions to the reaction–diffusion system

uttx=aΔutxguvm,txR+×Rn,E1
vttx=dΔvtx+λtxguvm,txR+×Rn,E2

with initial data

u0x=u0x,v0x=v0x,xRn.E3

Whe assume that.

(H1) The constants a, d are such that a,dR+.

(H2)λ:R+×RnR is a non-null, nonnegative and bounded function on R+×Rn such that λt.BUCRn for all tR+. We denote λ=supt0λt.

(H3)n and m are positive integers, i.e. n,mN.

(H4)g:BUCRnBUCRn is a function defined on BUCRn such that:

i. g0=0 and gu0 pour u0.

ii. g is of class C1 and dgudu is bounded on R.

(H5) The initial data u0, v0 are nonnegative and are in BUCRn.

One of the essential questions for (1)(3) is the existence of global solutions and possibly bounds uniform in time. Recently, Collet and Xin in their paper [3] published in 1996 have studied the system (1)(3) but with a=λ=1, d>1 and φu=u. In this particular case, this system describes the evolution of u the mass fraction of reactant A and that v of the product B for the autocatalytic chemical reaction of the form A+mBm+1B. They proved the existence of global solutions and showed that the L norm of v can not grow faster than Olnlnt for any space dimension.

If we replace guvm by uexpE/v where E>0 is a constant and take λ=1, there are many works on global solutions, see Avrin [4], Larrouturou [5] for results in one space dimension, among others.

It is worth mentioning here the result of S. Badraoui [6] who studied the system

ut=aΔuuvm,
vt=bΔu+dΔv+uvm,

where a>0, d>0, b0, xRn, nN,m2N is an even positive integer. He has proved that if u0, v0 are nonnegative and are in BUCRn that:

If a>d,b>0,v0badu0 on Rn, then the solution is global and uniformly bounded.

If a<d,b<0,v0badu0 on Rn, then the solution is global.

Our work here is a continuation of the work of Collet and Xin [3]. We treat the same question in a slightly general case. Inspired by the same ideas in [3] we prove that the system (1)(3) under the assumptions (H1) to (H5) has a unique global nonnegative classical solution.

The chapter is organized as follows: In section 2, we treat the existence of local solution and reveal its positivity using the maximum principle.

In section 3, firstly, we prove by a simple comparison argument that if ad, the solution is uniformly bounded and we give an upper bound of it. Afterwards, we attack the hard case in which a<d where we used the Lyapunov functional Luv=α+2uln1+ueεv(α,ε>0) and the cut-off function φx=1+x2n. We show that for α sufficiently large and ε small enough we can control the Lp-norms of vp>max1n/2 on every unit spacial cub in Rn from which we deduce the L-norm of v at any time t>0.

We emphazise here that I have engaged to calculate the constants encountered in all equations and inequalities exactly.

## 2. Existence of a local solution and its positivity

We convert the system (1)(3) to an abstract first order system in the Banach space XBUCRn×BUCRn of the form

wt=Awt+Fwt,t>0,w0=w0X.E4

Here wt=utvt; the operator A is defined as

AwaΔ00dΔw=aΔudΔv,

where DAw=uvX:ΔuΔvX. The function F is defined as Fwt=φutvmtλtφutvmt.

It is known that for c>0 the operator cΔ generates an analytic semigroup Gt in the space BUCRn:

Gtu=4πctn/2Rnexpxy24ctuydy.E5

Hence, the operator A generates an analytic semigroup defined by

St=S1t00S2t,E6

where S1t is the semigroup generated by the operator aΔ, and S2t is the semigroup generated by the operator dΔ.

Since the map F is locally Lipschitz in w in the space X, then proving the existence of a loacl classical solution on 0t1 where t1R+ is standard [7, 8].

For the positivity, let wt=utvt is a local solution of the problem (1)(3) under the assumptions Hjj=15 on the interval 0t1.

We can write the first equation as

utaΔu+vmddugξu=0,tx0t1×Rn,E7

for some ξR. Thanks to the assumption (H4)-ii we deduce that vmugξ is bounded on 0t1×Rn. Whence, by the theorem 9 on page 43 in [9], we obtain that

utx0,foralltx0t1×Rn,E8

The second equation can be written as

vtdΔv+λguvm1v,tx0t1×Rn.E9

By the same theorem we get

vtx0,foralltx0t1×Rn.E10

For the existence of a global solution, we use the contraposed of the characterization of the maximal existence time tmax ([8] on page 193) as follows

there existsamapC:R+R+such that:ut+vtCtforalltR+tmax=+.E11

## 3. Existence of a global solution

For this task we will use the fact that the solution is nonnegative.

Theorem 3.1. Let uv be the solution of the problem (1)(3) under the assumptions Hjj=15 and such that

Then, the solution is global and uniformly bounded on R+×Rn. More precisely, we have the estimates

utu0,foralltR+,E13

Proof. By the comparison principle we get (13).

The solution uv satisfies the integral equations

utx=S1tu00tS1tτguτvmτ,E15
vtx=S2tv0+0tS2tτλτguτvmτ.E16

Here S1t and S2t are the semigroups generated by the operators aΔ and dΔ in the space BUCRn respectively. As u is nonnegative, then from (15) we get

0tS1tτguτvmτS1tu0.E17

Since ad, using the explicit expression of S1tτguτvmτ and S2tτguτvmτ, one can observe that (see [10])

From (17) and (18) into (16) we get

This last inequality leads to the veracity of (14).

Thus, from (13) and (14), we deduce that the solution uv is global and uniformly bounded on R+×Rn.

In the case where d>a, it seems that the idea of comparison cannot be applied. Nevertheless, we can prove the existence of global classical solutions; but it appears that their boundedness is not assured.

Theorem 3.2. Let uv be the solution of the problem (1)(3) with the assumptions Hjj=15. If

a<d,E20

the solution uv is global. More precisely we have the estimates (13) and (83).

Proof. In this case, it is not easy to prove global existence. But can derive estimates of solutions independent of t1 by using the same method used in [3] and the same form of the functional used in [6] but with different coefficients.

We need some lemmas.

Lemma 3.3. Let uv be the solution of the problem (1)(3) under the assumptions Hjj=15 on the local interval time 0t1. Define the functional

Luv=α+2uln1+ueεvwithα,εR+.E21

Then for any φ=φxxRn a smooth nonnegative function with exponential spacial decay at infinity, we have

ddtRnφLdx=dRnΔφLdx+daRnL1φ.udxRnφaL11u2+a+dL12uv+dL22v2dx+RnφλL2L1guvmdx,E22

where

L1Lu=211+ueεv,L2Lv=εα+2uln1+ueεv,L112Lu2=11+u2eεv,L122Luv=ε211+ueεv,L222Lv2=ε2α+2uln1+ueεv.E23

Proof. Note that L>0, L1>0, L2>0, L11>0, L12>0 and L22>0. We can differentiate under the integral symbol

ddtRnφLdx=aRnφL1udx+dRnφL2Δvdx+RnφλL2L1guvmdx.E24

Using integration by parts, we get

RnφL1Δudx=RnφL1Δudx=RnφL1udx=RnL1φudxRnφL11u2dxRnφL12uvdx,E25

In fact, let ρR+, then we have by the Geen theorem

B0ρφL1Δudx=B0ρφL1Δudx=B0ρφL1.udx+S0ρφL1uνdx,E26

where uν is the derivative of u with respect to the unit outer normal ν to the boundary S0ρ.

We have

S0ρφL1tutνdx2eεvtutνS0ρφdx2eεvtutν11+ρ2n2πn/2ρn1Γn/2.E27

From (27) we obtain

limρSx0ρφL1tutνdx=0.E28

We pass to the limit for ρ in (26) taking into account (28) we obtain (25).

By the same way we get

RnφL2Δvdx=RnL2φ.vdxRnφL22v2dxRnφL12uvdx,E29
RnLΔφdx=RnL1φ.udxRnL2φ.vdx.E30

From (30) we find that

RnL2φ.vdx=RnL1φ.udxRnLΔφdx.E31

From (25), (29) and (31) into (24) we get our basic identity (22).

Lemma 3.4. There exist two positive real constants α=αadγ1u0 and ε=εadγ1γ2λu0 such that

ddtRnφLdxdRnLΔφdx+daRnL1φ.udxγ1RnφaL11u2+dL22v2dxγ2RnφL1guvmdx,E32

where γ1,γ201 are two arbitrary constants.

Proof. We seek L such that

aL11u2+a+dL12uv+dL22v2γ1aL11u2+dL22v2E33

and

λL2L1γ2L1E34

for γ1,γ201.

The inequality (33) is satisfied if

From (23); (35), then (33) is satisfied if

Also, (34) is satisfied if ελα+2u01γ21, i.e. ε1γ2λα+2u0, and from (36) we get

Whence, if α satisfies (36) and ε satisfies (37), we obtain (32).

As a consequence of (33) we have

ddtRnφLdxdRnLΔφdx+daRnL1φ.udxγ1aRnφL11u2dx.E38

Lemma 3.5. With the functional L defined in (21) and α, ε defined in (36) and (37) respectively and with the truncation function φ:RnR defined by

φx=11+xx02n.E39

We have

where

k1n=2n3n+2,k2n=2n.E41

Proof. Calulate Δφ and estimate it

Δφ=2n21+xx02n+14nn+1xx021+xx02n+2;

whence

Δφ2n3n+2φ.E42

Calulate φ and estimate it

φ2=4n2xx021+xx022n+2;

whence

φ2.E43

Using the Cauchy-Schwarz inequality φ.uφu and the inequalities (42) and (43) into (38) we get

ddtRnφLdxdk1nRnφLdx+dak2nRnφL1φdxγ1aRnφL11u2dx.E44

We pove that

dak2nφL1φγ1L11u214γ1da2ak22nφL12L11.E45

To do this, it sufficies to compute the discriminant of the trinoma in φ

Δ=γ1L11u2+dak2nφL1φ14γ1da2ak22nφL12L11.

From (45) into (42) we find the desired result (40).

Lemma 3.6. For α and ε defined in (36) and (37) respectively and for all real constant γ

γmax1a8u0+4,E46

we have

RnφLdxβeσt,foralltR+;E47

where

β=2nα+2u0ωneεv0,E48

and

Proof. We seek a constant γR+ such that

L12L11γL,forallu0u0.E50

The inequality (50) is equivalent to 2u+12eεvγα+2uln1+u. We prove that if γ satisfies (46) then (50) follows.

Whence, from (50) into (40) we obtain

As

RnφLt=0dx=Rnφα+2u0ln1+u0eεv0dx;E52

then, from (51) and (52) we get

RnφLdxα+2u0φ1expεv0eσt,foralltR+,E53

where σ is defined by (49).

Now, let us estimate φ1. We have ([11] on page 485)

φ1=Rnφdx=Rn11+x2ndx=ωn0rn111+r2ndr.

As

0rn11+r2ndr=01rn11+r2ndr+1rn11+r2ndr01rn1dr+11rn+1dr2n,

then

φ12nωn.E54

Thus, from (54) in (53) we get the estimate (47) with β and σ given by (48) and (49).

In the following step we trie to control the second component v of the solution on any unit spacial cube in the Lp norms with p1.

Let x0=x10xn0Rn be an arbitrary fixed point and

Q=x=x1xnRn:xkxk012forallk=1n.E55

Lemma 3.7. Let uv be the solution of the problem in consideration. For α and ε satisfying (36) above and (63) below respectively, then for any unit cube Q of Rn of the form (55) we have

Qvpdxβp+1p+1αεp+14+n4neσt,forallpt1×R+.E56

Proof. It’s obvious that

φx44+nn,forallxRn,E57

and

eεvεkk!vk,forallkN.E58

Then

RnφLdxαεkk!44+nnQvkdx.E59

Let us combine (47) and (59)

Qvkdxβk!αεk4+n4neσt,forallktN×R+.E60

By induction we prove that

k!pp,forallkNandpk.E61

Let p1 and k=p+1, then we have by the imbedding theorem for Lpspaces

QvpdxQvkdxp/k.E62

Taking ε enough small such that βk!αεk4+n4n1. Combining this with (37)

From (60), (61) and (63) into (62) we get (56).

Lemma 3.8. Let Qi et Qj be two different unit cubes of center xi=x1ixni and xj=x1jxnj respectively of the form

Qi=x=x1xnRn:xkxki1/2,forallk=1,,n,Qj=x=x1xnRn:xkxkj1/2,forallk=1,,n,E64

with xj=xi+l, where l=l1lnZn\0Zn. Then, there exists a positive constant

δn=2+n2,E65

such that

distxiQj2xiy2δndistxiQj2,forallyQj.E66

Proof. By Pythagorean theorem we have

xjyn2.E67

As xixj1, then from (67)

xjyn2xixj.E68

Also, it’s clear that distxiQj=distxiQj, but every point z=z1znQj is of the form

z=xj+s,E69

where s=s1sn0 and sk1212, for all k=1,,n with at least one of the sk1212.

It’s easy to prove that

xkjxki2xkjxki+sk,forallk=1,,n.E70

Then

xjxi2distxiQj.E71

As xiyxixj+xjy we get from (68) and (71) the estimate

xiy2distxiQj+n2xixj2+ndistxiQj.E72

We have obviously

xiydistxiQj.E73

From (71) and (73) we get (66).

Proof of theorem 3.2.

Let xRn an arbitrary point and QjjN be the family of pairwise disjoint measurable cubes of the form (64) covering Rn such that the center of Q0 is x0=x.

Firstly, using the fact that Rn=j=0Qj and applying the left-hand inequality in (66)

Rnexy24dtsλguvmdy=j=0Qjexy28dtsexy28dtsλguvmdyj=0edistxQj28dtsQjxy28dtsλguvmdy.E74

By Hölder ineguality with p>max1n2 and q=11p

Qjxy28dtsλguvmdyQjqxy28dtsdy1/qQjλpgpuvpmdy1/p.E75

As

Qjeqxy28dtsdyRnqxy28dtsdy=8πdqn/2tsn/2E76

and by (56) we have

Qjλpgpuvpmdyλpgpβpm+1pm+1αεpm+14+n4neσt,E77

where

g=supu0u0gu.E78

Then, from (76) and (77) into (75)

Qjxy28dtsλguvmdyK8πdqn211ptsn211pλgeσ/pt,E79

where

K=Kpmnαε=βpm+1pm+1αεpm+14+n4n1/p.E80

On the other hand, we deduce from the right-hand inequality in (66) that

Qjexy28ntsdyedistxQj28dts,foralljN.E81

Then

j=0edistxQj28dts1+j=1edistxQj28dts1+j=1Qjexy28ntsdy1+Rnexy28ntsdy1+8πdδnn/2tsn/2.E82

We have from (79) and (82) into (74)

14πdtsn/2Rnexy24dtsλguvmdy2n/211pn211ptsn2pKλgeσ/pt1+8πdδnn/2tsn/22n/211pn211pKλgeσ/pttsn2p+8πdδnn/2tsn211p.

Whence

0tS2tsλguvmds2n/211pn211pKλgeσ/pt2p2pnt1n2p+8πdδnn/22ppn+2+2ptn211p+1

and finally we have for all tR+

vtv0+2n/211pn211pKλgeσ/pt2p2pnt1n2p+8πdδnn/22ppn+2ntn211p+1E83

As p>max1n2, the function in t on the right-hand side of the estimate (83) is continuous on R+. As utu0 on 0tmax and v satisfied (83), we conclude from (11) that tmax=+. Whence, the solution is global.

Remark. We can extend the system to the case where instead of vm we put vhv provided that.

i. h:BUCRBUCR is a locally continuous Lipschitz function, namely: for all constant ρR+, there exists a constant cρR+ such that for all u,vBUCRn with uρ and vρ we have

hugvcρuv.

ii. There exist two constants MR+ and rN such that:

0hvMvr,forallvR+.

In this more general case, by examining the proof of the theorem 3.2; we see that under the same assumptions above, the system has also a global nonnegative classical solution.

## 4. Illustrative example

To illustrate the previous study about global existence, we give the following reaction–diffusion system

uttx=aΔutxc1u3c2+c3u2vm,txR+×Rn,vttx=dΔvtx+c4ec5tx2u3c2+c3u2vm,txR+×Rn,u0x=u0x,v0x=v0x,xRn,E84

where ck,k=1,,4 are real positive constants and c5 is a real nonnegative constant. If a,bR+,n,mN,u0,v0BUCRn and are nonnegative; the system (84) admits a unique global nonnegative classical solution uvCR+XC1R+X.

## 5. Conclusion and perspectives

We have prouved in the case where a<d that the solution is global, but it remains an interesting question that if it is uniformly bounded or not.

As perspectives, we will replace the function g=gu satisfying the hypothesis (H4) by the function gu=ur with r1 is a real constant and replace the term vm by eαv with α>0; namely that reaction term is of exponential growth. The system was studied on bounded domain by J. I. Kanel and M. Mokhtar in [12].

## References

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