Open access peer-reviewed chapter

Global Existence of Solutions to a Class of Reaction–Diffusion Systems onRn

By Salah Badraoui

Submitted: October 2nd 2020Reviewed: December 18th 2020Published: March 26th 2021

DOI: 10.5772/intechopen.95543

Downloaded: 60

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

For pR:pthe integer part of p.

For nNand x=x1xnRn:x2=j=1nxj2.

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

For x0Rnand ρR+,:

Bx0ρ=xRn:xx0ρthe closed ball of center x0and radius ρ.

Sx0ρ=xRn:xx0=ρthe boundary of Bx0ρ.

Let QRnnNa subset. Qdenote the boundary of Q.

ln: the natural logaritm function.

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

BUCRnthe Banach space of bounded and uniformly continuous functions on Rnwith the supremum norm u=supxRnux.

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

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

For u,v:RnRtwo regular functions, u=ux1uxnand 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,xΩwith ΩRnnNis an open set, Δis the Laplacian operator, a, dare two real positive constants called the coefficients of the diffusion. For a chemical reaction where two substances S1and S2, uand vrepresent their concentrations at time tand position xrespectively, and f1and f2represent 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, dare such that a,dR+.

(H2)λ:R+×RnRis a non-null, nonnegative and bounded function on R+×Rnsuch that λt.BUCRnfor all tR+. We denote λ=supt0λt.

(H3)nand mare positive integers, i.e. n,mN.

(H4)g:BUCRnBUCRnis a function defined on BUCRnsuch that:

i. g0=0and gu0pour u0.

ii. gis of class C1and dguduis bounded on R.

(H5)The initial data u0, v0are 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>1and φu=u.In this particular case, this system describes the evolution of uthe mass fraction of reactant Aand that vof the product Bfor the autocatalytic chemical reaction of the form A+mBm+1B.They proved the existence of global solutions and showed that the Lnorm of vcan not grow faster than Olnlntfor any space dimension.

If we replace guvmby uexpE/vwhere E>0is 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,m2Nis an even positive integer. He has proved that if u0, v0are nonnegative and are in BUCRnthat:

If a>d,b>0,v0badu0on Rn, then the solution is global and uniformly bounded.

If a<d,b<0,v0badu0on 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<dwhere we used the Lyapunov functional Luv=α+2uln1+ueεv(α,ε>0)and the cut-off function φx=1+x2n.We show that for αsufficiently large and εsmall enough we can control the Lp-norms of vp>max1n/2on every unit spacial cub in Rnfrom which we deduce the L-norm of vat any time t>0.

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

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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×BUCRnof the form

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

Here wt=utvt; the operator Ais defined as

AwaΔ00dΔw=aΔudΔv,

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

It is known that for c>0the operator cΔgenerates an analytic semigroup Gtin the space BUCRn:

Gtu=4πctn/2Rnexpxy24ctuydy.E5

Hence, the operator Agenerates an analytic semigroup defined by

St=S1t00S2t,E6

where S1tis the semigroup generated by the operator aΔ, and S2tis the semigroup generated by the operator dΔ.

Since the map Fis locally Lipschitz in win the space X, then proving the existence of a loacl classical solution on 0t1where t1R+is standard [7, 8].

For the positivity, let wt=utvtis a local solution of the problem (1)(3) under the assumptions Hjj=15on 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 vmugξis 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 uvbe the solution of the problem (1)(3) under the assumptions Hjj=15and such that

ad.E12

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

utu0,foralltR+,E13
vtv0+λadn/2u0,foralltR+.E14

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

The solution uvsatisfies the integral equations

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

Here S1tand S2tare the semigroups generated by the operators aΔand dΔin the space BUCRnrespectively. As uis nonnegative, then from (15) we get

0tS1tτguτvmτS1tu0.E17

Since ad, using the explicit expression of S1tτguτvmτand S2tτguτvmτ, one can observe that (see [10])

0tS2tτλτguτvmτadn/20tS1tτλτguτvmτλadn/20tS1tτguτvmτ.E18

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

vtS2tv0+λadn/2S1tu0.E19

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

Thus, from (13) and (14), we deduce that the solution uvis 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 uvbe the solution of the problem (1)(3) with the assumptions Hjj=15. If

a<d,E20

the solution uvis 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 t1by 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 uvbe the solution of the problem (1)(3) under the assumptions Hjj=15on the local interval time 0t1. Define the functional

Luv=α+2uln1+ueεvwithα,εR+.E21

Then for any φ=φxxRna 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>0and 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 uνis the derivative of uwith respect to the unit outer normal νto the boundary S0ρ.

We have

S0ρφL1tutνdx2eεvtutνS0ρφdx2eεvtutν11+ρ2n2πn/2ρn1Γn/2.E27

From (27) we obtain

limρSx0ρφL1tutνdx=0.E28

We pass to the limit for ρin (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=αadγ1u0and ε=εadγ1γ2λu0such that

ddtRnφLdxdRnLΔφdx+daRnL1φ.udxγ1RnφaL11u2+dL22v2dxγ2RnφL1guvmdx,E32

where γ1,γ201are two arbitrary constants.

Proof.We seek Lsuch that

aL11u2+a+dL12uv+dL22v2γ1aL11u2+dL22v2E33

and

λL2L1γ2L1E34

for γ1,γ201.

The inequality (33) is satisfied if

a+d2L1224ad1γ12L11L121.E35

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

αa+d21+2u024ad1γ12.E36

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

0<ε1γ2λ4ad1γ12a+d21+2u02+8ad1γ12u0.E37

Whence, if αsatisfies (36) and εsatisfies (37), we obtain (32).

As a consequence of (33) we have

ddtRnφLdxdRnLΔφdx+daRnL1φ.udxγ1aRnφL11u2dx.E38

Lemma 3.5.With the functional Ldefined in (21) and α, εdefined in (36) and (37) respectively and with the truncation function φ:RnRdefined by

φx=11+xx02n.E39

We have

ddtRnφLdxdk1nRnφLdx+14γ1ada2k22nRnφL12L11dx,E40

where

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

Proof.Calulate Δφand estimate it

Δφ=2n21+xx02n+14nn+1xx021+xx02n+2;

whence

Δφ2n3n+2φ.E42

Calulate φand estimate it

φ2=4n2xx021+xx022n+2;

whence

φ2.E43

Using the Cauchy-Schwarz inequality φ.uφuand 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 αand εdefined 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

σ=dk1n+γ4γ1ada2k22n.E49

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 γsatisfies (46) then (50) follows.

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

ddtRnφLdxdk1n+γ4γ1ada2k22nRnφLdx,foralltR+.E51

As

RnφLt=0dx=Rnφα+2u0ln1+u0eεv0dx;E52

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

RnφLdxα+2u0φ1expεv0eσt,foralltR+,E53

where σis 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 βand σgiven by (48) and (49).

In the following step we trie to control the second component vof the solution on any unit spacial cube in the Lpnorms with p1.

Let x0=x10xn0Rnbe an arbitrary fixed point and

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

Lemma 3.7.Let uvbe the solution of the problem in consideration. For αand εsatisfying (36) above and (63) below respectively, then for any unit cube Qof Rnof 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 p1and k=p+1,then we have by the imbedding theorem for Lpspaces

QvpdxQvkdxp/k.E62

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

0<εmin1γ2λ4ad1γ12a+d21+2u02+8ad1γ12u0,1αβk!4+n4n1/k.E63

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

Lemma 3.8.Let Qiet Qjbe two different unit cubes of center xi=x1ixniand xj=x1jxnjrespectively 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=z1znQjis of the form

z=xj+s,E69

where s=s1sn0and sk1212,for all k=1,,nwith at least one of the sk1212.

It’s easy to prove that

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

Then

xjxi2distxiQj.E71

As xiyxixj+xjywe 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 xRnan arbitrary point and QjjNbe the family of pairwise disjoint measurable cubes of the form (64) covering Rnsuch that the center of Q0is x0=x.

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

Rnexy24dtsλguvmdy=j=0Qjexy28dtsexy28dtsλguvmdyj=0edistxQj28dtsQjxy28dtsλguvmdy.E74

By Hölder ineguality with p>max1n2and 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 ton the right-hand side of the estimate (83) is continuous on R+. As utu0on 0tmaxand vsatisfied (83), we conclude from (11) that tmax=+. Whence, the solution is global.

Remark.We can extend the system to the case where instead of vmwe put vhvprovided that.

i. h:BUCRBUCRis a locally continuous Lipschitz function, namely: for all constant ρR+,there exists a constant cρR+such that for all u,vBUCRnwith uρand vρwe have

hugvcρuv.

ii.There exist two constants MR+and rNsuch 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,,4are real positive constants and c5is a real nonnegative constant. If a,bR+,n,mN,u0,v0BUCRnand 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<dthat 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=gusatisfying the hypothesis (H4) by the function gu=urwith r1is a real constant and replace the term vmby eαvwith 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].

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Salah Badraoui (March 26th 2021). <article-title xmlns:mml="http://www.w3.org/1998/Math/MathML">Global Existence of Solutions to a Class of Reaction–Diffusion Systems on <inline-formula id="I1"><mml:math id="m1"><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>n</mml:mi></m, Recent Developments in the Solution of Nonlinear Differential Equations, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.95543. Available from:

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