Open access peer-reviewed chapter

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

From the Edited Volume

Recent Developments in the Solution of Nonlinear Differential Equations

Edited by Bruno Carpentieri

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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,


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

1. Introduction

In the sequel, we use the notations.


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


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


with initial data


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


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


Here wt=utvt; the operator A is defined as


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:


Hence, the operator A generates an analytic semigroup defined by


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


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


The second equation can be written as


By the same theorem we get


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


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

The solution uv satisfies the integral equations


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


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


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


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




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


Using integration by parts, we get


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


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

We have


From (27) we obtain


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

By the same way we get


From (30) we find that


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


where γ1,γ201 are two arbitrary constants.

Proof. We seek L such that




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


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


We have




Proof. Calulate Δφ and estimate it




Calulate φ and estimate it




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


We pove that


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


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


we have






Proof. We seek a constant γR+ such that


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




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


where σ is defined by (49).

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






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


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


Proof. It’s obvious that






Let us combine (47) and (59)


By induction we prove that


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


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


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


such that


Proof. By Pythagorean theorem we have


As xixj1, then from (67)


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


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

It’s easy to prove that




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


We have obviously


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)


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




and by (56) we have




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




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




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




and finally we have for all tR+


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


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


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


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].


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Written By

Salah Badraoui

Submitted: 02 October 2020 Reviewed: 18 December 2020 Published: 26 March 2021