We consider a system of reaction-diffusion equations for which the uniform boundedness of the solutions can not be derived by existing methods. The system may represent, in particular, an epidemic model describing the spread of an infection disease within a population. We present an $L^{p}$ argument allowing to establish the global existence and the uniform boundedness of the solutions of the considered system

Melkemi, L.

Mokrane, A.Z.

Youkana, A.

Electronic journal of qualitative theory of differential equations

Repository of the Academy's Library

Electronic Journal of Qualitative Theory of Differential Equations2005, No. 24, 1-10; http://www.math.u-szeged.hu/ejqtde/On the uniform boundedness of the solutions ofsystems of reaction-diffusion equationsLamine MELKEMI, Ahmed Zerrouk MOKRANE, and Amar YOUKANAAbstract. We consider a system of reaction-diffusion equations forwhich the uniform boundedness of the solutions can not be derived byexisting methods. The system may represent, in particular, an epidemicmodel describing the spread of an infection disease within a population.We present an Lp argument allowing to establish the global existenceand the uniform boundedness of the solutions of the considered system.1. INTRODUCTIONIn this work we consider the following class of reaction diffusion systems(1){∂u∂t− d1∆u = c− f(u, v)− αu in R+ × Ω∂v∂t− d2∆v = g(u, v) − σv in R+ × Ωwith Neuman boundary conditions(2)∂u∂ν=∂v∂ν= 0 on R+ × Γwhere Ω is an open, bounded domain in Rn with boundary Γ = ∂Ω ofclass C1, d1, d2, α, σ are positive constants, c ≥ 0 and f, g are nonnegativefunctions of class C1(R+ × R+) satisfying the following assumptions(A1) f(0, .) = 0, g(., 0) ≥ 0,(A2) ξ ≥ 0, η ≥ 0 =⇒ 0 ≤ f(ξ, η) ≤ (1 + η)βϕ(ξ)and(A3) (∀ξ, η ≥ 0) g(ξ, η) ≤ ψ(η)f(ξ, η) with limη→+∞ψ(η)η= 0where β ≥ 1 and ϕ,ψ are nonnegative functions of class C(R+).2000 Mathematics Subject Classification. 35K57, 35K45.Key words and phrases. Reaction-Diffusion, Uniform Boundedness, Lyapunovfunction.EJQTDE, 2005 No. 24, p. 1We are interested in the problem of the existence of global solutions tothe system (1)-(2) in the class C(Ω¯) with initial data as well as their uniformboundedness(3) u(0, x) = u0, v(0, x) = v0with u0, v0 ∈ L∞(Ω) and u0 ≥ 0, v0 ≥ 0.We observe that the system (1)-(3) generalizes that when g = f sincethe assumption (A3) obviously holds in this case. In practice, when g = f ,the system (1)-(3) may represent an epidemic model describing the spreadof an infection disease within a population in which case c is the recruitmentratio into the susceptible class, α is the natural death rate and 1σdenotes theaverage infectious period (for details see for example [12], [3], [6],[8]). Whenc = 0 and g(u, v) = f(u, v) + αu+ σv we get a system of reaction-diffusionequations with balance law initially posed by R. H. Martin for which theproblem of the existence of global solutions and their uniform boundednesswas studied by many authors. The question was positively answered byAlikakos [1] and Masuda [10] under the assumption (A2) and later on byHaraux and Youkana [4], Youkana [13], Barabanova [2] and Kanel andKirane [9] when the vector field may be a function with exponential growth.When f = g and c > 0, it is clear that the problem of the uniformboundedness of the solutions of (1)-(3) is not as immediate as the globalexistence. In [7], Hollis, Martin and Pierre used duality arguments on Lptechniques allowing under certain conditions to derive the uniform bound-edness of solutions of (1) with mixed boundary conditions from uniformestimates. However we do not see how the method in [7] may be applied tothe problem (1)-(3). In order to establish the uniform boundedness of thesolutions of (1)-(3) in this case (that is when g = f and c > 0) one maypropose the method developed by Morgan in [11]. Indeed the method in[11] assures the uniform boundedness of the solutions of (1)-(3) once it isverified that(i) The vector field of (1) (i.e. the right hand side in (1)) is polynomialin v )(ii) c− f(u, v)− αu,(4) λ(c− f(u, v)− αu) + µ(g(u, v) − σv)EJQTDE, 2005 No. 24, p. 2for some positive constants λ, µ > 0 are uniformly upper boundedand(iii)∫Ω(u+ v) ≤ C independent of t.Using standard manipulations, one can check that concerning the system(1)-(3) considered here, the conditions (i)-(iii) are satisfied provided that ψis bounded. Unfortunately, when ψ is not bounded, the condition (ii) is notsatisfied.In this paper our main concern is to establish the global existence andthe uniform boundedness of the solutions of the system (1)-(3) under theassumptions (A1)-(A3). To this end we make use of the Lyapunov functiontechniques and present an approach similar to that developed by Harauxand Youkana [4] and Barabanova [2].2. STATEMENT AND PROOF OF THE MAIN RESULTIt is classical that for nonnegative u0, v0 ∈ L∞(Ω) there exists a uniquelocal nonnegative solution (u, v) of class C(Ω¯) of (1)-(3) on ]0, T ∗[, whereT ∗ is the eventual blowing-up time in L∞(Ω) (see [5]). By the comparisonprinciple one may also show that(5) 0 ≤ u(t, x) ≤ max(||u0||∞,cα) =: K.As a consequence our problem amounts to establish the uniform bound-edness of v. To do so, we will make use of the result established in [5] fromwhich the uniform boundedness of v is derived once||g(u, v) − σv||p ≤ C(where C is a nonnegative constant independent of t) for some p > n2 . Be-cause of the assumptions (A2) and (A3), we are led to establish the uniformboundedness of the ||v||p on ]0, T∗[ in order to get that of ||v||∞ on ]0, T∗[.For p ≥ 2, we letγ =(d1 − d2)24d1d2, γ(p) =pγ + 1p− 1, Mp = K +cγ(p)σ.The main result of this paper is stated in what follows:EJQTDE, 2005 No. 24, p. 3Proposition 1. Assume p ≥ 2 and let(6) Gb(t) =∫Ω[bu+ exp(−p− 1pγ + 1ln(γ(p)[Mp − u]))vp]dtwhere (u, v) is the solution of (1)-(3) on ]0, T ∗[. Then under the assumptions(A1)-(A3) there exist two positive constants a > 0 and b > 0 such that(7)ddtGb ≤ −(p− 1)σGb + a .Before proving this theorem we first need the following two technicallemmas:Lemma 1. Let (u, v) be a solution of (1)-(3). Then(8)ddt∫Ωudx+∫Ωf(u, v)dx+ α∫Ωudx = c|Ω|.Proof. We integrate the both sides of the equationdudt− d1∆u = c− f(u, v)− αusatisfied by u on Ω. We find∫Ωf(u, v)dx = c|Ω| − α∫Ωu−ddt∫Ωu(t, x)dx.Lemma 2. Assume that p ≥ 2. Then under the assumptions (A1)-(A3)there exists b1 > 0 such that(9) [pηp−1g(ξ, η)−1γ(p)Mpηpf(ξ, η)] ≤ b1f(ξ, η)for all 0 ≤ ξ ≤ K and η ≥ 0.Proof. According to the assumption (A3), we have[pηp−1g(ξ, η)−1γ(p)Mpηpf(ξ, η)] ≤ [pψ(η)η−1γ(p)Mp]ηpf(ξ, η).Since ψ(η)ηgoes to 0 as η → +∞, there exists η0 > 0 such that(0 ≤ ξ ≤ K, η ≥ η0) =⇒ [pψ(η)η−1γ(p)Mp]ηpf(ξ, η) ≤ 0EJQTDE, 2005 No. 24, p. 4On the other hand(ξ, η) 7−→ pηp−1ψ(η)−1γ(p)Mpηpbeing continuous is bounded on the compact interval [0, η0] so that (9) im-mediately follows.We proceed now to the proof of Proposition 1Proof of Proposition 1. Leth(u) = −p− 1pγ + 1ln(γ(p)(Mp − u))so thatGb(t) = b∫Ωudx+ L(t)whereL(t) =∫Ωeh(u)vpdx.Differentiating L(t) with respect to t and using the Green formula oneobtainsddtL = H + SwhereH = −d1∫Ω(h′2(u) + h′′(u))eh(u)vp(∇u)2dx−p(d1 + d2)∫Ωh′(u)eh(u)vp−1∇u∇vdx−d2∫Ωp(p− 1)eh(u)vp−2(∇v)2dxandS = c∫Ωh′(u)eh(u)vpdx+∫Ω[pvp−1g(u, v) − h′(u)vpf(u, v)]eh(u)dx−α∫Ωh′(u)ueh(u)vpdx− σ∫Ωpeh(u)vpdx.We observe that H involves a quadratic form with respect to ∇u and ∇vQ = d1(h′2(u)+h′′(u))vp(∇u)2+p(d1+d2)h′(u)vp−1∇u∇v+d2p(p−1)vp−2(∇v)2EJQTDE, 2005 No. 24, p. 5which is nonnegative if(10) [p(d1 + d2)h′(u)vp−1]2 − 4d1d2p(p− 1)(h′2(u) + h′′(u))v2p−2 ≤ 0.We have chosen h(u) in such a way thath′(u) =1γ(p)(Mp − u), h′′(u) =γ(p)[γ(p)(Mp − u)]2.The left hand side of (10) can be writtenv2p−2{p2[(d1 − d2)2h′2(u)− 4d1d2h′′(u)] + 4d1d2p(h′2(u) + h′′(u))}= 4d1d2pv2p−2{p[γ1[γ(p)(Mp − u)]2−γ(p)[γ(p)(Mp − u)]2]+1 + γ(p)[γ(p)(Mp − u)]2}= 0since pγ−pγ(p)+1+γ(p) = 0. Therefore (10) holds, Q ≥ 0 and consequentlyH = −∫ΩQeh(u)dx ≤ 0.Concerning the second term S we observe thatS ≤∫Ω(ch′(u)− σp)eh(u)vpdx+∫Ω[pvp−1g(u, v) − h′(u)vpf(u, v)]eh(u)dx≤ −(p− 1)σ∫Ωeh(u)vpdx+∫Ω[pvp−1g(u, v) − h′(u)vpf(u, v)]eh(u)dxsinceh′(u) =1γ(p)(Mp − u)≤1γ(p)(Mp −K)=σcOn the other hand−h′(u) = −1γ(p)(Mp − u)≤ −1γ(p)Mph(u) ≤ −1γ(p)lncσ.EJQTDE, 2005 No. 24, p. 6Therefore by virtue of Lemma 2 (in particular (9)) and the fact that v ≥ 0pvp−1g(u, v) − h′(u)vpf(u, v) ≤ pvp−1g(u, v) −1γ(p)Mpvpf(u, v)≤ b1f(u, v).As a consequenceS ≤ −(p− 1)σL+ b1∫Ωeh(u)f(u, v)dx≤ −(p− 1)σL+ b1e−1γ(p)ln cσ∫Ωf(u, v)dx.Letb = b1e− 1γ(p)ln cσwhere b1 > 0 is a positive constant satisfying (9). Using (8) one obtainsS ≤ −(p− 1)σL+ b(c|Ω| −ddt∫Ωu(t, x)dx)≤ −(p− 1)σL+ bc|Ω| − bddt∫Ωu(t, x)dx.Since Gb(t) = b∫Ω udx+ L(t), it follows thatS ≤ −(p− 1)σGb + (p− 1)σb∫Ωudx+ bc|Ω| − bddt∫Ωu(t, x)dx≤ −(p− 1)σGb + b((p− 1)σK + c)|Ω| − bddt∫Ωu(t, x)dxfrom which we conclude thatddtGb(t) ≤ −(p− 1)σGb(t) + awith a = b((p− 1)σK + c)|Ω|.We are now ready to establish the global existence and uniform bound-edness of the solutions of (1)-(3)Theorem 1. Under the assumptions (A1)-(A3), the solutions of (1)-(3)are global and uniformly bounded on [0,+∞[.EJQTDE, 2005 No. 24, p. 7Proof. By a multiplication of the inequality (7) by e(p−1)σt and thenintegrating, we deduce that there exists a positive constant C1 > 0 indepen-dent of t such that:Gb(t) ≤ C1.Sinceeh(u) ≥ e− 1γ(p)ln γ(p)Mpit follows that for all p ≥ 2∫Ωvpdx ≤ e1γ(p)ln[Kγ(p)+ cσ]Gb(t)≤ C1.[Kγ(p) +cσ]1γ(p) =: C(p).Select now p > n2 and proceed to bound ||g(u, v)−σv||p. Beforehand letA = max0≤η≤η0ψ(η), B = max0≤ξ≤Kϕ(ξ)where η0 is such thatη ≥ η0 =⇒ ψ(η) ≤ ηAccording to (A2)-(A3), we haveg(u, v) ≤ ψ(v)f(u, v)≤ ψ(v)ϕ(u)(1 + v)β≤ Bψ(v)(1 + v)βsince 0 ≤ u ≤ K. Therefore∫Ωg(u, v)p ≤ Bp∫Ωψ(v)p(1 + v)βp= Bp(∫v≤η0ψ(v)p(1 + v)βp +∫v≥η0ψ(v)p(1 + v)βp)as a consequence∫Ωg(u, v)p ≤ (AB(1 + η0)β)p|Ω|+Bp∫Ωvp(1 + v)βp.Using the fact that for x, y ≥ 0 and r ≥ 1(x+ y)r ≤ 2r−1(xr + yr)EJQTDE, 2005 No. 24, p. 8we obtain∫Ωg(u, v)p ≤ (AB(1 + η0)β)p|Ω|+ 2βp−1Bp(C(p) + C((β + 1)p))= : Rp.Finally||g(u, v) − σv||p ≤ ||g(u, v)||p + σ||v||p≤ R+ σ p√C(p).Using the result of [5, p.39, Thm 1.6.1] we conclude that the solutions of(1)-(3) are indeed global and uniformly bounded on ]0,+∞[×Ω.Acknowledgement 1. We have benefited from helpful discussions withProfessor M.Ziane of South California University. We gratefully thank himfor his advises. We also thank the anonymous referee for his valuable com-ments.References[1] N. Alikakos, Lp-Bounds of reaction-diffusion equations, Comm. P.D.E 4(1979). 827-828[2] A. Barabanova, On the global existence of solutions of a reaction-diffusion equationwith exponential nonlinearity, Proc. A.M.S vol.122 no3 (1992). 827-831[3] C. Castillo-Chavez, K. Cook, W. Huang and S. A. Levin, On the role of long incu-bation periods in the dynamics of acquires immunodeficiency syndrome (AIDS), J.Math. Biol. 27(1989) 373-398[4] A. Haraux and A. Youkana., On a result of K. Masuda concerning reaction-diffusionequations, To¨hoku Math. J. 40(1988) 159-163[5] D. Henry, Geometric theory of semilinear parabolic equations, Lecture notes in math-ematics 840, Springer Verlag, New York (1981)[6] Y. Hamaya, On the asymptotic behavior of a diffusive epidemic model (AIDS), Non-linear Analysis 36(1999) 685-696.[7] S. I. Hollis, R. H. Martin and M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal. vol. 18 no3 (1987) 744-761[8] W. E. Fitzgibbon, M. Langlais, F. Marpeau and J. J. Morgan, Modeling the circula-tion of a disease between two host populations on non coincident spatial domains, toappear in J. Biological Invasions.[9] J. I. Kanel and M. Kirane, Global solutions of reaction-diffusion systems with a balancelaw and nonlinearities of exponential growth, J. Diff. Equations 165 no1 (2000) 24-41[10] K. Masuda, On the global existence and asymptotic behavior of solutions of reaction-diffusion equations, Hokkaido Math. J. 12(1983) 360-370EJQTDE, 2005 No. 24, p. 9[11] J. Morgan, Boundedness and decay results for reaction-diffusion systems, SIAM J.Math. Anal. 21(1990) 1172-1189[12] G. F. Webb, A reaction diffusion model for a deterministic diffusive epidemic, J.Math. Anal. Appl.84(1981) 150-161[13] A.Youkana, The`se de 3e`me cycle, Chapitre 3, Universite´ de Paris 6.(1986)(Received July 1, 2005)Department of Mathematics, Faculty of Science, University of BatnaE-mail address: lmekmi@yahoo.frDepartment of Mathematics, Faculty of Science, University of BatnaE-mail address: ahmed mokr@hotmail.comDepartment of Mathematics, Faculty of Science, University of BatnaE-mail address: youkana amar@yahoo.frEJQTDE, 2005 No. 24, p. 10

On the uniform boundedness of the solutions of systems of reaction-diffusion equations

L. Melkemi

Ahmed Mokrane

A. Youkana

Directory of Open Access Journals

On the uniform boundedness of the solutions of systems of reaction-diffusion equations

Abstract

Similar works

Full text

Available Versions

Repository of the Academy's Library

Directory of Open Access Journals