Existence and uniqueness of periodic solutions for a nonlinear reaction-diffusion problem

Publicacions Matem`atiques, Vol. 44 (2000), 295–307 EXISTENCE AND UNIQUENESS OF PERIODIC SOLUTIONS FOR A NONLINEAR REACTION-DIFFUSION PROBLEM Maurizio Badii Abstract We consider a class of degenerate reaction-diffusion equations on a bounded domain with nonlinear flux on the boundary. These problems arise in the mathematical modelling of flow through porous media. We prove, under appropriate hypothesis, the existence and uniqueness of the nonnegative weak periodic solution. To establish our result, we use the Schauder fixed point theorem and some regularizing arguments. 1. Introduction This paper deals with the existence and uniqueness of periodic solutions for the following nonlinear reaction-diffusion problem ut= div(∇ϕ(u)) + c(x, t, u),in Q:= Ω ×R(1) −∂ϕ(u)/∂ν =g(ϕ(u)),on ∂Ω×R(2) u(x, t +ω)=u(x, t) and u≥0inQ(3) where Ω is a bounded domain in RNwith smooth boundary ∂Ω and νdenotes the outward unit normal to ∂Ω. Equation (1) models the filtration of a fluid in a homogenous, isotropic, rigid and unsaturated porous medium, with the lower order term c(x, t, u)≥0inΩ×R×R+, accounts for a reaction taking place in the medium. We consider the Partially supported by G.N.A.F.A. and M.U.R.S.T. 40% Equazioni Differenziali. 296 M. Badii following assumptions on the data (Hϕ)     ϕ∈C([0,∞)) ∩C2+α loc ((0,∞)),ϕ(0)=0 and ϕ(s)>0 for s>0 and there exist r0>0,α 0>0,α 1>0,0≤m0≤m1<1 such that α0ϕ(r)m0≤ϕ(r)≤α1ϕ(r)m1,for any r≥r0. (Hc)         i) c∈C(Ω×R×R+),c(x, t +ω,s)=c(t, s), cu∈C(Ω ×R×R+\{0}) ii) cu∈L∞(Q×[−M,M]) for every M>0 iii) c(x, t, r)≤C0ϕ(r)β,C 0>0,β∈[0,1) and r≥r0. (Hg)a) g∈C(R+),g≥0,g∈C1(R+),|g(r)|<∞for all r≥0 b) there exists d0>0 such that g(ϕ(r))≥d0ϕ(r) for all r≥r0. Remark. Hypotesis (Hc) iii) is enough to have the global existence result for the associated initial-boundary value problem (see [2]). In problem (1)-(3), udenotes the moisture content in the soil represented by the domain Ω, therefore we require the condition u≥0. Assumption (Hϕ) includes the case of degenerate equation i.e. ϕ(0) = 0, thus classical solution doesn’t exist and a concept of weak solution has to be introduced; the weak solution is continuous but not smooth. In this paper we prove the existence and uniqueness of the periodic weak solution to (1)-(3) under the above assumptions. To prove the existence of periodic weak solutions we use as a preliminear step the Schauder fixed point theorem for the Poincar´e map of a nondegenerate initial-boundary value problem associated to (1)-(3). To this purpose, we will consider a sequence of approximated nondegenerate problems which can be solved in a classical sense. The uniqueness of the periodic weak solution will be established using an adaptation of the method of [4] and the assumption that g(ϕ(u)) = d0ϕ(u). Initial and Dirichlet’s boundary value problems have been studied to equation (1) by many authors; we quote for example [6], [7], [12] and [13]. Recently, [2] and [1] have studied the asymptotic behavior and the blow-up in finite time of solutions for equations of type (1) with various boundary conditions. Periodic solutions for a nonlinear problem 297 To the knowledge of the author, it seems that the topic considered in the present paper has not been discussed previously. We only are aware of the paper [8] which treats the periodic case for (1)-(3) but for g=0, ϕ(u)=um,m>1 and a zero order term which is linear w.r.t.u. Our assumptions, allows to consider more general ϕand zero order terms. 2. Existence of periodic solutions Since our equation (1) may degenerates, we make the following definition of periodic solution Definition 1. A function u∈C(I;L1(Ω))∩L∞(QI)(QI:= Ω×(t0,t 1)) is said to be a periodic weak solution of (1)-(3) if for any compact interval I=[t0,t 1]∈R, satisfies −∂ϕ(u(x, t))/∂ν =g(ϕ(u)) on ∂Ω×I, u(x, t +ω)=u(x, t), ϕ(u)∈L2((t0,t 1); H1(Ω)) and t1 t0Ω (uζt+ϕ(u)∆ζ+c(x, t, u)ζ(x, t)) dx dt =Ω (u(x, t1)ζ(x, t1)−u(x, t0)ζ(x, t0)dx +t1 t0∂Ω g(ϕ(u))ζ(x, t)dx dt +t1 t0∂Ω ϕ(u)∂ζ(x, t)/∂ν dσ dt, for any ζ,ζt,∆ζ∈L2(Ω ×(t0,t 1)), and ∂ζ/∂ν ∈L2(∂Ω×(t0,t 1)). To show the existence of periodic weak solutions to (1)-(3), we begin to show the existence of a periodic solution for the approximated problem uεt = div(∇ϕε(uε)) + cε(x, t, uε),in Q(4) −∂ϕε(uε)/∂ν =gε(ϕε(uε)),on ∂Ω×(0,∞)(5) uε(x, t +ω)=uε(x, t) and uε≥0,in Q(6) where ϕε,cεand gεare smooth approximations of ϕ,cand g, constructed by convolutions with mollifiers functions. 298 M. Badii Construct ϕε(s), cε(x, t, s) and gε(s) such that (Hε)     ϕε∈C2+α loc ([0,∞)),ϕ ε(0) = 0,ϕ ε(s)=ϕ(s),for s≥ε/2 ϕ ε(s)≥ε, for all s≥0, ϕε→ϕuniformly on compact subsets of R+. (Hcε)     cε∈C1(QT×R+),c ε(x, t, 0) = 0,c ε(x, t, s)=c(x, t, s), for all (x, t)∈QTand s≥ε/2, cε→cuniformly on compact subsets of QT×R+. (Hgε)α)gε∈C∞(R+),g ε→guniformly on compact subsets of R+, β)gε(ϕε(r)) ≥d0ϕε(r) for all r≥r0, γ)gεis uniformly Lipschitz continuous on R+. We shall show for (4)-(6) the existence of periodic weak solutions as fixed points for the Poincar´e map of a suitable initial-boundary value problem. Hence, we need construct a closed, convex and nonempty set where to find these fixed points for the following sequence of nondegenerate initialboundary value problems. uεt = div(∇ϕε(uε)) + cε(x, t, uε),in QT,T≥ω(7) −∂ϕε(uε)/∂ν =gεϕε(uε),on ∂Ω×(0,T)(8) uε(x, 0) = u0ε(x),on Ω(9) where (H0ε)u0ε∈C2(Ω), such that 0 ≤u0ε(x) for all xin Ω and satisfying the compatibility condition −∂ϕε(u0ε)/∂ν =gε(ϕε(u0ε)) on ∂Ω. Problem (7)-(9) is an approximation of the following problem ut= div(∇ϕ(u)) + c(x, t, u),in QT,T≥ω(10) −∂ϕ(u)/∂ν =g(ϕ(u)),on ∂Ω×(0,T)(11) u(x, 0) = u0(x),on Ω(12) where u0∈L∞(Ω),0≤u0(x),a.e. in Ω,u0ε∞≤u0∞and u0ε→u0in L2(Ω) as ε→0+. (13) Periodic solutions for a nonlinear problem 299 It is well known that there exists a unique classical solution uεof (7)-(9) (see [9]). Moreover, by a result of [2], the following estimates hold uε∞≤C1(u0∞)(14) T 0Ω |∇ϕε(uε)|2dx dt ≤C2(u0∞),for all ε>0.(15) In [2] is showed that the set of stationary solutions of (10)-(12) is bounded in L∞(Ω) i.e. the solutions u(x)of div(∇ϕ(u)) + C0ϕ(u)β= 0 in Ω −∂ϕ(u)/∂ν =g(ϕ(u)) on ∂Ω (SP) satisfy u∞=M. It is easy to verify that εis a subsolution of (SP), hence u(x)≥εa.e. in Ω. If u0εis chosen in such way that besides satisfies ε≤u0ε(x)≤u(x)≤M, a.e. in Ω, then u(x) is a supersolution and εis a subsolution to (7)-(9). Thus we have ε≤uε(x, t)≤u(x)≤M, for a.e. (x, t)∈QT.(16) Applying the continuity result of [5], the following regularity property for the solutions of (7)-(9) holds. Proposition 1 ([5]).Since u0εis continuous on Ω, the sequence {uε} of the solutions to (7)-(9) is equicontinuous in QTi.e. there exists ω0: R+→R+,ω0(0) = 0, continuous and nondecreasing such that |uε(x1,t 1)−uε(x2,t 2)|≤ω0(|x1−x2|+|t1−t2|1/2) for any (x1,t 1),(x2,t 2)∈Ω×[0,T]. The function s→ω0(s), depends on the essential bound of uεin QT. If consider the Poincar´e map associated to problem (7)-(9) and defined by F(u0ε(·)) = uε(·,ω) where uεis the unique solution of (7)-(9) and introduce the closed, nonempty, bounded and convex set Kε:= {w∈C(Ω):ε≤w(x)≤M, for any x∈Ω} then, by (16) and Proposition 1, we get 300 M. Badii 1. i) F(Kε)⊂Kε. 2. ii) F(Kε) is relatively compact in C(Ω). Remains to prove that 1. iii) F|Kεis continuous. To this purpose, we show Proposition 2. If un 0ε,u0ε∈Kεand un 0ε→u0εuniformly in Ωas n→∞, then if un εand uεare solutions to (7)-(9) of initial data un 0εand u0εrespectively, we have that un ε(·,t)converges to uε(·,t)uniformly as n→∞, for any t∈[0,T]. Proof: Multiplying (7) by sgn(un ε−uε) and integrating on Qtone has Ω |un ε(x, t)−uε(x, t)|dx ≤Ω |un 0ε(x)−u0ε(x)|dx +Lt 0Ω |un ε(x, t)−uε(x, t)|dx dt, because of the local Lipschitz continuity of cε(x, t, ·) with Lipschitz constant L. Applying Gronwall’s lemma, it is easy to see that un ε(x, t) converges to uε(x, t) strongly in L1(Ω) as ngoes to infinity. Consequently, for a subsequence, we have that un ε(x, t) converges to uε(x, t) for a.e. x∈Ω. Since un ε(x, t)≤M, by the Lebesgue theorem, we conclude that un ε(x, t)→uε(x, t)inLp(Ω) for any 1 ≤p≤∞. Since un ε(·,t), uε(·,t)∈C(Ω) the uniform convergence holds. Thus, by the Schauder fixed point theorem it follows that there exists a fixed point for the Poincar´e map, which is a periodic solution to (4)-(6). From (16) we get, for a subsequence if necessary, that uε%u, in L2(Ω).(17) Since the set of ϕε(uε) is relatively compact in L2(QT), we have that uεstrongly converges to uin L2(QT). In fact set vε=ϕ(uε), vεis bounded in L2(0,T;H1(Ω)) ⊂L2(0,T;Ws,2(Ω)), 0 <s<1. If we suppose that ϕ−1is H¨older continuous of order θ∈(0,1)(18) Periodic solutions for a nonlinear problem 301 for a classical result (see [3]) one has uε(t)1/θ wθs,2/θ(Ω) ≤vε(t)ws,2(Ω)ϕ−11/θ H¨older that integrate with respect to t, gives uε2/θ L2/θ(0,T ;wθs,2/θ(Ω)) ≤vεL2(0,T ;ws,2(Ω))ϕ−12/θ H¨older. Moreover, (see [3]) Wθs,2/θ(Ω) ⊂L2(Ω) with compact injection. Then, uε→u, in L2(QT) and a.e. .(19) From (18) and the Lebesgue theorem one has ϕε(uε)→ϕ(u),in L2(ΩT)(20) thus, we get by (15) and (20) that ∇ϕε(uε)%∇ϕ(u), in L2(QT) and ϕε(uε)%ϕ(u),in H1(Ω).(21) Theorem 3.4.5 of [11] states that if (21) holds, then ϕε(uε) converges to ϕ(u)inL2(0,T;L1(∂Ω)). Now, by the uniform convergence of gεon compact set of R+and its uniformly Lipschitz continuity, it is easy to see that gε(ϕε(uε)) →g(ϕ(u)) in L2(0,T;L2(∂Ω)). In fact T 0∂Ω |gε(ϕε(uε)) −g(ϕ(u))|2dσ dt ≤LT 0∂Ω |ϕε(uε)−ϕ(u)|2dσ dt +T 0∂Ω |gε(ϕ(u)) −g(ϕ(u))|2dσ dt→0asε→0. Finally, in [2] is proven that u∈C([0,T]; L1(Ω)), ϕ(u)∈L2(0,T;H1(Ω)) thus, uis a periodic weak solution to (1)-(3). 302 M. Badii 3. Uniqueness To get the uniqueness result, suppose that uεand vare periodic solutions of (4)-(6) with boundary data gε, respectively, gsuch that ε≤max{uε(x, t),v(x, t)}≤M then T 0Ω [(uε−v)ζt+(ϕ(uε)−ϕ(v))∆ζ+(c(x, t, uε)−c(x, t, u))ζ]dx dt =Ω (uε(x, T)−v(x, T))ζ(x, T)dx −Ω (uε(x, 0) −v(x, 0))ζ(x, 0) dx +T 0∂Ω (gε(ϕε(uε)) −g(ϕ(v)))ζ(x, t)dσ dt +T 0∂Ω (ϕ(uε)−ϕ(v))∂ζ(x, t)/∂ν dσ dt (22) for any ζ∈L2(QT) such that ζt,∆ζ∈L2(QT) and ∂ζ/∂ν ∈L2(∂Ω× (0,T)). Proceeding as in [4], define Φε(x, t):=1 0 ϕu(θuε(x, t)+(1−θ)v(x, t)) dθ and Cε(x, t):=1 0 cu(x, t, θuε(x, t)+(1−θ)v(x, t)) dθ then (uε−v)Φε(x, t)=ϕ(uε)−ϕ(v) and (uε−v)Cε(x, t)=c(x, t, uε)−c(x, t, v). Periodic solutions for a nonlinear problem 303 Hence, T 0Ω (uε−v)(ζt+Φ ε∆ζ+Cεζ)dx dt =Ω (uε(x, T)−v(x, T))ζ(x, T)dx −Ω (uε(x, 0) −v(x, 0))ζ(x, 0) dx +T 0∂Ω (gε(ϕε(uε)) −g(ϕ(v)))ζ(x, t)dσ dt +T 0∂Ω (ϕ(uε)−ϕ(v))∂ζ(x, t)/∂ν dσ dt. (23) From [7], there exist some positive constants α1,α2depending only on εand M, such that α1(ε)≤Φε(x, t)≤α2(M),for any (x, t)∈QT |Cε(x, t)|≤L1(M),for any (x, t)∈QT. Let ζε,m denotes the solution of the backward linear parabolic problem with smooth coefficients ζε,mt +Φ ε,m∆ζε,m +Cε,mζε,m =f, in QT (24) ζε,m(x, T)=θ(x),in Ω(25) ∂ζε,m(x, t)/∂ν =−d0ζε,m,on ∂Ω×(0,T)(26) with Φε,m,Cε,m,f∈C∞(QT), Φε,m →Φε,Cε,m →Cεuniformly in QT as mgoes to infinity and θ∈C∞ 0(Ω), 0 ≤θ(x)≤1. Also for Φε,m and Cε,m holds α1(ε)≤Φε,m(x, t)≤α2(M),for any (x, t)∈QT |Cε,m(x, t)|≤L1(M),for any (x, t)∈QT. The existence, uniqueness and regularity of ζε,m(x, t)isprovenin[8]. The following esimates will be need Lemma 3 ([7]).Let ζ(x, t):=ζε,m(x, t)be the solution to (24)-(26). Then, T 0Ω |∇ζ(x, t)|2dx dt ≤k1;T 0Ω |∆ζ(x, t)|2dx dt ≤k1.(27)