EXISTENCE AND UNIQUENESS OF PERIODIC SOLUTIONS FOR A NONLINEAR REACTION-DIFFUSION PROBLEM

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.

Remark.Hypotesis (H c ) iii) is enough to have the global existence result for the associated initial-boundary value problem (see [2]).
In problem ( 1)-( 3), u denotes the moisture content in the soil represented by the domain Ω, therefore we require the condition u ≥ 0. Assumption (H ϕ ) 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 Poincaré 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)) = d 0 ϕ(u).
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.
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) = u m , m > 1 and a zero order term which is linear w.r.t.u.
Our assumptions, allows to consider more general ϕ and zero order terms.

Existence of periodic solutions
Since our equation (1) may degenerates, we make the following definition of periodic solution ) is said to be a periodic weak solution of (1)-( 3) if for any compact interval 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 where ϕ ε , c ε and g ε are smooth approximations of ϕ, c and g, constructed by convolutions with mollifiers functions.
We shall show for ( 4)-( 6) the existence of periodic weak solutions as fixed points for the Poincaré 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.
If u 0ε is chosen in such way that besides satisfies then u(x) is a supersolution and ε is a subsolution to ( 7)-( 9).Thus we have Applying the continuity result of [5], the following regularity property for the solutions of ( 7)-( 9) holds.

Proposition 1 ([5]
).Since u 0ε is continuous on Ω, the sequence {u ε } of the solutions to ( 7)-( 9) is equicontinuous in Q T i.e. there exists ω 0 : R + → R + , ω 0 (0) = 0, continuous and nondecreasing such that If consider the Poincaré map associated to problem (7)-( 9) and defined by where u ε is the unique solution of ( 7)-( 9) and introduce the closed, nonempty, bounded and convex set then, by ( 16) and Proposition 1, we get Remains to prove that To this purpose, we show Proposition 2. If u n 0ε , u 0ε ∈ K ε and u n 0ε → u 0ε uniformly in Ω as n → ∞, then if u n ε and u ε are solutions to ( 7)-( 9) of initial data u n 0ε and u 0ε respectively, we have that Proof: Multiplying (7) by sgn(u n ε − u ε ) and integrating on Q t one has because of the local Lipschitz continuity of c ε (x, t, •) with Lipschitz constant L. Applying Gronwall's lemma, it is easy to see that u n ε (x, t) converges to u ε (x, t) strongly in L 1 (Ω) as n goes to infinity.Consequently, for a subsequence, we have that u n ε (x, t) converges to u ε (x, t) for a.e.x ∈ Ω.Since u n ε (x, t) ≤ M , by the Lebesgue theorem, we conclude that Thus, by the Schauder fixed point theorem it follows that there exists a fixed point for the Poincaré map, which is a periodic solution to (4)- (6).From (16) we get, for a subsequence if necessary, that for a classical result (see [3]) one has Hölder that integrate with respect to t, gives Hölder .
Moreover, (see [3]) with compact injection.Then, From ( 18) and the Lebesgue theorem one has thus, we get by ( 15) and ( 20) Theorem 3.4.5 of [11] states that if (21) holds, then ϕ ε (u ε ) converges to ϕ(u) in L 2 (0, T ; L 1 (∂Ω)).Now, by the uniform convergence of g ε on compact set of R + and its uniformly Lipschitz continuity, it is easy to see that In fact thus, u is a periodic weak solution to (1)-(3).

Uniqueness
To get the uniqueness result, suppose that u ε and v are periodic solutions of ( 4)-( 6) with boundary data g ε , respectively, g such that Proceeding as in [4], define Hence, From [7], there exist some positive constants α 1 , α 2 depending only on ε and M , such that Let ζ ε,m denotes the solution of the backward linear parabolic problem with smooth coefficients in Ω (25) for any (x, t) ∈ Q T .
The existence, uniqueness and regularity of ζ ε,m (x, t) is proven in [8].
The following esimates will be need Lemma 3 ([7]).Let ζ(x, t) := ζ ε,m (x, t) be the solution to ( 24)-( 26).Then, where k The main result of this section is the following By Lemma 3, one concludes (31) Going to the limit as m → ∞ in (31), we obtain the desired result.