GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOUR FOR A DEGENERATE DIFFUSIVE SEIR MODEL

In this paper we analyze the global existence and asymptotic behavior of a reaction diusion system with degenerate diusion arising in modeling the spatial spread of an epidemic disease.


Introduction
In this paper we shall be concerned with a degenerate parabolic system of the form (1.1) in Ω × (0, +∞), subject to the initial conditions and to the Neumann boundary conditions Herein, Ω is an open, bounded and connected domain in IR N , N ≥ 1, with a smooth boundary ∂Ω; ∆ is the Laplace operator in IR N .Powers m i verify m i > 1, i = 1..4.
In the spatially homogeneous case and for ν = µ = α = m = 0 and π = 1 this problem reduces to one of the models of propagation of an epidemic disease devised in Kermack and McKendricks [21], namely    S = −γSI, I = +γSI − λI, R = +λI.In that setting it is known, loc.cit., that I(t) → 0 as t → +∞, while the large time behavior of S(t) and R(t) depends on the initial state (S 0 , I 0 , R 0 ); note that for t > 0, S(t) + I(t) + R(t) = S 0 + I 0 + R 0 .term or number of susceptible individuals infected by contact with an infective individual I per time unit and becoming exposed E, while R is the density of removed or resistant (immune) individuals.Then λ (resp.α) is the inverse of the duration of the exposed stage (resp.infective stage) or rate at which exposed individuals enter the infective class (resp.infective individuals who do not die from the disease recover), m is the death-rate induced by the disease.The last two parameters are control parameters : first ν is a vaccination rate; next, for a population of animals, as it is considered here as in Anderson et al [5], Fromont et al [17], Courchamp et al [10] or Langlais and Suppo [23], µ is an elimination rate of exposed and infective individuals.Lastly, as it is suggested by the FeLV, a retrovirus of domestic cats (Felis catus) see [17], one also introduces a parameter π measuring the proportion of exposed individuals which actually develop the disease after the exposed stage, the remaining proportion 1 − π becoming resistant.
The nonlinear incidence term γ takes various forms as it can be found from the literature; at least two of them are widely used in applications [5,8,21], mass action in [7,8] , or pseudo-mass action in [20,12] .
We refer to De Jong et al, [20] and Diekmann et al [12] for a discussion supporting the second one in populations of varying size and Fromont et al [18] for a specific discussion in the case of a cat population.See Capasso and Serio [9] and Capasso [8] for more general incidence terms.Note that no demographical effect is considered in our model.
A mathematical analysis of the model of Kermack and McKendricks for spatially structured populations with linear diffusion, i.e. m i = 1, i = 1..4, is performed in Webb [27].Nonlinear but nondegenerate diffusion terms are introduced in Fitzgibbon et al [16].Global existence and large time behavior results are derived therein.Homogeneous Neumann boundary conditions correspond to isolated populations.
A comprehensive analysis of generic (S − E − I − R) models with linear diffusion is initiated in Fitzgibbon and Langlais [14] and Fitzgibbon et al [15].These models include a logistic effect on the demography, yielding L 1 (Ω) a priori estimates on solutions independent of the initial data for large time; this allows to use a bootstrapping argument to show global existence and exhibit a global attractor in (C(Ω)) 4 .
For degenerate reaction-diffusion equations, a similar approach is followed in Le Dung [13].In our case, L 1 (Ω) a priori estimates can be established for nonegative solutions upon integrating over Ω × (0, t) U i,0 (x)dx for all t > 0, EJQTDE, 2005 No. 2, p. 2 but they cannot be found to be independent of the initial data.Moreover, generally speaking, the large time behavior of solutions depends on these initial datas, as it can be already seen for spatially homogeneous problems see § §5.3.This can also be checked on the disease free model: assuming U i,0 (x) ≡ 0 in Ω i = 2..4, the uniqueness result given in Theorem 1 implies U i (x, t) ≡ 0 in Ω × (0, +∞) i = 2..4.Then, it should be clear that γ(U 1 , 0, 0, 0) = 0 for any reasonable incidence term so that the equation for U 1 reads +∞); this is the so-called porous medium equation.Now U 1 verifies homogeneous Neumann boundary conditions and it is well-known (see Alikakos [1]) that as The case of mass action incidence was studied by Aliziane and Moulay [4] and they established the long time behavior of the solution of the SIS model, Aliziane and Langlais [3] study the case of models include a logistic effect on the demography and they established global existence result of the solution and existence of periodic solution.We also obtain the existence of the global attractor.Finally Hadjadj et al [19] study the case where the source term depends on gradient of solution, they study existence of globally bounded weak solutions or blow-up, depending on the relations between the parameters that appear in the problem.

Main results
2.1.Basic assumptions and notations.Herein, Ω is an open, bounded and connected domain of the N -dimensional Euclidian space IR N , N ≥ 1, with a smooth boundary ∂Ω, a (N − 1)-dimensional manifold so that locally Ω lies on one side of ∂Ω; x = (x 1 , ..., x N ) is the generic element of IR N .Next we shall denote the gradient with respect to x by ∇ and the Laplace operator in IR N by ∆.Then we set Ω × (0, T ) = Q T and for 0 Next we shall assume throughout this paper (H0) Powers In the limiting case λ + µ = 0 the equations for U 3 and U 4 do not depend on U 2 , the equation for U 3 being a porous medium type equation as in (1.4).This condition also implies λ = 0 which is not relevant if one goes back to our motivating problem.
The assumption γ(0, U 2 , U 3 , U 4 ) = 0 is required to make sure that the nonnegative orthant IR 4  + is forward invariant by (1.1); this is a natural assumption for our motivating problem : no new exposed individuals when there is no susceptible ones.(H4) removes mass action incidence terms; in that case one can also get global existence results, but no L ∞ (Q 0,∞ ) bounds for U 2 and U 3 .
2.2.Main results.System (1.1) is degenerate : when U i = 0 the equation for U i degenerates into first order equation.Hence classical solutions cannot be expected for Problem (1.1) − (1.3).A suitable notion of generalized solutions is required : we adopt the notion of weak solution introduced in Oleinik et al [25].
We are now ready to state our first result.
Theorem 1.For each quadruple of continuous nonnegative initial functions (U 1,0 , U 2,0 , U 3,0 , U 4,0 ) there exists a unique weak solution The proof is found in Section §4.Now we look at the large time behavior of weak solutions.
Theorem 2. There exist nonnegative constants The proof is found in Section §5.
Remark 2. In the non degenerate case m 4 = 1 one has that U 4 (., t) −→ U * 4 in C( Ω).More generally, this still holds provided that U 4 lies in L ∞ (Q ∞ ), the proof being similar to the one for U 1 when ν = 0, see subsection § §5.2.

Auxiliary problem and a priori estimates
In this section we consider an auxiliary problem depending on a small parameter ε, with 0 < ε ≤ 1. Namely let us introduce in Ω × (0, +∞) the quasilinear nondegenerate initial and boundary value problem Herein (r) + is the nonnegative part of the real number r; for each i = 1..4 d i : IR −→ ( ε 2 , +∞) is a smooth and increasing functions with (U i,0,ε ) i=1..4 is a quadruple of smooth functions over Ω such that we refer to [2,19] for a construction of such a set of initial data.From standard results, i.e. [22] or [26], local existence and uniqueness of a quadruple (U Next, from the maximum principle and the nonnegativity of γ, ν and Then one can apply results in [16] in Ω × (0, +∞), together with (3.2).
Along the same lines, from the equation for U 3,ε one gets for 0 < ε ≤ 1 Hence, going back to the equation for U 4,ε one can derive the a priori estimate upon multiplying it by p (U 2,ε − ε) p−1 , p ≥ 1 and using (3.13) − (3.

14).
Lemma 2. There exist constants M i,3 , i = 1..3 and a nondecreasing function F 2 , independent of ε, 0 < ε ≤ 1 such that Proof.The estimate for ∇U m 1 1,ε is obtained upon multiplying the equation for U 1,ε by U m 1 1,ε , integrating over Ω × (0, T ) and using the nonnegativity of γ and U 1,ε − ε.One finds Proceeding along the same lines for U 2,ε one gets Using the properties of U 2,0,ε , the uniform estimate for U 2,ε in Lemma 1 and the L 1 estimate for γ in (3.7) we obtain A similar computation supplies the estimate for U 3,ε .The estimate for U 4,ε then follows.
Lemma 3.For all t > 0 (3.17) Qτ,t Next, for any suitably smooth and nonnegative function U and any m > 1 one gets The last term on the right hand side of this inequality is bounded from above by Integrating this inequality in τ over ( t 2 , t) and using the explicit value for M 1,3 found in the proof of Lemma 2 we deduce the desired result.Lemma 4.There exists a constant M 1 and non decreasing function F 1 , independent of ε, 0 < ε ≤ 1 such that Proof.The estimate for U 1,ε is immediatly deduced from (3.18) keeping in mind that ) 2 t (x, t).And one can establish such estimates for U 2,ε , U 3,ε and U 4,ε in the same way.

Existence and uniqueness: proofs
In this section we supply a quick proof of Theorem 1.

4.1.
Existence.Let us fix T > 0. From the estimates established in the previous section one has : for each i = 1..4 (U i,ε − ε) 0<ε≤1 and (∇U m i i,ε ) 0<ε≤1 are respectively bounded in L 2 (Q T ) and (L 2 (Q T )) N .Then there exists two sequences which one still denotes (U i,ε − ε) 0<ε≤1 and (∇U m i i,ε ) 0<ε≤1 such that for i = 1..4 as ε → 0 : (U i,ε − ε) 0<ε≤1 is weakly convergent to some U i in L 2 (Q T ) and (∇U m i i,ε ) 0<ε≤1 is weakly convergent to some V i in (L 2 (Q T )) N .On the other hand (U i,ε ) 0<ε≤1 is bounded in L ∞ (Q T ); using a weak formulation of the equation for U i,ε one can invoke the results in Di Benedetto [11] to get : As a first consequence one has : V i = ∇U m i i ; next one also has : From standard arguments one may conclude that the quadruple (U 1 , U for every ϕ i ∈ C 1 ( QT ), such that ∂ϕ i ∂η = 0 on ∂Ω × (0, T ) and ϕ i > 0.
We follow an idea of [24] and introduce a function ψ i as follows (4.2) Let us consider a sequence of smooth functions (ψ i,ε ) ε≥0 such that For any 0 < ε ≤ 1, σ > 0 let us introduce the adjoint nondegenerate boundary value problem For any smooth χ i with 0 ≤ χ i (x, t) ≤ 1, i = 1..4, any 0 < ε ≤ 1 and any σ > 0 this problem has a unique classical solution ϕ i,ε such that see [24] (4.4) 0 ≤ ϕ i,ε (x, t) ≤ 1 (4.5) If in (4.1) we replace ϕ i by ϕ i,ε , which is the solution of problem (4.3) with Using the local lipschitz continuity of f i and the properties of ψ i,ε and ϕ i,ε we deduce by letting ε → 0 (4.7) In a similar fashion we establish an analogous inequality for (U i − V i ) − and deduce (4.8) Uniqueness follows from Gronwall's Lemma.