Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 2, 1-15; http://www.math.u-szeged.hu/ejqtde/

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOUR FOR A DEGENERATE DIFFUSIVE SEIR MODEL

TARIK ALIZIANE AND MICHEL LANGLAIS

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

1. Introduction

In this paper we shall be concerned with a degenerate parabolic system of the form

(1.1)  \{ \begin{matrix}∂_{t}U_{1} − ΔU_{1}^{m_{1}} = −γ(U_{1}, U_{2}, U_{3}, U_{4}) − νU_{1} & = f_{1}(U_{1}, U_{2}, U_{3}, U_{4}), \\ ∂_{t}U_{2} − ΔU_{2}^{m_{2}} = γ(U_{1}, U_{2}, U_{3}, U_{4}) − (λ + μ)U_{2} & = f_{2}(U_{1}, U_{2}, U_{3}, U_{4}), \\ ∂_{t}U_{3} − ΔU_{3}^{m_{3}} = λπU_{2} − (α + m + μ)U_{3} & = f_{3}(U_{1}, U_{2}, U_{3}, U_{4}), \\ ∂_{t}U_{4} − ΔU_{4}^{m_{4}} = (1 − π)λU_{2} + αU_{3} + νU_{1} & = f_{4}(U_{1}, U_{2}, U_{3}, U_{4}). \end{matrix}

in Ω × (0, +∞), subject to the initial conditions

(1.2)  U_{i}(x, 0) = U_{i,0}(x) ≥ 0, x ∈ Ω;  i = 1..4.

and to the Neumann boundary conditions

(1.3)  \frac{∂U_{i}^{m_{i}}}{∂η} (x, t) = 0, x ∈ ∂Ω, t > 0,  i = 1..4.

Herein, Ω is an open, bounded and connected domain in IR^{N}, N ≥ 1, with a smooth boundary ∂Ω; Δ 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

\{ \begin{matrix}S′ & = −γSI, \\ I′ & = +γSI − λI, \\ R′ & = +λI. \end{matrix}

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}.

This basic model served as a starting point for many further developments, both from epidemiological or mathematical point of vue : see the books of Busenberg and Cooke [7] or Capasso [8] and their references. These lead to so-called (S − E − I − R) models : S is the distribution of susceptible individuals in a given population, γ(S, E, I, R) is the incidence

EJQTDE, 2005 No. 2, p. 1 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.

1991 Mathematics Subject Classification. Primary 74G25, 35B40; Secondary 35K65, 35K57, 92D30.

Key words and phrases. Global existence, asymptotic behavior, degenerate equation, reaction-diffusion equations, population dynamics.

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

γ(S, E, I, R) = \{ \begin{matrix}γSI, & [5, 8, 21], & mass action in [7, 8] , or pseudo-mass action in [20, 12] . \\ γ \frac{SI}{S + E + I + R}, & [10, 17, 23], & proportionate mixing in [7] or true mass action in [20, 12] . \end{matrix}

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(\overline{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)

∑_{i=1}^{4} ∫_{Ω} U_{i}(x, t)dx ≤ ∑_{i=1}^{4} ∫_{Ω} U_{i,0}(x)dx for all t > 0,

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

(1.4)  ∂_{t}U_{1} − ΔU_{1}^{m_{1}} + νU_{1} = 0 in Ω × (0, +∞);

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 t ⟶ +∞

\{ \begin{matrix}U_{1}(., t) ⟶ 0 & if ν > 0, \\ U_{1}(., t) ⟶ \frac{1}{mes(Ω)} ∫_{Ω} U_{1,0}(x)dx & if ν = 0. \end{matrix}

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.

2. 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 Δ. Then we set Ω × (0, T) = Q_{T} and for 0 ≤ τ < T, Ω × (τ, T) = Q_{τ,T}. The norm in L^{p}(Ω) is ∥ ∥_{p,Ω} and the norm in L^{p}(Q_{τ,T}) is ∥ ∥_{p,Q_{τ,T}} for 1 ≤ p ≤ +∞.

Next we shall assume throughout this paper (H0) Powers m_{i} verify m_{i} > 1, i = 1..4. (H1) μ, α, ν, m, λ, π are nonnegative constants, λ + μ > 0, α + m + μ > 0 and 0 ≤ π ≤ 1. (H2) U_{i,0} ∈ C(Ω̄), U_{i,0}(x) ≥ 0, x ∈ Ω, i = 1..4. (H3) γ : IR_{+}^{4} ⟶ IR_{+} is a locally lipschitz continuous function with polynomial growth

and γ(0, U_{2}, U_{3}, U_{4}) = 0 on IR_{+}^{3}. (H4) There exists nonnegative constants C_{1}, C_{2} and r such that γ(U_{1}, U_{2}, U_{3}, U_{4}) ≤ C_{1} +

C_{2}U_{1}^{r} on IR_{+}^{4}.

Remark 1. 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.

EJQTDE, 2005 No. 2, p. 3 In the limiting case α + m + μ = 0 one could not get L^{∞}(Q_{0,_{∞}}) bounds for U_{3}, but one still has global existence.

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].

Definition 1. A quadruple (U_{1}, U_{2}, U_{3}, U_{4}) of nonnegative and continuous functions U_{i} : Ω × [0, +∞) → [0, +∞), i = 1..4, is a weak solution of Problem (1.1) − (1.3) in Q_{T} , T > 0

if for each i = 1..4 and for each φ_{i} ∈ C^{1}(Q̄_{T}), such that \frac{∂φ_{i}}{∂η} = 0 on ∂Ω × (0, T).

(i) ∇U^{m_{i}} exists in the sense of distribution and ∇U_{i}^{m_{i}} ∈ L^{2}(Q_{T}); (ii) U_{i} verifies the identity

(2.1)  \begin{matrix}∫_{Ω} U_{i}(x, T)φ_{i}(x, T)dx + ∫_{Q_{T}} ∇U_{i}^{m_{i}}∇φ_{i}(x, t)dxdt \\ = ∫_{Q_{T}} (∂_{t}φ_{i}U_{i} − f_{i}φ_{i})(x, t)dxdt + ∫_{Ω} U_{i,0}(x)φ_{i}(x, 0)dx. \end{matrix}

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 (U_{1}, U_{2}, U_{3}, U_{4}) of Problem (1.1) − (1.3) on Q_{∞}; furthermore

(i) For all i = 1..3, U_{i} ∈ L^{1} ∩ L^{∞}(Q_{∞}) and ∇U_{i}^{m_{i}}, ∂_{t}U_{i}^{m_{i}} ∈ L^{2}(Q_{τ,∞}), τ > 0; (ii) U_{4} ∈ L^{1} ∩ L^{∞}(Q_{T}) and ∇U_{4}^{m_{4}}, ∂_{t}U_{4}^{m_{4}} ∈ L^{2}(Q_{τ,T}), τ > 0.

The proof is found in Section §4.

Now we look at the large time behavior of weak solutions.

Theorem 2. There exist nonnegative constants U_{1}^{∗}, U_{4}^{∗} such that

U_{2}(., t), U_{3}(., t) ⟶ 0 , U_{1}(., t) ⟶ U_{1}^{∗} in C(Ω̄) as t ⟶ +∞
and \overline{U_{4}}(t) ⟶ U_{4}^{∗} in L^{p}(Ω) forall p ≥ 1 as t ⟶ +∞; .

moreover, if ν > 0 then U_{1}^{∗} = 0.

EJQTDE, 2005 No. 2, p. 4

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(Ω̄). 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.

3. 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

(3.1)  \{ \begin{matrix}∂_{t}U_{1} − Δd_{1}(U_{1}) = −γ((U_{1} − ε)^{+}, U_{2}, U_{3}, U_{4}) − ν(U_{1} − ε), \\ ∂_{t}U_{2} − Δd_{2}(U_{2}) = γ((U_{1} − ε)^{+}, U_{2}, U_{3}, U_{4}) − (λ + μ)(U_{2} − ε), \\ ∂_{t}U_{3} − Δd_{3}(U_{3}) = λπ(U_{2} − ε) − (α + m + μ)(U_{3} − ε), \\ ∂_{t}U_{4} − Δd_{3}(U_{4}) = (1 − π)λ(U_{2} − ε) + α(U_{3} − ε) + ν(U_{1} − ε). \end{matrix}

(3.2)  \{ \begin{matrix}U_{i,ε}(x, 0) = U_{i,0,ε}(x) ≥ 0, x ∈ Ω; \\ \frac{∂d_{i}(U_{i,ε})}{∂η}(x, t) = 0, x ∈ ∂Ω, t > 0, \end{matrix} i = 1..4.

Herein (r)^{+} is the nonnegative part of the real number r; for each i = 1..4 d_{i} : IR ⟶ (\frac{ε}{2}, +∞) is a smooth and increasing functions with

(3.3)  d_{i}(u) = u^{m_{i}}, ε ≤ u;

(U_{i,0,ε})_{i=1..4} is a quadruple of smooth functions over Ω̄ such that

(3.4)  \{ \begin{matrix}U_{i,0,ε}(x) ≥ ε, x ∈ Ω, 0 < ε ≤ 1; \\ ∫_{Ω}(U_{i,0,ε}(x) − ε)dx = ∫_{Ω} U_{i,0}(x)dx \\ U_{i,0,ε} ⟶ U_{i,0} in C(Ω̄), as ε ⟶ 0; \end{matrix} i = 1..4;

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_{1,ε}, U_{2,ε}, U_{3,ε}, U_{4,ε}), a classical solution of (3.1) − (3.2) in some maximal interval [0, T_{max,ε}) is granted.

Looking at the equation for U_{i,ε} it is checked that ε is a subsolution, thus 0 < ε ≤ U_{i,ε}(x, t), x ∈ Ω, 0 < t < T_{max,ε} . Next, from the maximum principle and the nonnegativity of γ, ν and U_{1,ε} − ε, it follows U_{1,ε}(x, t) ≤ ∥U_{1,ε,0}∥_{∞,Ω}, x ∈ Ω, 0 < t < T_{max,ε}. As a consequence one has

(3.5)  \{ \begin{matrix}0 < ε ≤ U_{1,ε}(x, t) ≤ ∥U_{1,ε,0}∥_{∞,Ω}, x ∈ Ω, t < T_{max,ε} \\ 0 < ε ≤ U_{i,ε}(x, t), x ∈ Ω, t < T_{max,ε}, i = 2..4 \end{matrix}

Then one can apply results in [16] to show global existence, i.e. T_{max,ε} = +∞, of a classical solution for (3.1) − (3.2). Using (3.3) and (3.5) this yields global existence for the initial and

EJQTDE, 2005 No. 2, p. 5 boundary value problem

(3.6)  \{ \begin{matrix}∂_{t}U_{1} − ΔU_{1}^{m_{1}} = −γ(U_{1} − ε, U_{2}, U_{3}, U_{4}) − ν(U_{1} − ε), \\ ∂_{t}U_{2} − ΔU_{2}^{m_{2}} = γ(U_{1} − ε, U_{2}, U_{3}, U_{4}) − (λ + μ)(U_{2} − ε), \\ ∂_{t}U_{3} − ΔU_{3}^{m_{3}} = λπ(U_{2} − ε) − (α + m + μ)(U_{3} − ε), \\ ∂_{t}U_{4} − ΔU_{4}^{m_{4}} = (1 − π)λ(U_{2} − ε) + α(U_{3} − ε) + ν(U_{1} − ε). \end{matrix}

in Ω × (0, +∞), together with (3.2).

We derive a priori estimates. First, using the L^{1} property of U_{1,0,ε} in (3.4) and the nonnegativity of U_{1,ε} − ε, a straightforward integration of the equation for U_{1,ε} over Ω × (0, +∞) gives:

(3.7)  ∫_{Q_{T}} (γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε}) + ν(U_{1,ε} − ε))(x, t)dxdt ≤ ∫_{Ω} U_{1,0}(x)dx.

In what follows T is a positive number, M_{1}, .., M_{n} are positive constants independent of T and ε, 0 < ε ≤ 1, and F_{1}, .., F_{n} are non decreasing functions of T independent of ε, 0 < ε ≤ 1.

Lemma 1. There exists a constant M_{1} and nondecreasing function F_{1}, independent of ε, 0 < ε ≤ 1 such that

(3.8)  0 < ε ≤ U_{i,ε}(x, t) ≤ M_{1}, x ∈ Ω, t > 0, i = 1..3;
(3.9)  ε ≤ U_{4,ε}(x, t) ≤ F_{1}(T), x ∈ Ω, 0 < t < T.

Proof. The estimate for U_{1,ε} follows from (3.5) and the choice of (U_{1,0,ε})_{ε>0}. Multiplying the equation for U_{2,ε} by p (U_{2,ε} − ε)^{p−1}, p ≥ 1, and integrating over Ω one has

\begin{matrix}\frac{d}{dt}∥U_{2,ε}(., t) − ε∥_{p,Ω}^{p} + p(λ + μ) ∥U_{2,ε}(., t) − ε∥_{p,Ω}^{p} \\ ≤ p ∫_{Ω} γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε})(U_{2,ε} − ε)^{p−1}(x, t)dx; \end{matrix}

keeping in mind λ + μ > 0 from (H1), one gets from Young’s inequality

(3.10)  \frac{d}{dt}∥U_{2,ε}(., t) − ε∥_{p,Ω}^{p} ≤ (\frac{1}{λ + μ})^{p−1} ∫_{Ω}[γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε})]^{p}(x, t)dx.

A further integration over (0, T) leads to

∥U_{2,ε}(., T) − ε∥_{p,Ω}^{p} ≤ ∥U_{2,0,ε} − ε∥_{p,Ω}^{p} + (\frac{1}{λ + μ})^{p−1} ∫_{Q_{T}} [γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε})]^{p}(x, t)dxdt.

Using the already known L^{∞} estimate for U_{1,ε}, assumption (H4) and inequality (3.7) one arrives at : for each T > 0

(3.11)  ∥U_{2,ε}(., T) − ε∥_{p,Ω}^{p} ≤ ∥U_{2,0,ε} − ε∥_{p,Ω}^{p} + (\frac{1}{λ + μ})^{p−1}(C_{1} + C_{2}M_{1}^{r})^{p−1}∥U_{1,0}∥_{1,Ω}.

EJQTDE, 2005 No. 2, p. 6 To conclude, one observes that U_{2,ε} − ε being continuous on Ω̄ × [0, +∞) it follows

lim_{p→+∞} ∥U_{2,ε}(., t) − ε∥_{p,Ω} = ∥U_{2,ε}(., t) − ε∥_{∞,Ω}.

Hence for some constant M_{2} independent of ε, 0 < ε ≤ 1, one gets

(3.12)  0 < ε ≤ U_{2,ε}(x, t) ≤ M_{2}, x ∈ Ω, t > 0.

Now, integrating the equation for U_{2,ε} over Ω × [0, T), using the L^{1} property of U_{2,0,ε} in (3.4), the nonnegativity of U_{2,ε} − ε and (3.7) one has for 0 < ε ≤ 1

(3.13)  (λ + μ) ∫_{Q_{T}} (U_{2,ε} − ε)(x, t)dxdt ≤ ∫_{Ω}[U_{1,0,ε}(x) + U_{2,0,ε}(x)]dx.

The estimate for U_{3,ε} follows from computations similar to the ones for U_{2,ε} above, carried over the equation for U_{3,ε} and getting help from (3.13) and from the positivity of α + m + μ.

Along the same lines, from the equation for U_{3,ε} one gets for 0 < ε ≤ 1

(3.14)  (α + m + μ) ∫_{Q_{T}} (U_{3,ε} − ε)dxdt ≤ ∫_{Ω}[U_{1,0,ε}(x) + U_{2,0,ε}(x) + U_{3,0,ε}(x)]dx.

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

(3.15)  ∫_{Q_{T}} ∥∇U_{i,ε}^{m_{i}}∥^{2}(x, t)dxdt ≤ M_{i,3}, T > 0, i = 1..3;

(3.16)  ∫_{Q_{T}} ∥∇U_{4,ε}^{m_{i}}∥^{2}(x, t)dxdt ≤ F_{2}(T), T > 0.

Proof. The estimate for ∇U_{1,ε}^{m_{1}} is obtained upon multiplying the equation for U_{1,ε} by U_{1,ε}^{m_{1}}, integrating over Ω × (0, T) and using the nonnegativity of γ and U_{1,ε} − ε. One finds

M_{1,3} = \frac{1}{m_{1} + 1} ∫_{Ω} U_{1,ε}^{m_{1}+1}(x, 0)dx

Proceeding along the same lines for U_{2,ε} one gets

\begin{matrix}\frac{1}{m_{2} + 1} ∫_{Ω} U_{2,ε}^{m_{2}+1}(x, T)dx + ∫_{Q_{T}} ∥∇U_{2,ε}^{m_{2}}(x, t)∥^{2}dxdt ≤ \\ \frac{1}{m_{2} + 1} ∫_{Ω} U_{2,ε}^{m_{2}+1}(x, 0)dx + ∫_{Q_{T}} γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε})[U_{2,ε}]^{m_{2}}(x, t)dxdt. \end{matrix}

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

∫_{Q_{T}} ∥∇U_{2,ε}^{m_{2}}(x, t)∥^{2}dxdt ≤ M_{2,3}, T > 0.

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) \begin{matrix}∥∇U_{1,ε}^{m_{1}}(., t)∥_{2,Ω}^{2} ≤ \frac{2}{t(m_{1} + 1)} ∫_{Ω} U_{1,0,ε}^{m_{1}+1}(x)dx \\ +m_{1}^{2}ν^{2}∥U_{1,0}∥_{∞,Ω}^{m_{1}−1} ∫_{Q_{\frac{t}{2} ,t}}(U_{1,ε} − ε)^{2}(x, s)dxds \\ +m_{1}^{2}∥U_{1,0}∥_{∞,Ω}^{m−1}(C_{1} + C_{2}∥U_{1,0}∥_{∞,Ω}^{r}) ∫_{Q_{\frac{t}{2} ,t}} γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε})(x, s)dxds \end{matrix}

Proof. Let us multiply the equation for U_{1,ε} by ∂_{t}U_{1,ε}^{m_{1}} and integrate over Ω×(τ, t), \frac{t}{2} ≤ τ ≤ t; then one finds (3.18)

\begin{matrix}(\frac{2}{m_{1} + 1})^{2} ∫_{Q_{τ,t}}(∂_{t}U1,ε^{\frac{m_{1}+1}{2}} )^{2}(x, s)dxds + ∥∇U_{1,ε}^{m_{1}}(., t)∥_{2,Ω}^{2} \\ ≤ ∫_{Q_{τ,t}}(−γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε}) − ν(U_{1,ε} − ε))∂_{t}U_{1,ε}^{m_{1}}(x, s)dxds + ∥∇U_{1,ε}^{m_{1}}(., τ)∥_{2,Ω}^{2}. \end{matrix}

Next, for any suitably smooth and nonnegative function U and any m > 1 one gets

∂_{t}U^{m} = \frac{2m}{m + 1}U^{\frac{m−1}{2}} ∂_{t}U^{\frac{m+1}{2}} so that

(3.19)  \begin{matrix}∫_{Q_{τ,t}}(−γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε}) − ν(U_{1,ε} − ε))∂_{t}U_{1,ε}^{m_{1}}(x, s)dxds \\ ≤ \frac{2}{(m_{1} + 1)^{2}} ∫_{Q_{\frac{t}{2} ,t}}(∂_{t}U_{1,ε}^{( \frac{m_{1}+1}{2} )})^{2}(x, s)dxds \\ +m_{1}^{2}ν^{2} ∫_{Q_{\frac{t}{2} ,t}}[(U_{1,ε} − ε)U1,ε^{\frac{m−1}{2}} ]^{2}(x, s)dxds \\ +m_{1}^{2} ∫_{Q_{\frac{t}{2} ,t}}[γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε})U1,_{e}psilon^{\frac{m−1}{2}}]^{2}(x, s)dxds. \end{matrix}

The last term on the right hand side of this inequality is bounded from above by

m_{1}^{2}∥U_{1,ε}∥_{∞,Q_{∞}}^{m−1}∥γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε})∥_{∞,Q_{∞}} ∫_{Q_{\frac{t}{2} ,t}} γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε})(x, s)dxds. putting this estimate in (3.18) one obtains

(3.20) \begin{matrix}∥∇U_{1,ε}^{m_{1}}(., t)∥_{2,Ω}^{2} ≤ ∥∇U_{1,ε}^{m_{1}}(., τ)∥_{2,Ω}^{2} + m_{1}^{2}ν^{2}∥U_{1,0}∥_{∞,Ω}^{m_{1}−1} ∫_{Q_{\frac{t}{2} ,t}}(U_{1,ε} − ε)^{2}(x, s)dxds \\ +m_{1}^{2}∥U_{1,0}∥_{∞,Ω}^{m_{1}−1}(C_{1} + C_{2}∥U_{1,0}∥_{∞,Ω}^{r}) ∫_{Q_{\frac{t}{2} ,t}} γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε})(x, t)dxdt. \end{matrix}

Integrating this inequality in τ over (\frac{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

(3.21)  ∫_{Q_{T}} |(U_{i,ε}^{m_{i}})_{t}|^{2}(x, t)dxdt ≤ M_{2}, T > 0, i = 1..3;

(3.22)  ∫_{Q_{T}} |(U_{4,ε}^{m_{i}})_{t}|^{2}(x, t)dxdt ≤ F_{2}(T), T > 0.

Proof. The estimate for U_{1,ε} is immediatly deduced from (3.18) keeping in mind that

|(U_{1,ε}^{m_{1}})_{t}|^{2}(x, t) ≤ \frac{m_{1}^{2}}{2} ∥U_{1,ε}∥_{∞,Ω}^{m_{1}−1}(U1,ε^{\frac{m_{1}+1}{2}} )_{t}^{2}(x, t).

And one can establish such estimates for U_{2,ε}, U_{3,ε} and U_{4,ε} in the same way.

□

4. 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_{i,ε}^{m_{i}})_{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_{i,ε}^{m_{i}})_{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_{i,ε}^{m_{i}})_{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 : (U_{i,ε})_{0<ε≤1} is a relatively compact subset of C(\overline{Ω}×[0, T]). It follows that actually (U_{i,ε}−ε)_{0<ε≤1} is convergent to U_{i} in C(\overline{Ω} × [0, T]) and (U_{i,ε}^{m_{i}})_{0<ε≤1} is convergent to U_{i}^{m_{i}} in C(\overline{Ω} × [0, T]). As a first consequence one has : V_{i} = ∇U_{i}^{m_{i}}; next one also has :

γ(U_{1,ε} − ε, U_{2,ε}, U_{3,ε}, U_{4,ε}) → γ(U_{1}, U_{2}, U_{3}, U_{4}) in C(\overline{Ω} × [0, T]) as ε → 0.

From standard arguments one may conclude that the quadruple (U_{1}, U_{2}, U_{3}, U_{4}) is a desirable weak solution.

The regularity results for ∇U_{i}^{m_{i}} and ∂_{t}U_{i}^{m_{i}} follow from the a priori estimates in Lemma 2 and Lemma 4.

EJQTDE, 2005 No. 2, p. 9 4.2. Uniqueness. Assume there exist two quadruples (U_{j,1}, U_{j,2}, U_{j,3}, U_{j,4})_{j=1,2}, both weak solutions of Problem (1.1) − (1.3). They verify the integral identity, for i = 1..4 (4.1)

\begin{matrix}∫_{Ω}(U_{1,i} − U_{2,i})(x, T)φ_{i}(x, T)dx + ∫_{Q_{T}} ∇(U_{1,i}^{m_{i}} − U_{2,i}^{m_{i}})∇φ_{i}(x, t)dxdt \\ = ∫_{Q_{T}} [∂_{t}φ_{i}(U_{1,i} − U_{2,i}) − (f_{i}(U_{1,1}, U_{1,2}, U_{1,3}, U_{1,4}) − f_{i}(U_{2,1}, U_{2,2}, U_{2,3}, U_{2,4}))φ_{i}](x, t)dxdt \end{matrix}

for every φ_{i} ∈ C^{1}(Q̄_{T}), such that \frac{∂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)  ψ_{i}(x, t) = \{ \begin{matrix}\frac{U_{1,i}^{m_{i}} − U_{2,i}^{m_{i}}}{U_{1,i} − U_{2,i}} & if U_{1,i} ≠ U_{2,i}, \\ 0 & otherwise. \end{matrix} i = 1..4.

Let us consider a sequence of smooth functions (ψ_{i,ε})_{ε≥0} such that ψ_{i,ε} ≥ ε, ψ_{i,ε} is uniformly bounded in L^{∞}(Q_{T}) and

lim_{ε→0} ∥(ψ_{i,ε} − ψ_{i})/\sqrt{ψ_{i,ε}}∥_{L^{2}(Q_{T})} = 0.

For any 0 < ε ≤ 1, σ > 0 let us introduce the adjoint nondegenerate boundary value problem

(4.3)  \{ \begin{matrix}∂_{t}φ_{i} + ψ_{i,ε}Δφ_{i} = 0 & in Ω × (0, T) \\ \frac{∂φ_{i}}{∂η} (x, t) = 0 & in ∂Ω × (0, T) \\ φ_{i}(x, T) = χ_{i} & in Ω \end{matrix} i = 1..4.

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)  ∫_{Q_{T}} ψ_{i,ε}(Δφ_{i,ε})^{2}dxdt ≤ K_{1},

If in (4.1) we replace φ_{i} by φ_{i,ε}, which is the solution of problem (4.3) with χ_{i} = sign((U_{i}− V_{i})^{+}) we obtain.

(4.6)  \begin{matrix}∫_{Ω}(U_{1,i} − U_{2,i})^{+}(x, T)φ_{i,ε}(x, T)dx + ∫_{Q_{T}} (ψ_{i} − ψ_{i,ε})(U_{1,i} − U_{2,i})Δφ_{i,ε}dxdt \\ = ∫_{Q_{T}} (f_{i}(U_{1,1}, U_{1,2}, U_{1,3}, U_{1,4}) − f_{i}(U_{2,1}, U_{2,2}, U_{2,3}, U_{2,4}))φ_{i,ε}dxdt \end{matrix}

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)  ∫_{Ω}(U_{1,i} − U_{2,i})^{+}(x, T)dx ≤ K ∫_{Q_{T}} ∑_{i=1}^{4} |U_{1,i} − U_{2,i}|(x, t)dxdt

In a similar fashion we establish an analogous inequality for (U_{i} − V_{i})^{−} and deduce

(4.8)  ∫_{Ω} ∑_{i=1}^{4} |U_{1,i} − U_{2,i}|(x, T)dx ≤ K ∫_{Q_{T}} ∑_{i=1}^{4} |U_{1,i} − U_{2,i}|(x, t)dxdt

Uniqueness follows from Gronwall’s Lemma.

5. Large time behavior: proofs

The semi-orbit {(U_{1}(., t), U_{2}(., t), U_{3}(., t)), t ≥ 0} is relatively compact in (C(Ω̄))^{3} : it is actually bounded in (L^{∞}(Q_{∞}))^{3} by (3.15) and then one may use a result of [11].

5.1. Case ν > 0. A convergence and continuity argument allows to deduce from (3.7)

(5.1)  ∫_{Q_{T}} γ(U_{1}, U_{2}, U_{3}, U_{4})(x, t)dxdt + ν ∫_{Q_{T}} U_{1} (x, t)dxdt ≤ ∥U_{1,0}∥_{1,Ω}, T > 0.

Hence U_{1} ∈ L^{1}(Ω × (0, +∞)) and there is a sequence (τ_{j})_{j≥0} such that τ_{j} ⟶ +∞ as j ⟶ +∞ and ∫_{Ω} U_{1}(x, τ_{j})dx ⟶ 0 as j ⟶ +∞. Next, given any t > τ_{j}, one has

(5.2)  0 ≤ ∫_{Ω} U_{1}(x, t)dx ≤ ∫_{Ω} U_{1}(x, τ_{j})dx;

actually such an identity holds for U_{1,ε} from a straightforward integration over Ω × (τ_{j}, t) and is preserved upon letting ε ⟶ 0 because U_{1,ε} → U_{1} in C^{0}(\overline{Ω} × (0, +∞) as ε → 0. This shows that U_{1}(., t) ⟶ 0 in L^{1}(Ω) as t ⟶ +∞ and also in C(Ω̄).

Then, along the same lines, from (3.13) and (3.14) one has for T > 0

(5.3)  \begin{matrix}(λ + μ) ∫_{Q_{T}} U_{2} (x, t)dxdt + (α + m + μ) ∫_{Q_{T}} U_{3} (x, t)dxdt \\ ≤ 2∥U_{1,0}∥_{1,Ω} + 2∥U_{2,0}∥_{1,Ω} + ∥U_{3,0}∥_{1,Ω}. \end{matrix}

Again, for some sequence (τ_{j})_{j≥0} such that τ_{j} ⟶ +∞ one has ∫_{Ω} U_{2}(x, τ_{j})dx ⟶ 0 as j ⟶ +∞. Integrating over Ω × (τ_{j}, t) the equation in (3.6) for U_{2,ε} and letting ε ⟶ 0 one finds

(5.4)  0 ≤ ∫_{Ω} U_{2}(x, t)dx ≤ ∫_{τ_{j}}^{t} ∫_{Ω} γ(U_{1}, U_{2}, U_{3}, U_{4})(x, τ)dxdτ + ∫_{Ω} U_{2}(x, τ_{j})dx;

thus again U_{2}(., t) ⟶ 0 in L^{1}(Ω) and in C(Ω̄) because γ lies in L^{1}(Ω × (0, +∞). The conclusion for U_{3}(., t) is derived in the same fashion, using the third equation in (3.6).

Now we will establish the long time behavior of U_{4}, to do this let us consider for any τ > 0 the following problem

(5.5)  \{ \begin{matrix}∂_{t}V − ΔV^{m_{4}} = 0, & (x, t) ∈ Ω × (0, +∞) \\ V (x, 0) = U_{4}(x, τ), & x ∈ Ω; \\ \frac{∂V^{m_{4}}}{∂η} (x, t) = 0. & x ∈ ∂Ω, t > 0. \end{matrix}

It is well known see [1] that lim_{t→+∞} V (., t) = \overline{V} (0) = \overline{U_{4}}(τ) in L^{p}(Ω), for all p ≥ 1, and in

another hand from [6] we have for all p ≥ 1

(5.6)  ∥U_{4}(x, τ + h) − V (x, h)∥_{p,Ω} ≤ ∫_{h}^{τ+h} ∥f_{4}(x, s)∥_{p,Ω} ds,

with f_{4}(x, t) = (1 − π)λU_{2}(x, t) + αU_{3}(x, t) + νU_{1}(x, t). Set τ = h = \frac{t}{2}, we can write

∥U_{4}(x, t) − \overline{U_{4}}(\frac{t}{2})∥_{p,Ω} ≤ ∥U_{4}(x, t) − V (x, \frac{t}{2})∥_{p,Ω} + ∥V (x, t) − \overline{U_{4}}(\frac{t}{2})∥_{p,Ω},
≤ ∫ t_{\frac{t}{2}} ∥f_{4}(x, s)∥_{p,Ω}ds + ∥V (x, t) − \overline{U_{4}}(\frac{t}{2})∥_{p,Ω}; p ≥ 1,

since f_{4} ∈ L^{1}(Q_{∞}) ∩ L^{∞}(Q_{∞}) we deduce that lim_{t→+∞} ∥U_{4}(x, t) − \overline{U_{4}}(\frac{t}{2})∥_{p,Ω} = 0, furthermore f_{4} ≥ 0 allow to show that t ⟶ \overline{U_{4}}(t) is bounded and nondecreasing and then converges to some nonnegative constant U_{4}^{∗} and this yields lim_{t→+∞} U_{4}(., t) = lim_{t→+∞} \overline{U_{4}}(t) = U_{4}^{∗}

in L^{p}(Ω) for all p ≥ 1.

5.2. Case ν = 0. The analysis of the behavior of {U_{1}(., t), t > 0} requires modifications because it is not known, and actually it is not true, that U_{1} ∈ L^{1}(Ω × (0, +∞)). Set

ϕ̄(t) = \frac{1}{mes(Ω)} ∫_{Ω} ϕ(x, t)dx;

then multiplying the equation for U_{1,ε} in (3.7) by \frac{1}{m_{1}} U_{1,ε}^{m_{1}−1} and integrating over Ω×(τ, τ +t) yields

(5.7)  \overline{U_{1,ε}^{m_{1}}}(τ) ≥ \overline{U_{1,ε}^{m_{1}}}(τ + t) ≥ 0, τ > 0, t > 0;

so that upon letting ε ⟶ 0, the average Ū_{1}^{m_{1}} is a nonincreasing function of time. From the inequality of Poincaré-Wirtinger one can conclude the existence of a constant K(Ω) such that for t > 0

(5.8)  ∥U_{1}^{m_{1}}(., t) − \overline{U_{1}^{m_{1}}}(t)∥_{2,Ω} ≤ K(Ω) ∥∇U_{1}^{m_{1}}(., t)∥_{2,Ω}.

Now, one gets from Lemma 3 with ν = 0 that

(5.9)  \begin{matrix}∥∇U_{1}^{m_{1}}(., t)∥_{2,Ω}^{2} ≤ \frac{2}{t(m_{1} + 1)} ∫_{Ω} U_{1,0}^{m_{1}+1}(x)dx \\ +m_{1}^{2}∥U_{1,0}∥_{∞,Ω}^{m−1}(C_{1} + C_{2}∥U_{1,0}∥_{∞,Ω}^{r}) ∫_{Q_{\frac{t}{2} ,t}} γ(U_{1}, U_{2}, U_{3}, U_{4})(x, s)dxds \end{matrix}

It follows that ∥∇U_{1}^{m_{1}}(., t)∥_{2,Ω} ⟶ 0 as t ⟶ +∞, so that ∥U_{1}^{m_{1}}(., t) − Ū_{1}^{m_{1}}(t)∥_{2,Ω} ⟶ 0 by (5.8). The monotonicity of t ⟶ \overline{U_{1}^{m_{1}}}(t) yields lim_{t→+∞} U_{1}^{m_{1}}(., t) = lim_{t→+∞} Ū_{1}^{m_{1}}(t) = U_{1}^{∗} in

L^{2}(Ω) and also in C(Ω̄).

EJQTDE, 2005 No. 2, p. 12 5.3. An elementary spatially homogeneous system. Let us consider the system of ordinary differential equation

(5.10)  \{ \begin{matrix}U′_{1} & = −γ(U_{1}, U_{2}, U_{3}, U_{4}) \\ U′_{2} & = γ(U_{1}, U_{2}, U_{3}, U_{4}) − λU_{2} − μU_{2}, \\ U′_{3} & = λπU_{2} − αU_{3} − μU_{3} − mU_{3}, \\ U′_{4} & = (1 − π)λU_{2} + αU_{3}. \end{matrix}

With U_{i}(0) ≥ 0, i = 1..4, U_{1}(0) > 0, U_{3}(0) ≥ 0 and,

\{ \begin{matrix}λ > 0, α > 0, m ≥ 0, μ ≥ 0, \\ γ(U_{1}, U_{2}, U_{3}, U_{4}) = σ(U_{2}, U_{3}, U_{4}) U_{1}, \end{matrix}

γ having either a masse action or a proportionate mixing form : see the introduction.

Then U_{1}(t) = U_{1}(0)exp (− ∫_{0}^{t} σ(U_{2}, U_{3}, U_{4})(τ)dτ) so that U_{1}(t) ↘ U_{1}^{∗} ≥ 0 as t ⟶ +∞

and U_{1}^{∗} = 0 if and only if ∫_{0}^{+∞} σ(U_{2}, U_{3}, U_{4})(τ)dτ = +∞.

Next U_{1} + U_{2} = −(λ + μ)U_{2}(t) and upon integrating over (0, +∞) one gets U_{2} lies in L^{1}(0, +∞) so that U_{2}(t) ⟶ 0 as t ⟶ +∞ because U′_{2} is bounded.

A similar argument yields U_{3} lies in L^{1}(0, +∞) and U_{3}(t) ⟶ 0 as t ⟶ +∞. Then one

has U_{4}(t) = U_{4}(0)+(1−π)λ ∫_{0}^{t} U_{2}(τ)dτ +α ∫_{0}^{t} U_{3}(τ)dτ. Here U_{4}(t) ↗ U_{4}^{∗} > 0 as t ⟶ +∞.

To conclude that U_{1}^{∗} > 0 note that • When γ(U_{1}, U_{2}, U_{3}, U_{4}) = γU_{1}U_{3} then σ(U_{2}, U_{3}, U_{4}) = γU_{3} lies in L^{1}(0, +∞).

• When γ(U_{1}, U_{2}, U_{3}, U_{4}) = γ \frac{U_{1}U_{3}}{U_{1} + U_{2} + U_{3} + U_{4}} then σ(U_{2}, U_{3}, U_{4}) = γ \frac{U_{3}}{U_{1} + U_{2} + U_{3} + U_{4}}

Now (U_{1} + U_{2} + U_{3} + U_{4})(t) ⟶ U_{1}^{∗} + U_{4}^{∗} as t ⟶ +∞ and U_{1}^{∗} + U_{4}^{∗} > 0, because U_{4}^{∗} > 0 and U_{1}^{∗} ≥ 0; hence for t ≥ t_{0} one has

\frac{1}{2}(U_{1}^{∗} + U_{4}^{∗}) ≤ (U_{1} + U_{2} + U_{3} + U_{4})(t) ≤ (U_{3} + U_{4})(0)

which implies

\frac{U_{3}(t)}{(U_{3} + U_{4})(0)} ≤ σ(U_{2}, U_{3}, U_{4})(t) ≤ 2\frac{U_{3}(t)}{U_{1}^{∗} + U_{4}^{∗}} , t ≥ t_{0}.

As a conclusion σ(U_{2}, U_{3}, U_{4}) lies in L^{1}(0, +∞) and U_{1}^{∗} > 0.

Last when m = μ = 0, U_{1}^{∗} + U_{4}^{∗} = (U_{1} + U_{2} + U_{3} + U_{4})(0).

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(Received November 26, 2004)

University of Sciences and Technology Houari Boumediene, Faculty of Mathematics, Po.Box 32 El Alia, Algiers, Algeria.

E-mail address: taliziane@usthb.dz

UPRES A 5466, ”Mathématiques Appliquées de Bordeaux”U.F.R M.I.2S, B.P 26, Université Victor Segalen, 33076 Bordeaux Cedex. FRANCE

E-mail address: langlais@sm.u-bordeaux2.fr