# Weak Solution to a Parabolic Nonlinear System Arising in Biological Dynamic in the Soil

Authors: Côme Goudjo; Babacar Lèye; Mamadou Sy
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

We study a nonlinear parabolic system governing the biological dynamic in the soil. We prove global existence (in time) and uniqueness of weak and positive solution for this reaction-diffusion semilinear system in a bounded domain, completed with homogeneous Neumann boundary conditions and positive initial conditions.


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## Body

## 1. Introduction

Modelling biological dynamic in the soil is of great interest during these last years. Several attempts are made in1D, 2D, and rarely in 3D.  For more details, readers are referred to [1–3]. We deal here with the mathematical study of the model described in [2].LetT>0 be a fixed time, Ω⊂ℝ3 an open smooth bounded domain, QT=]0,T[×Ω, and ΓT=]0,T[×∂Ω.The set of equations describing the organic matter cycle of decomposition in the soil is given by the following system:(1.1)(S){∂ui∂t-DiΔui+qi(u)ui=fi(u)inQT,∂ui∂n=0overΓT,ui(0,x)=u0i(x)inΩ,
for i=1,…,6.We have noticedu=(u1,u2,…,u6)T with u1 is the density of microorganisms (MB), u2 is the density of DOM, u3 is the density of SOM, u4 is the density of FOM, u5isthe density of enzymes, and u6 is the density of CO2,(1.2)q1(u)=-ku2Ks+u2+μ+r+ν,q2(u)=ku1Ks+u2,q3(u)=c1u5Km+u5,q4(u)=c2u5Km+u5,q5(u)=ζ,q6(u)=0,f1(u)=0,f2(u)=u5Km+u5(c1u3+c2u4)+ζu5+μu12,f3(u)=ζu5+μu12,f4(u)=0,f5(u)=νu1,f6(u)=ru1,
with μ mortality rate, r is the breathing rate, ν is the enzymes production rate, ζ is the transformation rate of deteriorated enzymes, c1 is the maximal transformation rate of SOM, c2 is the maximal transformation rate of FOM, k maximal growth rate, Km and Ks represent half-saturation constants, and Di, i=1 to 6, are strictly positive constants.System(𝒮) is introduced in [2]. To our knowledge, it is the first time that diffusion is used to model biological dynamics and linking it to real soil structure described by a 3D computed tomography image.Similar systems to(𝒮) operate in other situations. It comes in population dynamics as Lotka-Voltera equation which corresponds to the case f=0, ui denoting the densities of species present and qi growth rate. This system is also involved in biochemical reactions. In this case, the ui are the concentrations of various molecules, qi is the rate of loss, and fi represents the gains.For models in biology, interested reader can consult with profit [4] where the author presents some models based on partial differential equations and originating from various questions in population biology, such as physiologically structured equations, adaptative dynamics, and bacterial movement. He describes original mathematical methods like the generalized relative entropy method, the description of Dirac concentration effects using a new type of Hamilton-Jacobi equations, and a general point of view on chemotaxis including various scales of description leading to kinetic, parabolic, or hyperbolic equations.Theoretical study of semilinear equations is widely investigated. Some interesting mathematical difficulties arise with these equations because of blowup in finite time, nonexistence and uniqueness of solution, singularity of the solutions, and noncontinuity of the solution regarding data.In [5], the authors prove the blowup in finite time for the system in 1D, (1.3)ut=uxx-a(x,t)f(u)0<x<1,t∈(0,T),ux(0,t)=0t∈(0,T),ux(1,t)=b(t)g(u(1,t))t∈(0,T).A sufficient condition for the blowup of the solution of parabolic semilinear second-order equation is obtained in [6] with nonlinear boundary conditions, and so the set in which the explosion takes place. He also gives a sufficient condition for the solution of this equation  which tends to zero, and its asymptotic behavior.Existence and uniqueness of weak solutions for the following system are considered in [7]:(1.4)(∂t+L)u+F(t,x,u,∇u)=0∀(t,x)∈[0,T[×Rd,u(x,T)=g(x)∀x∈Rd,
with(1.5)F:[0,T]×Rd×Rm×Rm×d⟶Rm,b∈Cb2(Rd;Rd),σ∈Cb3(Rd;Rd×d),a=σσt,L=∑i=1dbi∂∂xi+12∑i,j=1dai,j∂2∂xi∂xj,
with obstacles, giving a probabilistic interpretation of solution. This problem is solved using a probabilistic method under monotony assumptions.By using bifurcation theory, in [8], authors determine the overall behavior of the dynamic system(1.6)∂u∂t=DΔu+uf(x,u)x∈Ω,t>0,u(x,0)=u0(x)≥0x∈Ω.A Cauchy problem for parabolic semilinear equations with initial data inHps(ℝn) is studied in [9]. Particularly the author solves local existence using distributions data.Michel Pierre’s paper, see [10], presents few results and open problems on reaction-diffusion systems similar to the following one:(1.7)∀i=1,…,m,∂tui-diΔui=fi(u1,…,um)in(0,T)×Ω,αi∂ui∂n+(1-αi)ui=βion⁡(0,T)×∂Ω,ui(0)=ui0inΩ,
where the fi:ℝm→ℝ are C1 functions of u=(u1,…,um), and di∈(0,∞), αi∈[0,1], βi∈C2([0,T]×Ω¯), βi≥0.The systems usually satisfy the two main properties:(i)
the positivity of the solutions is preserved for all time,(ii)
the total mass of the components is uniformly controlled in time.He recalls classical local existence result [11–13] under the above hypothesis.It is assumed throughout the paper that(i)
all nonlinearities are quasipositives,(ii)
they satisfy a “mass-control structure”(1.8)∀r=(r1,…,rn),∑i=1mf(r)≤C(1+∑i=1mri),C>0.
It follows that the total mass is bounded on any interval. Few examples of reactions-diffusion systems for which these properties hold are studied.Systems where the nonlinearities are bounded inL1((0,T)×Ω)  are also considered, for instance, for fi in L1((0,T),Ω) whose growth rate is less than |u|(N+2)/N when N tends to +∞ [14].Other situations are investigated, namely, when the growth of the nonlinearities is not small. But many questions are still unsolved, so several open problems are indicated.A global existence result for the following system:(1.9)∂tu-d1Δu=f(u,v),∂tv-d2Δv=g(u,v),u(0,⋅)=u0(⋅)≥0,v(0,⋅)=v0(⋅)≥0with  either:∂u∂n=β1,∂v∂n=β2on⁡(0,+∞)×∂Ω,or:u=β1,v=β2on⁡(0,+∞)×∂Ω,
where d1,d2∈(0,+∞),β1,β2∈[0,+∞), and f,g:[0,+∞)2→ℝ are C1, holds for the additional following hypothesis:(1.10)∀u≥U,∀v≥0,f(u,v)≤C[1+u+v],U,C≥0,∃r≥1,∀u,v≥0,|g(u,v)|≤C[1+ur+vr].This approach has been extended tom×m systems for which f1,f1+f2,f1+f2+f3,… are all bounded by a linear of the ui (see [15]).However,L∞(Ω)-blow up may occur in finite time for polynomial 2×2 systems as proved in [16, 17].A very general result for systems which preserves positivity and for which the nonlinearities are bounded inL1 may be found in [18]. It is assumed that, for all i=1,…,m,(1.11)fi:]0,T[×Ω×[0,+∞)m⟶Ris measurable,fi(⋅,0)∈L1(]0,T[×Ω)∃K:]0,T[×Ω×[0,+∞)⟶[0,+∞)with∀M>0,K(⋅,M)∈L1(]0,T[×Ω)and a.e.(t,x)∈]0,T[×Ω,∀r,r̃∈[0,+∞)mwith|r|,|r̃|≤M,|fi(t,x,r)-fi(t,x,r̃)|≤K(t,x,M)|r-r̃|,∀r∈[0,+∞)m,fi(t,x,r1,…,ri-1,0,ri+1,…,rm)≥0,
there is a sequence which converges in L1(]0,T[×Ω) to a supersolution of (1.7).One consequence is that global existence of weak solutions for systems whose nonlinearities are at most quadratic withu0∈L2(Ω)m can be obtained.Results are also obtained in the weak sense for systems satisfyingWt-ΔZ≤H, where W=∑iui, Z=∑idiui, and H∈L2(]0,T[×Ω).The aim of our paper is to study the global existence in time of solution for the system(𝒮). In our work, we use an approach based both on variational method and semigroups method to demonstrate existence and uniqueness of weak solution.The difficulty is thatu being in the denominator of some qi(u) and fi(u), it is necessary to guarantee that u is nonnegative to avoid explosion of these expressions, whereas the classical methods assume that these expressions are bounded.For instance, to show that weak solution is positive with an initial positive datum, Stampachia’s method uses majoration ofqi(u) by a function of t.In our work, we show existence and unicity of a global positive weak solution of System(𝒮) for an initial positive datum.The work is organized as follows. In the first part, we recall some preliminary results concerning variational method and semigroups techniques. In the second part, we prove, using these methods, existence, uniqueness, and positivity of weak solution under assumptions of positive initial conditions.


## 2. Preliminary Results

### 2.1. Variational Method (See [19])

We consider two Hilbert spacesH and V such that V is embedded continuously and densely in H.Then, we have dualityH′↪V′. Using Riesz theorem, we identify H and H′. So we get V↪H↪V′.Definition 2.1.
We define the Hilbert space(2.1)W(0,T,V,V′)={u∈L2(]0,T[,V)such that∂u∂t∈L2(]0,T[,V′)},
equipped with the norm
(2.2)‖u‖W2=‖u‖L2(]0,T[,V)2+‖∂u∂t‖L2(]0,T[,V′)2.We assume the two following lemmas, see [19].Lemma 2.2.
There exists a continuous prolongation operatorP from W(0,T,V,V′) to W(-∞,+∞,V,V′) such that
(2.3)Pu∣]0,T[=u∀u∈W(0,T,V,V′).Lemma 2.3.
𝒟(ℝ,V) is dense in W(-∞,+∞,V,V′).Corollary 2.4.
C∞([0,T],V) is dense in W(0,T,V,V′).Proof.
Ifu∈W(0,T,V,V′), one takes a sequence un of 𝒟(ℝ,V) which converges in W(-∞,+∞,V,V′) toward Pu, and then (un)∣]0,T[ converges toward u and (un)∣]0,T[∈C∞([0,T],V) for all n∈ℕ.Proposition 2.5.
Every elementu∈W(0,T,V,V′) is almost everywhere equal to a function in C0([0,T],H).
Furthermore, the injection ofW(0,T,V,V′) into C0([0,T],H) is continuous when C0([0,T],H) is equipped with the supnorm.Proof.
See [19].Application
For allt∈[0,T], a bilinear form (u,v)→a(t;u,v) is given on V×V such that for u and v fixed, t→a(t;u,v) is measurable and
(2.4)∃M>0,suchthat|a(t;u,v)|≤M‖u‖V‖v‖V∀u,v∈V.
For each fixedt, one defines a continuous linear application A(t)∈ℒ(V,V′) by
(2.5)〈A(t)u,v〉=a(t;u,v).
Then, we have(2.6)‖A(t)‖L(V,V′)≤M.
Also we associate, for all fixedt, an unbounded operator in H whose domain is the set of u∈V such that v→a(t;u,v) is continuous on V for the induced norm by H. It is exactly the set of u∈V such that A(t)u∈H and then
(2.7)a(t;u,v)=(A(t)u,v)H.
To simplify the writting the unbounded operator is notedA(t).
Letψ∈C∞([0,T],V), we have, for v∈V,
(2.8)ddt(ψ(t),v)H=(∂ψ∂t,v)H=〈ψ′(t),v〉,
where the bracket is the duality between V′ and V because V↪V′. By density, if u∈W(0,T,V,V′), one has, for all v∈V,
(2.9)ddt(u(t),v)H=〈u′(t),v〉fora.e.t.
The variational parabolic problem associated to the triple(H,V,a(t;·,·)) is the following.
Givenf(t)∈L2(]0,T[;V′) and u0∈H, find u∈W(0,T,V,V′) such that
(2.10)(P){ddt(u(t),v)H+a(t;u(t),v)=(f(t),v)∀v∈V,u(0)=u0.
This problem is equivalent to(2.11)du(t)dt+A(t)u(t)=f(t),u(0)=u0.Definition 2.6.
The forma is coercive or V coercive if α>0 exists such that
(2.12)a(t;u,u)≥α‖u‖V2∀(t,u)∈[0,T]×V.Theorem 2.7.
If the form is coercive, then the problem(𝒫) admits a unique solution.Proof.
See Dautray-Lions [19].Definition 2.8.
The form isH coercive if there exist two constants λ and α>0 such that
(2.13)a(t;u,u)+λ‖u‖H2≥α‖u‖V2,∀t∈[0,T],∀u∈V.If we setu(t)=eλtw(t), then u is solution of (𝒫) if and only if w is solution of(2.14)(P′){ddt(w(t),v)H+a(t;w,v)+λ(w,v)H=〈e-λtf(t),v〉∀v∈V,w(0)=u0.Writing(2.15)b(t;u,v)=a(t;u,v)+λ(u,v)H,b is a coercive form, and then (𝒫′) admits a unique solution, and therefore (𝒫) too. We apply Theorem 2.7 in the following case: (2.16)H=L2(Ω),V=H1(Ω),
and defining(2.17)a(t;u,v)=∑i,j=13∫Ωaij(t,x)∂u∂xi∂v∂xjdx,
we assume that(2.18)aij(t,x)∈L∞(]0,T[×Ω)∀i,j,
and there exists α>0 such that, for all ζ=(ζi)i=1,2,3, we have(2.19)∑i,j=13aij(t,x)ζiζj≥α‖ζ‖R32a.e.in]0,T[×Ω.Then, we deduce that(2.20)a(t;u,u)≥α∑i=13‖∂u∂xi‖L2(Ω)2.The form is thenH coercive, and it suffices to take λ=α.In addition, let us takea0∈L∞(]0,T[×Ω) with a0(t,x)≥0 for all t, x.The form(2.21)b(t;u,v)=a(t;u,v)+(a0(t)u,v)H
is still H coercive. We have the following theorem.Theorem 2.9.
Under the previous hypothesis, problem(𝒫) associated to the triple (H,V,b) admits a unique solution for all u0∈H and f∈L2(]0,T[;V′).
Moreover, ifu0≥0 and
(2.22)〈f(t),v〉≥0∀v∈Vsuchthatv≥0,t∈]0,T[a.e.,
one has
u(t)≥0 for all
t∈[0,T].Proof
It remains to show that the solution is nonnegative.
Givenu∈L2(Ω), we set
u+=max⁡(0,u(x)) and
u-=max⁡(0,-u(x)). If
u∈H1(Ω), then we have
u+ and
u-∈H1(Ω).
By replacingv by
u-(t) in
(P), we obtain
(2.22)〈dudt,u-(t)〉+a(t;u(t),u-(t))+(a0(t)u,u-)H=〈f(t),u-〉.
One gets(2.24)u=u+-u-,u+u-=0,du+dtu-=0,
and by linearity, we obtain
(2.25)-〈du-dt,u-〉-a(t;u-(t),u-(t))-(a0(t)u-,u-)H=〈f(t),u-〉,
but(2.26)〈du-dt,u-〉=12ddt(u-(t),u-(t))H.
Since(2.27)(a(t)u-,u-)≥0,(a0(t)u-,u-)H≥0,〈f(t),u-(t)〉≥0,
it comes that
(2.28)ddt‖u-(t)‖H2≤0.
By integration over]0,t[, we deduce
(2.29)‖u-(t)‖H2≤‖u-(0)‖H2,∀t.
Butu0≥0, then u0-=0, so u-(0)=0.
Hence, we conclude thatu-(t)=0 for all t.
Instead ofa0(t,x)≥0, assume that
(2.30)∃C>0suchthata0(t,x)≥-C∀t,x.
As previously mentioned, if we setu(t)=eλtw(t), w(t) is solution of
(2.31)ddt(w(t),v)H+a(t;w(t),v)+((a0(t)+λ)w,v)H=〈e-λtf(t),v〉,
with
(2.32)a0(t,x)+λ≥λ-C.
It suffices to takeλ≥C to reduce to the previous case, and w(t)≥0 implies u(t)≥0.
Then, we get.Corollary 2.10.
Consider the triple(H,V,a) satisfying assumptions of Theorem 2.9.
Ifa0∈L∞(]0,T[×Ω) and
(2.33)∃C>0such thata0(t,x)≥-Cfora.e.(t,x)∈]0,T[×Ω,
then the variational problem
(2.34)ddt(u(t),v)+a(t;u,v)+(a0(t)u,v)H=〈f(t),v〉∀v∈V,u(0)=u0
admits a unique solution in W(0,T,V,V′) for all u0∈H and f∈L2(]0,T[,V′).
Moreover, ifu0≥0 and
(2.35)〈f(t),v〉≥0∀v∈V,t∈]0,T[a.e.,
then one has
u(t)≥0 for all
t∈[0,T].Equivalence of the Variational Solution with the Initial Problem
We have(2.35)∂QT=({0}×Ω)∪({T}×Ω)∪(]0,T[×∂Ω).
For the sake of simplicity, we setaij=δij which is the Kronecker symbol.
Then,A=-Δ over 𝒟(Ω), and we have
(2.37)∂u∂t-Δu+a0u=fin]0,T[×Ω.
We assume thatf∈L2(]0,T[,H), then
(2.38)∂u∂t-Δu∈L2(]0,T[,H)
if u∈L2(]0,T[,V). We set
(2.39)v=(u,-grad(u))T=(u,-∂u∂x1,-∂u∂x2,-∂u∂x3)T.
Consequently, we havev∈(L2(]0,T[×Ω))4 and
(2.40)div⁡(v)=∂u∂t-Δu∈L2(]0,T[×Ω).
Thenu(0,x), u(T,x), and ∂u/∂n∣]0,T[×∂Ω are well defined.
It remains to show that∂u/∂n|]0,T[×∂Ω=0.
Letϕ∈𝒟(]0,T[), and we multiply (2.34)1 by ϕ, and by integration over ]0,T[, one gets
(2.41)∫0T∫Ωϕv∂u∂tdtdx+∑i=13∫0T∫Ωϕ∂u∂xi∂v∂xidtdx+∫0T∫Ωϕa0uvdtdx=∫0T∫Ωfvϕdtdx∀v∈H1(Ω).
Using Green formula with(u,-grad(u))T, we have
(2.42)∫0T∫Ωϕv(∂u∂t-Δu+a0u)dxdt=∫0T∫Ωfϕvdxdt-∫0T∫∂Ωϕv∂u∂ndsdt.
As(2.43)∂u∂t-Δu+a0u=f,
we can conclude the following statement:
(2.44)∫0T∫∂Ωϕv∂u∂ndtds=0∀(ϕ,v)∈D(]0,T[)×H1(Ω).
We deduce that(2.45)〈v,∂u∂n〉H1/2H-1/2=0∀v∈H1(Ω)andta.e.
Functionv→v|∂Ω from H1(Ω) into H1/2(∂Ω) being surjective, we deduce that
(2.46)∂u∂n=0inH-1/2(Γ)forta.e.


### 2.2. Semigroup Method

Consider the variational triple(H,V,a) where a is independent of t. We associate operators A∈ℒ(V,V′) and AH in H with(2.47)D(AH)={u∈VsuchthatAu∈H}.Assume thata is H coercive, then AH is the infinitesimal generator of semigroup t↦G(t) of class 𝒞0 over H, and G(t) operates over V and V′. If we note 𝒢(t)̃ the extension of G(t) by 0 for t<0, then the Laplace transform of 𝒢(t)̃ is the resolvent of AH.Proposition 2.11.
Foru0∈H and f∈L2(]0,T[,V′), problem (𝒫) which consists in finding u∈W(0,T,V,V′) such that
(2.48)dudt+Au=fwithu(0)=u0
admits a unique solution given by
(2.49)u(t)=G(t)u0+∫0tG(t-s)f(s)ds.Proof.
Noteũ and f̃ the extensions by 0 of u and f outside ]0,T[, then we have
(2.50)dũdt=dudt+u(0)δ(t)-u(T)δ(T-t),
with δ(t) the Dirac measure on ℝ.
Thus,(2.51)dũdt+Aũ=dudt+Ãu+u(0)δ(t)-u(T)δ(T-t)=f̃+u(0)δ(t)-u(T)δ(T-t).
Hence, an equation of the form(2.52)dUdt+AU=FinD+′(V′),
where 𝒟+′(V′) is the space of distributions over ℝ into V′whose support is in [0,+∞[. By Laplace transform, one is reduced to
(2.53)(A+PI)L(U)=L(F),
where
(2.54)L(U)=(A+PI)-1L(F),
and therefore, we have U=G̃*F.
But since(2.55)supp⁡(δ(T-t))=T,supp⁡(G̃*(U(T)δ(T-t)))⊂[T,+∞[,
we have
(2.56)U∣]0,T[=u=(G̃*F)∣]0,T[.
Hence, we get the result.


## 2.1. Variational Method (See [19])

We consider two Hilbert spacesH and V such that V is embedded continuously and densely in H.Then, we have dualityH′↪V′. Using Riesz theorem, we identify H and H′. So we get V↪H↪V′.Definition 2.1.
We define the Hilbert space(2.1)W(0,T,V,V′)={u∈L2(]0,T[,V)such that∂u∂t∈L2(]0,T[,V′)},
equipped with the norm
(2.2)‖u‖W2=‖u‖L2(]0,T[,V)2+‖∂u∂t‖L2(]0,T[,V′)2.We assume the two following lemmas, see [19].Lemma 2.2.
There exists a continuous prolongation operatorP from W(0,T,V,V′) to W(-∞,+∞,V,V′) such that
(2.3)Pu∣]0,T[=u∀u∈W(0,T,V,V′).Lemma 2.3.
𝒟(ℝ,V) is dense in W(-∞,+∞,V,V′).Corollary 2.4.
C∞([0,T],V) is dense in W(0,T,V,V′).Proof.
Ifu∈W(0,T,V,V′), one takes a sequence un of 𝒟(ℝ,V) which converges in W(-∞,+∞,V,V′) toward Pu, and then (un)∣]0,T[ converges toward u and (un)∣]0,T[∈C∞([0,T],V) for all n∈ℕ.Proposition 2.5.
Every elementu∈W(0,T,V,V′) is almost everywhere equal to a function in C0([0,T],H).
Furthermore, the injection ofW(0,T,V,V′) into C0([0,T],H) is continuous when C0([0,T],H) is equipped with the supnorm.Proof.
See [19].Application
For allt∈[0,T], a bilinear form (u,v)→a(t;u,v) is given on V×V such that for u and v fixed, t→a(t;u,v) is measurable and
(2.4)∃M>0,suchthat|a(t;u,v)|≤M‖u‖V‖v‖V∀u,v∈V.
For each fixedt, one defines a continuous linear application A(t)∈ℒ(V,V′) by
(2.5)〈A(t)u,v〉=a(t;u,v).
Then, we have(2.6)‖A(t)‖L(V,V′)≤M.
Also we associate, for all fixedt, an unbounded operator in H whose domain is the set of u∈V such that v→a(t;u,v) is continuous on V for the induced norm by H. It is exactly the set of u∈V such that A(t)u∈H and then
(2.7)a(t;u,v)=(A(t)u,v)H.
To simplify the writting the unbounded operator is notedA(t).
Letψ∈C∞([0,T],V), we have, for v∈V,
(2.8)ddt(ψ(t),v)H=(∂ψ∂t,v)H=〈ψ′(t),v〉,
where the bracket is the duality between V′ and V because V↪V′. By density, if u∈W(0,T,V,V′), one has, for all v∈V,
(2.9)ddt(u(t),v)H=〈u′(t),v〉fora.e.t.
The variational parabolic problem associated to the triple(H,V,a(t;·,·)) is the following.
Givenf(t)∈L2(]0,T[;V′) and u0∈H, find u∈W(0,T,V,V′) such that
(2.10)(P){ddt(u(t),v)H+a(t;u(t),v)=(f(t),v)∀v∈V,u(0)=u0.
This problem is equivalent to(2.11)du(t)dt+A(t)u(t)=f(t),u(0)=u0.Definition 2.6.
The forma is coercive or V coercive if α>0 exists such that
(2.12)a(t;u,u)≥α‖u‖V2∀(t,u)∈[0,T]×V.Theorem 2.7.
If the form is coercive, then the problem(𝒫) admits a unique solution.Proof.
See Dautray-Lions [19].Definition 2.8.
The form isH coercive if there exist two constants λ and α>0 such that
(2.13)a(t;u,u)+λ‖u‖H2≥α‖u‖V2,∀t∈[0,T],∀u∈V.If we setu(t)=eλtw(t), then u is solution of (𝒫) if and only if w is solution of(2.14)(P′){ddt(w(t),v)H+a(t;w,v)+λ(w,v)H=〈e-λtf(t),v〉∀v∈V,w(0)=u0.Writing(2.15)b(t;u,v)=a(t;u,v)+λ(u,v)H,b is a coercive form, and then (𝒫′) admits a unique solution, and therefore (𝒫) too. We apply Theorem 2.7 in the following case: (2.16)H=L2(Ω),V=H1(Ω),
and defining(2.17)a(t;u,v)=∑i,j=13∫Ωaij(t,x)∂u∂xi∂v∂xjdx,
we assume that(2.18)aij(t,x)∈L∞(]0,T[×Ω)∀i,j,
and there exists α>0 such that, for all ζ=(ζi)i=1,2,3, we have(2.19)∑i,j=13aij(t,x)ζiζj≥α‖ζ‖R32a.e.in]0,T[×Ω.Then, we deduce that(2.20)a(t;u,u)≥α∑i=13‖∂u∂xi‖L2(Ω)2.The form is thenH coercive, and it suffices to take λ=α.In addition, let us takea0∈L∞(]0,T[×Ω) with a0(t,x)≥0 for all t, x.The form(2.21)b(t;u,v)=a(t;u,v)+(a0(t)u,v)H
is still H coercive. We have the following theorem.Theorem 2.9.
Under the previous hypothesis, problem(𝒫) associated to the triple (H,V,b) admits a unique solution for all u0∈H and f∈L2(]0,T[;V′).
Moreover, ifu0≥0 and
(2.22)〈f(t),v〉≥0∀v∈Vsuchthatv≥0,t∈]0,T[a.e.,
one has
u(t)≥0 for all
t∈[0,T].Proof
It remains to show that the solution is nonnegative.
Givenu∈L2(Ω), we set
u+=max⁡(0,u(x)) and
u-=max⁡(0,-u(x)). If
u∈H1(Ω), then we have
u+ and
u-∈H1(Ω).
By replacingv by
u-(t) in
(P), we obtain
(2.22)〈dudt,u-(t)〉+a(t;u(t),u-(t))+(a0(t)u,u-)H=〈f(t),u-〉.
One gets(2.24)u=u+-u-,u+u-=0,du+dtu-=0,
and by linearity, we obtain
(2.25)-〈du-dt,u-〉-a(t;u-(t),u-(t))-(a0(t)u-,u-)H=〈f(t),u-〉,
but(2.26)〈du-dt,u-〉=12ddt(u-(t),u-(t))H.
Since(2.27)(a(t)u-,u-)≥0,(a0(t)u-,u-)H≥0,〈f(t),u-(t)〉≥0,
it comes that
(2.28)ddt‖u-(t)‖H2≤0.
By integration over]0,t[, we deduce
(2.29)‖u-(t)‖H2≤‖u-(0)‖H2,∀t.
Butu0≥0, then u0-=0, so u-(0)=0.
Hence, we conclude thatu-(t)=0 for all t.
Instead ofa0(t,x)≥0, assume that
(2.30)∃C>0suchthata0(t,x)≥-C∀t,x.
As previously mentioned, if we setu(t)=eλtw(t), w(t) is solution of
(2.31)ddt(w(t),v)H+a(t;w(t),v)+((a0(t)+λ)w,v)H=〈e-λtf(t),v〉,
with
(2.32)a0(t,x)+λ≥λ-C.
It suffices to takeλ≥C to reduce to the previous case, and w(t)≥0 implies u(t)≥0.
Then, we get.Corollary 2.10.
Consider the triple(H,V,a) satisfying assumptions of Theorem 2.9.
Ifa0∈L∞(]0,T[×Ω) and
(2.33)∃C>0such thata0(t,x)≥-Cfora.e.(t,x)∈]0,T[×Ω,
then the variational problem
(2.34)ddt(u(t),v)+a(t;u,v)+(a0(t)u,v)H=〈f(t),v〉∀v∈V,u(0)=u0
admits a unique solution in W(0,T,V,V′) for all u0∈H and f∈L2(]0,T[,V′).
Moreover, ifu0≥0 and
(2.35)〈f(t),v〉≥0∀v∈V,t∈]0,T[a.e.,
then one has
u(t)≥0 for all
t∈[0,T].Equivalence of the Variational Solution with the Initial Problem
We have(2.35)∂QT=({0}×Ω)∪({T}×Ω)∪(]0,T[×∂Ω).
For the sake of simplicity, we setaij=δij which is the Kronecker symbol.
Then,A=-Δ over 𝒟(Ω), and we have
(2.37)∂u∂t-Δu+a0u=fin]0,T[×Ω.
We assume thatf∈L2(]0,T[,H), then
(2.38)∂u∂t-Δu∈L2(]0,T[,H)
if u∈L2(]0,T[,V). We set
(2.39)v=(u,-grad(u))T=(u,-∂u∂x1,-∂u∂x2,-∂u∂x3)T.
Consequently, we havev∈(L2(]0,T[×Ω))4 and
(2.40)div⁡(v)=∂u∂t-Δu∈L2(]0,T[×Ω).
Thenu(0,x), u(T,x), and ∂u/∂n∣]0,T[×∂Ω are well defined.
It remains to show that∂u/∂n|]0,T[×∂Ω=0.
Letϕ∈𝒟(]0,T[), and we multiply (2.34)1 by ϕ, and by integration over ]0,T[, one gets
(2.41)∫0T∫Ωϕv∂u∂tdtdx+∑i=13∫0T∫Ωϕ∂u∂xi∂v∂xidtdx+∫0T∫Ωϕa0uvdtdx=∫0T∫Ωfvϕdtdx∀v∈H1(Ω).
Using Green formula with(u,-grad(u))T, we have
(2.42)∫0T∫Ωϕv(∂u∂t-Δu+a0u)dxdt=∫0T∫Ωfϕvdxdt-∫0T∫∂Ωϕv∂u∂ndsdt.
As(2.43)∂u∂t-Δu+a0u=f,
we can conclude the following statement:
(2.44)∫0T∫∂Ωϕv∂u∂ndtds=0∀(ϕ,v)∈D(]0,T[)×H1(Ω).
We deduce that(2.45)〈v,∂u∂n〉H1/2H-1/2=0∀v∈H1(Ω)andta.e.
Functionv→v|∂Ω from H1(Ω) into H1/2(∂Ω) being surjective, we deduce that
(2.46)∂u∂n=0inH-1/2(Γ)forta.e.


## 2.2. Semigroup Method

Consider the variational triple(H,V,a) where a is independent of t. We associate operators A∈ℒ(V,V′) and AH in H with(2.47)D(AH)={u∈VsuchthatAu∈H}.Assume thata is H coercive, then AH is the infinitesimal generator of semigroup t↦G(t) of class 𝒞0 over H, and G(t) operates over V and V′. If we note 𝒢(t)̃ the extension of G(t) by 0 for t<0, then the Laplace transform of 𝒢(t)̃ is the resolvent of AH.Proposition 2.11.
Foru0∈H and f∈L2(]0,T[,V′), problem (𝒫) which consists in finding u∈W(0,T,V,V′) such that
(2.48)dudt+Au=fwithu(0)=u0
admits a unique solution given by
(2.49)u(t)=G(t)u0+∫0tG(t-s)f(s)ds.Proof.
Noteũ and f̃ the extensions by 0 of u and f outside ]0,T[, then we have
(2.50)dũdt=dudt+u(0)δ(t)-u(T)δ(T-t),
with δ(t) the Dirac measure on ℝ.
Thus,(2.51)dũdt+Aũ=dudt+Ãu+u(0)δ(t)-u(T)δ(T-t)=f̃+u(0)δ(t)-u(T)δ(T-t).
Hence, an equation of the form(2.52)dUdt+AU=FinD+′(V′),
where 𝒟+′(V′) is the space of distributions over ℝ into V′whose support is in [0,+∞[. By Laplace transform, one is reduced to
(2.53)(A+PI)L(U)=L(F),
where
(2.54)L(U)=(A+PI)-1L(F),
and therefore, we have U=G̃*F.
But since(2.55)supp⁡(δ(T-t))=T,supp⁡(G̃*(U(T)δ(T-t)))⊂[T,+∞[,
we have
(2.56)U∣]0,T[=u=(G̃*F)∣]0,T[.
Hence, we get the result.


## 3. System(𝒮) Resolution

In this part, we go back to system(𝒮) with assumptions and will analyze this problem by using the framework described in the previous section.We defineH=L2(Ω) and V=H1(Ω) and the following hypothesis for initial conditions:(3.1)u01∈L∞(Ω),u0i∈Hfori∈{2,…,6},u0i≥0fori∈{1,…,6}.We will make a resolution component by applying Theorem2.7 with, for each i, the form (3.2)a(t;u,v)=∑j=13Di∫Ω∂u∂xj∂v∂xjdx.One approaches the solution by a sequence of solutions of linear equations.


### 3.1. Recursive Sequence of Solutions

Forn=0, we note that ui0 is the solution of(3.3)∂ui0∂t-DiΔui0=0inQT,ui0(0)=u0iinΩ,∂ui0∂n∣∂Ω=0.This equation admits strong solution andui0≥0.By induction, we note thatuin is solution of equation(3.4)∂uin∂t-DiΔuin+qi(un-1)uin=fi(un-1)inQT,uin(0)=u0iinΩ,∂uin∂n∣∂Ω=0.It is a linear equation within the framework of Corollary2.10 with a0=qi(un-1) and f(t)=fi(un-1(t)). Let us suppose that there exists a unique nonnegative solution un-1. Assuming by induction that uij≥0 for 0≤j≤n-1, we have (3.5)0≤ku2n-1Ks+u2n-1≤k,
which implies that (3.6)μ+r+ν-k≤q1(un-1)≤μ+r+ν.un-1 is nonnegative also that implies that there are two positive constants C1, C2 such that (3.7)0≤q3(un-1)≤C1,0≤q4(un-1)≤C2.
For the rest, we notice that q5 and q6 are constant.We have shown thatqi(un-1)∈L∞(]0,T[×Ω) for i≠2. It remains to prove that the same property is satisfied by q2(un-1).To prove thatq2(un-1) is bounded, we need to show that u1n∈L∞(0,T;L∞(Ω)).Case ofu10
Letk∈ℕ*, we multiply (3.3)1 by (u10)2k-1 and integrate it over Ω, and it comes that
(3.8)12kddt∫Ω(u10)2kdx+(2k-1)D1∫Ω(u10)2(k-1)|∇u10|2dx=0.
The second term is nonnegative, then we have
(3.9)12kddt∫Ω(u10)2kdx≤0.
By integrating over ]0,t[, we obtain
(3.10)‖u10(t)‖L2k(Ω)≤‖u01‖L2k(Ω).
When k tends to +∞, it comes that,
(3.11)‖u10(t)‖L∞(Ω)≤‖u01‖L∞(Ω),
which implies that u10∈L∞(0,T;L∞(Ω)).Case ofu1n with n∈ℕ*
By induction, we suppose thatu10,u11,…,u1n-1 are in L∞(0,T;L∞(Ω)).Remark 3.1.
We make the following change:win=e-λtuin,i=1,…,6. We obtain
(3.12)∂w1n∂t-D1Δw1n+(λ+q1((eλtwin-1)i))w1n=0.
The functionq1 being undervalued, we can choose λ≥0 such that
(3.13)λ+q1((eλtwin-1)i)≥0.
We multiply (3.12) by (w1n)2k-1 and integrate it over Ω. We obtain
(3.14)12kddt∫Ω(w1n)2kdx+(2k-1)D1∫Ω(w1n)2(k-1)|∇w1n|2dx+∫Ω(λ+q1((eλtwin-1)i))(w1n)2kdx=0.
The second and third term being nonnegative, we can conclude as in the previous case that
(3.15)‖w1n(t)‖L2k(Ω)≤‖u01‖L2k(Ω).
Since k tends to ∞, it follows that
(3.16)‖w1n(t)‖L∞(Ω)≤‖u01‖L∞(Ω).
As a result, we have proved that w1n∈L∞(0,∞;L∞(Ω)), and since u1n=eλtw1n, we have u1n∈L∞(0,T;L∞(Ω)).Conclusion 1.
With the previous demonstration, we obtain by induction that ifu01∈L∞(Ω) with u01≥0, then qi(un)∈L∞(]0,T[×Ω) for all n and i=1,…,6.
We also havefi(un-1)≥0 and fi(un-1)∈L2(]0,T[;V′). Then by means of Corollary 2.10, there exists a unique solution uin∈W(0,T,V,V′) with uin≥0.


### 3.2. Boundedness of the Solution

Let us show that the sequence is bounded.uin satisfies (2.34), so(3.17)∂∂t(uin,v)H+Dia(uin,v)+(qi(un-1)uin,v)H=〈fi(un-1),v〉∀v∈V.We remark that(3.18)∂∂t(uin,v)H=〈∂uin∂t,v〉.Forψ∈C∞([0,T],V), (3.19)〈∂ψ(t)∂t,ψ(t)〉=12ddt(ψ(t),ψ(t))H.
By density and choosing v=uin(t), we have (3.20)〈∂uin(t)∂t,uin(t)〉=12ddt(uin(t),uin(t))H.Hence,(3.21)12ddt‖uin‖H2+Dia(uin,uin)+(qi(un-1)uin,uin)H=〈fi(un-1),uin〉.We have seen that we can obtain problem(𝒫′) replacing u(t) by w(t)=e-λtu(t), and since 0≤t≤T, if w is bounded, u is also bounded.We take thenλ=β+δ, and one is reduced to (3.22)12ddt‖uin‖H2+Dia(uin,uin)+β‖uin‖H2+((δ+qi(un-1))uin,uin)H=〈fi(un-1),uin〉.The formDia is H coercive, so we take β such that (3.23)Dia(u,u)+β‖u‖H2≥α‖u‖V2∀u∈V.Theqi are bounded, so(3.24)∃δ>0suchthatδ+qi(u)≥l∀i,u≥0.
Therefore, (3.25)12ddt‖uin‖H2+α‖uin‖V2+l‖uin‖H2≤‖fi(un-1)‖V′‖uin‖V≤ϵ2‖uin‖V2+12ϵ‖fi(un-1)‖V′2ϵ>0.We takeϵ small enough such that α-ϵ/2=γ>0. Hence, (3.26)12‖uin‖H2+γ‖uin‖V2+l‖uin‖H2≤12ϵ‖fi(un-1)‖V′2.Fori=1 and i=4, fi=0. Therefore, by integration, (3.27)12‖uin(t)‖H2+γ∫0t‖uin(s)‖V2ds+l∫0t‖uin(s)‖H2≤12‖uin(0)‖H2.We deduce that(u1n) and (u4n) remain bounded in C0([0,T],H) and L2(]0,T[,V).f5(u)=νu1, thus, f5(un-1)=νu1n-1 remains bounded in L2(]0,T[,V); therefore, u5n has the same property as u1n, u4n. It is the same for u6n because f6(u)=ru1.We havef3(u)=(ζu5+μu1)/2; therefore, we have the same conclusion for u3n and finally for u2n because f2(u) depends on u1, u2, u4, and u5.


### 3.3. Convergence of the Sequence

We deduce at this stage that the sequence(uin)n≥0 (one can extract subsequence (uim)m≥0) converges weakly in L2(]0,T[,V) to ui and weakly star in L∞(]0,T[,H) to ui.But it is not enough to pass to the limit in the equation, we need the pouctual convergence for almost allt to deduce that ui≥0 for all i and to pass to the limit in qi(un-1) and fi(un-1). To pass to the limit, we need strong compactness. Using Proposition 2.11, for all n, we have (3.28)uin(t)=∫0tGi(t-s)gin(s)ds+Gi(t)u0i,
where Gi(t) is the semigroup generated by the unbounded operator -DiAH. Let us denote (3.29)gin(s)=-qi(un-1(s))uin(s)+fi(un-1(s)),
and we deduce gin∈L2(]0,T[,V).Moreover, the sequence(uin)n≥0 is bounded in C0([0,T],H) which implies that the sequence (gin)n≥0 is bounded in C0([0,T],H) for all i.Then, we can conclude showing that operator𝒢i from C0([0,T],H) into C0([0,T],H) defined by (3.30)Gi(f)(t)=∫0tGi(t-s)f(s)ds
is compact.One takes the triple(L2(Ω),H1(Ω),a) with(3.31)a(u,v)=∑j=1n∫Ω∂u∂xj∂v∂xjdx,
where Ω is regular and bounded. The unbounded variational operator AH associated to a is a positive symmetric operator with compact resolvent. It admits a sequence (λk)k of positive eigenvalues with limk→+∞λk=+∞ and a Hilbert basis (ek)k of H consisting of eigenvectors of AH. If (G(t))t>0 is the semigroup generated by -AH, then for all u0∈H,(3.32)G(t)u0=∑k=0+∞e-tλk(u0,ek)ek,which proves that the operator is compact for allt>0 because (3.33)lim⁡k→+∞⁡e-tλk=0.
We have the same formula for Gi(t), and it suffices to replace λk by Diλk.If we set(3.34)GN(t)u=∑k=0Ne-tλk(u,ek)ek,
then GN(t) is an operator with finite rank which converges to G(t).Theorem 3.2.
Lett→G(t) be an application from [0,∞[ into ℒ(H). One assumes that there exists a sequence of operators (GN(t))N≥0 of H with the following properties:  (1)
for allN and all t>0, GN(t) is finite rank independent of t,(2)
t→GN(t) is continuous from [0,∞[ into ℒ(H) for all N,(3)
forN→∞, GN(t) converges to G(t) in L1(]0,T[,ℒ(H)) for all T>0,
then the operator𝒢 is compact from 𝒞0([0,T],H) into 𝒞0([0,T],H) for all T>0.Proof.
Regarding property (3) of Theorem3.2, 𝒢 is well defined on 𝒞0([0,T],H), and we have
(3.35)‖∫0tG(s)f(t-s)ds-∫0tGN(s)f(t-s)ds‖≤(∫0T‖G(s)-GN(s)‖ds)‖f‖∞.
This proves that 𝒢N converges to 𝒢 in ℒ([0,T],H) using property  (3) of Theorem 3.2.
To show that𝒢 is compact, it suffices to show that for all N, 𝒢N is compact.
Letℬ be a bounded set of 𝒞0([0,T],H), 𝒢N(ℬ) is bounded in 𝒞0([0,T],H), using Ascoli result, it will be relatively compact if 𝒢N(ℬ) is equicontinuous and if for all t0 in [0,T], 𝒢N(ℬ)(t0) is relatively compact in H.
But𝒢N(ℬ)(t0) being bounded and embedded in a subspace of finite dimension of H is relatively compact in H. Then, let us show the equicontinuity on t0.
LetM and CN such that
(3.36)‖f‖∞≤M,∀f∈B,‖GN(t)‖≤CN∀t∈[0,T].
For 0≤t0≤t≤T, one has
(3.37)GN(f)(t)-GN(f)(t0)=∫0t0(GN(t-s)-GN(t0-s))f(s)ds+∫t0tGN(t-s)f(s)ds.
We have
(3.38)‖∫t0tGN(t-s)f(s)ds‖≤MCN|t-t0|,‖∫0t0(GN(t-s)-GN(t0-s))f(s)ds‖≤M∫0t0‖GN(t-s)-GN(t0-s)‖ds,
which tend to 0 when t→t0 using property (2) of Theorem 3.2 and the continuity under the integral.
We apply Theorem3.2 to the semigroup generated by the Laplacien, and we obtain(1)
GN(t) is of rank N+1 for all t≥0,(2)
∥GN(t)-GN(t0)∥2≤∑k=0N|e-tλk-e-t0λk|2 which tends to 0 if t→t0,(3)
∥G(t)u-GN(t)u∥2≤∑k=N+1∞e-2tλk|uk|2≤e-2tλN+1∥u∥2 if we take an increasing sequence (λk), then one gets
(3.39)‖G(t)-GN(t)‖≤e-tλN+1.
Thus, we have
(3.40)∫0T‖G(t)-GN(t)‖dt≤1λN+1,
which tends to 0 if N→∞.
Thus,𝒢i is compact for all i. We have
(3.41)uin(t)=Gi(t)ui0+Gi(gin)(t),
where (gin)n≥0 is bounded in C0([0,T],H), then (uin)n≥0 belongs to a relatively compact set of C0([0,T],H).
Therefore, from the sequence(uin)n≥0, we can extract a subsequence (uim)m≥0 which converges uniformly to ui∈𝒞0([0,T],H) for each i.
Moreover, we can assume thatuim converges weakly to ui in L2(]0,T[,V).
ButAi∈ℒ(V,V′) and (uin) bounded in L2(]0,T[,V) imply
(3.42)∫0T‖Aiuin‖V′2dt≤‖Ai‖2∫0T‖uin‖V2dt.
Thus, the sequence(Aiuin)n≥0 is bounded in L2(]0,T[,V′). We can assume that Aiuim converges weakly to Aiui in L2(]0,T[,V′).
Remark that for allT<∞, we have
(3.43)C0([0,T],H)↪L2(]0,T[,V′).Conclusion 2.
One has(3.44)uim⟶uiinC0([0,T],H).
Thus, ui≥0 and ui(0)=u0i.
We check that(3.45)qi(um-1)⟶qi(u),qi(um-1)uim⟶qi(u)uiinC0([0,T],H)∀i.
We have also(3.46)fi(um-1)⟶fi(u)inC0([0,T],H)∩L2(]0,T[,V′)∀i.
For alli, uim is solution of
(3.47)〈∂uim∂t,v〉+〈Aiuim,v〉+(qi(um-1)uim,v)=〈fi(um-1),v〉∀v∈V.
We takeϕ∈𝒟(]0,T[), and therefore ϕv∈L2(]0,T[,V),
(3.48)∫0T〈∂uim∂t,ϕv〉dt+∫0T〈Aiuim,ϕv〉dt+∫0T(qi(um-1)uim,ϕv)dt=∫0T〈fi(um-1),ϕv〉dt.
The second term in the left side and the right side of the egality converges due to the weak convergence inL2(]0,T[,V′). The third term in the left-hand side also passes to the limit, due to the convergence in 𝒞0([0,T],H).
We deduce that∂uim/∂t converges weakly in L2(]0,T[,V′).
But we have(3.49)uim⟶uiinC0([0,T],H),
then
(3.50)∂uim∂t⟶∂ui∂tinD′(]0,T[,H).
Therefore, we obtain
(3.51)∂uim∂t⟶∂ui∂tweaklyinL2(]0,T[,V′),∫0T〈∂ui∂t,ϕv〉dt+∫0T〈Aiui,ϕv〉dt+∫0T(qi(u)ui,ϕv)Hdt=∫0T〈fi(u),ϕv〉dt.
This formulation being true for all ϕ, we have
(3.52)〈∂ui∂t,v〉+〈Aiui,v〉+(qi(u)ui,v)H=〈fi(u),v〉∀v∈V,
that is to say,
(3.53)ddt(ui,v)H+a(ui,v)+(qi(u)ui,v)H=〈fi(u),v〉∀v∈V,i,∂ui∂t=fi(u)-Aiui-qi(u)ui∈L2(]0,T[,V′)∀i.
For alli, one has
(3.54)uim(t)=Gi(t)u0i+∫0tGi(t-s)(-qi(um-1)uim+fi(uim-1))(s)ds,
and as qi(um-1)uim and fi(um-1) converge in C0([0,T],H), we have
(R)ui(t)=Gi(t)u0i+∫0tGi(t-s)(-qi(u)ui+fi(u))(s)ds.


### 3.4. Main Result

Theorem 3.3.
If the initial condition satisfies (3.1), then system (𝒮) admits a unique nonnegative solution ui∈W(0,T,V,V′) for all i.
Moreover, for alli, ui satisfies relation (3.62).Proof.
We have already shown existence of solution; thus, it remains to show uniqueness.
Letv be another solution of system (𝒮)(3.55)vi∈W(0,T,V,V′)⟹vi∈C0([0,T],H),vi≥0.
Consequently, we obtain
(3.56)qi(v)vi+fi(v)∈L2(]0,T[,V′).
Thus, by Proposition 2.11, we have
(3.57)vi(t)=Gi(t)u0i+∫0tGi(t-s)(-qi(v)vi+fi(v))(s)ds.
By subtraction, we have(3.58)ui(t)-vi(t)=∫0tGi(t-s)(-qi(u)ui+qi(v)vi+fi(u)-fi(v))(s)ds,
with
(3.59)qi(u)ui-qi(v)vi=qi(u)(ui-vi)+(qi(u)-qi(v))vi.
ui being positive, we have
(3.60)‖ujK+ui‖≤1K‖uj‖∞,
where we have defined
(3.61)‖uj‖∞=‖uj‖L∞(]0,T[,H).
If we define(3.62)‖u‖∞=∑j=16‖uj‖∞,
there is M1>0 such that
(3.63)‖qi(u)‖∞≤M1‖u‖∞∀u.
So the numerator ofqi(u)-qi(v) is the sum of terms of form (uk-vk)vj or (uj-vj)uk, and we can find M2>0 such that
(3.64)|qi(u)-qi(v)|H(s)≤M2(∑j=16|uj(s)-vj(s)|H).
Also we can findM3>0 such that
(3.65)|fi(u)-fi(v)|H(s)≤M3(∑j=16|uj(s)-vj(s)|H).
Summing and noting that∥Gi(t-s)∥≤NieΩiT with Ni, Ωi>0, we can find M such that
(3.66)∑i=16|ui(t)-vi(t)|H≤M‖u-v‖∞.
Replacing in (3.58), we obtain
(3.67)∑i=16|ui(t)-vi(t)|H≤M2‖u-v‖∞∫0tsds=M2t22‖u-v‖∞.
By induction, we have(3.68)∑j=16|uj(t)-vj(t)|H≤Mnn!Tn‖u-v‖∞,lim⁡n→+∞⁡Mnn!Tn‖u-v‖∞=0.


## 3.1. Recursive Sequence of Solutions

Forn=0, we note that ui0 is the solution of(3.3)∂ui0∂t-DiΔui0=0inQT,ui0(0)=u0iinΩ,∂ui0∂n∣∂Ω=0.This equation admits strong solution andui0≥0.By induction, we note thatuin is solution of equation(3.4)∂uin∂t-DiΔuin+qi(un-1)uin=fi(un-1)inQT,uin(0)=u0iinΩ,∂uin∂n∣∂Ω=0.It is a linear equation within the framework of Corollary2.10 with a0=qi(un-1) and f(t)=fi(un-1(t)). Let us suppose that there exists a unique nonnegative solution un-1. Assuming by induction that uij≥0 for 0≤j≤n-1, we have (3.5)0≤ku2n-1Ks+u2n-1≤k,
which implies that (3.6)μ+r+ν-k≤q1(un-1)≤μ+r+ν.un-1 is nonnegative also that implies that there are two positive constants C1, C2 such that (3.7)0≤q3(un-1)≤C1,0≤q4(un-1)≤C2.
For the rest, we notice that q5 and q6 are constant.We have shown thatqi(un-1)∈L∞(]0,T[×Ω) for i≠2. It remains to prove that the same property is satisfied by q2(un-1).To prove thatq2(un-1) is bounded, we need to show that u1n∈L∞(0,T;L∞(Ω)).Case ofu10
Letk∈ℕ*, we multiply (3.3)1 by (u10)2k-1 and integrate it over Ω, and it comes that
(3.8)12kddt∫Ω(u10)2kdx+(2k-1)D1∫Ω(u10)2(k-1)|∇u10|2dx=0.
The second term is nonnegative, then we have
(3.9)12kddt∫Ω(u10)2kdx≤0.
By integrating over ]0,t[, we obtain
(3.10)‖u10(t)‖L2k(Ω)≤‖u01‖L2k(Ω).
When k tends to +∞, it comes that,
(3.11)‖u10(t)‖L∞(Ω)≤‖u01‖L∞(Ω),
which implies that u10∈L∞(0,T;L∞(Ω)).Case ofu1n with n∈ℕ*
By induction, we suppose thatu10,u11,…,u1n-1 are in L∞(0,T;L∞(Ω)).Remark 3.1.
We make the following change:win=e-λtuin,i=1,…,6. We obtain
(3.12)∂w1n∂t-D1Δw1n+(λ+q1((eλtwin-1)i))w1n=0.
The functionq1 being undervalued, we can choose λ≥0 such that
(3.13)λ+q1((eλtwin-1)i)≥0.
We multiply (3.12) by (w1n)2k-1 and integrate it over Ω. We obtain
(3.14)12kddt∫Ω(w1n)2kdx+(2k-1)D1∫Ω(w1n)2(k-1)|∇w1n|2dx+∫Ω(λ+q1((eλtwin-1)i))(w1n)2kdx=0.
The second and third term being nonnegative, we can conclude as in the previous case that
(3.15)‖w1n(t)‖L2k(Ω)≤‖u01‖L2k(Ω).
Since k tends to ∞, it follows that
(3.16)‖w1n(t)‖L∞(Ω)≤‖u01‖L∞(Ω).
As a result, we have proved that w1n∈L∞(0,∞;L∞(Ω)), and since u1n=eλtw1n, we have u1n∈L∞(0,T;L∞(Ω)).Conclusion 1.
With the previous demonstration, we obtain by induction that ifu01∈L∞(Ω) with u01≥0, then qi(un)∈L∞(]0,T[×Ω) for all n and i=1,…,6.
We also havefi(un-1)≥0 and fi(un-1)∈L2(]0,T[;V′). Then by means of Corollary 2.10, there exists a unique solution uin∈W(0,T,V,V′) with uin≥0.


## 3.2. Boundedness of the Solution

Let us show that the sequence is bounded.uin satisfies (2.34), so(3.17)∂∂t(uin,v)H+Dia(uin,v)+(qi(un-1)uin,v)H=〈fi(un-1),v〉∀v∈V.We remark that(3.18)∂∂t(uin,v)H=〈∂uin∂t,v〉.Forψ∈C∞([0,T],V), (3.19)〈∂ψ(t)∂t,ψ(t)〉=12ddt(ψ(t),ψ(t))H.
By density and choosing v=uin(t), we have (3.20)〈∂uin(t)∂t,uin(t)〉=12ddt(uin(t),uin(t))H.Hence,(3.21)12ddt‖uin‖H2+Dia(uin,uin)+(qi(un-1)uin,uin)H=〈fi(un-1),uin〉.We have seen that we can obtain problem(𝒫′) replacing u(t) by w(t)=e-λtu(t), and since 0≤t≤T, if w is bounded, u is also bounded.We take thenλ=β+δ, and one is reduced to (3.22)12ddt‖uin‖H2+Dia(uin,uin)+β‖uin‖H2+((δ+qi(un-1))uin,uin)H=〈fi(un-1),uin〉.The formDia is H coercive, so we take β such that (3.23)Dia(u,u)+β‖u‖H2≥α‖u‖V2∀u∈V.Theqi are bounded, so(3.24)∃δ>0suchthatδ+qi(u)≥l∀i,u≥0.
Therefore, (3.25)12ddt‖uin‖H2+α‖uin‖V2+l‖uin‖H2≤‖fi(un-1)‖V′‖uin‖V≤ϵ2‖uin‖V2+12ϵ‖fi(un-1)‖V′2ϵ>0.We takeϵ small enough such that α-ϵ/2=γ>0. Hence, (3.26)12‖uin‖H2+γ‖uin‖V2+l‖uin‖H2≤12ϵ‖fi(un-1)‖V′2.Fori=1 and i=4, fi=0. Therefore, by integration, (3.27)12‖uin(t)‖H2+γ∫0t‖uin(s)‖V2ds+l∫0t‖uin(s)‖H2≤12‖uin(0)‖H2.We deduce that(u1n) and (u4n) remain bounded in C0([0,T],H) and L2(]0,T[,V).f5(u)=νu1, thus, f5(un-1)=νu1n-1 remains bounded in L2(]0,T[,V); therefore, u5n has the same property as u1n, u4n. It is the same for u6n because f6(u)=ru1.We havef3(u)=(ζu5+μu1)/2; therefore, we have the same conclusion for u3n and finally for u2n because f2(u) depends on u1, u2, u4, and u5.


## 3.3. Convergence of the Sequence

We deduce at this stage that the sequence(uin)n≥0 (one can extract subsequence (uim)m≥0) converges weakly in L2(]0,T[,V) to ui and weakly star in L∞(]0,T[,H) to ui.But it is not enough to pass to the limit in the equation, we need the pouctual convergence for almost allt to deduce that ui≥0 for all i and to pass to the limit in qi(un-1) and fi(un-1). To pass to the limit, we need strong compactness. Using Proposition 2.11, for all n, we have (3.28)uin(t)=∫0tGi(t-s)gin(s)ds+Gi(t)u0i,
where Gi(t) is the semigroup generated by the unbounded operator -DiAH. Let us denote (3.29)gin(s)=-qi(un-1(s))uin(s)+fi(un-1(s)),
and we deduce gin∈L2(]0,T[,V).Moreover, the sequence(uin)n≥0 is bounded in C0([0,T],H) which implies that the sequence (gin)n≥0 is bounded in C0([0,T],H) for all i.Then, we can conclude showing that operator𝒢i from C0([0,T],H) into C0([0,T],H) defined by (3.30)Gi(f)(t)=∫0tGi(t-s)f(s)ds
is compact.One takes the triple(L2(Ω),H1(Ω),a) with(3.31)a(u,v)=∑j=1n∫Ω∂u∂xj∂v∂xjdx,
where Ω is regular and bounded. The unbounded variational operator AH associated to a is a positive symmetric operator with compact resolvent. It admits a sequence (λk)k of positive eigenvalues with limk→+∞λk=+∞ and a Hilbert basis (ek)k of H consisting of eigenvectors of AH. If (G(t))t>0 is the semigroup generated by -AH, then for all u0∈H,(3.32)G(t)u0=∑k=0+∞e-tλk(u0,ek)ek,which proves that the operator is compact for allt>0 because (3.33)lim⁡k→+∞⁡e-tλk=0.
We have the same formula for Gi(t), and it suffices to replace λk by Diλk.If we set(3.34)GN(t)u=∑k=0Ne-tλk(u,ek)ek,
then GN(t) is an operator with finite rank which converges to G(t).Theorem 3.2.
Lett→G(t) be an application from [0,∞[ into ℒ(H). One assumes that there exists a sequence of operators (GN(t))N≥0 of H with the following properties:  (1)
for allN and all t>0, GN(t) is finite rank independent of t,(2)
t→GN(t) is continuous from [0,∞[ into ℒ(H) for all N,(3)
forN→∞, GN(t) converges to G(t) in L1(]0,T[,ℒ(H)) for all T>0,
then the operator𝒢 is compact from 𝒞0([0,T],H) into 𝒞0([0,T],H) for all T>0.Proof.
Regarding property (3) of Theorem3.2, 𝒢 is well defined on 𝒞0([0,T],H), and we have
(3.35)‖∫0tG(s)f(t-s)ds-∫0tGN(s)f(t-s)ds‖≤(∫0T‖G(s)-GN(s)‖ds)‖f‖∞.
This proves that 𝒢N converges to 𝒢 in ℒ([0,T],H) using property  (3) of Theorem 3.2.
To show that𝒢 is compact, it suffices to show that for all N, 𝒢N is compact.
Letℬ be a bounded set of 𝒞0([0,T],H), 𝒢N(ℬ) is bounded in 𝒞0([0,T],H), using Ascoli result, it will be relatively compact if 𝒢N(ℬ) is equicontinuous and if for all t0 in [0,T], 𝒢N(ℬ)(t0) is relatively compact in H.
But𝒢N(ℬ)(t0) being bounded and embedded in a subspace of finite dimension of H is relatively compact in H. Then, let us show the equicontinuity on t0.
LetM and CN such that
(3.36)‖f‖∞≤M,∀f∈B,‖GN(t)‖≤CN∀t∈[0,T].
For 0≤t0≤t≤T, one has
(3.37)GN(f)(t)-GN(f)(t0)=∫0t0(GN(t-s)-GN(t0-s))f(s)ds+∫t0tGN(t-s)f(s)ds.
We have
(3.38)‖∫t0tGN(t-s)f(s)ds‖≤MCN|t-t0|,‖∫0t0(GN(t-s)-GN(t0-s))f(s)ds‖≤M∫0t0‖GN(t-s)-GN(t0-s)‖ds,
which tend to 0 when t→t0 using property (2) of Theorem 3.2 and the continuity under the integral.
We apply Theorem3.2 to the semigroup generated by the Laplacien, and we obtain(1)
GN(t) is of rank N+1 for all t≥0,(2)
∥GN(t)-GN(t0)∥2≤∑k=0N|e-tλk-e-t0λk|2 which tends to 0 if t→t0,(3)
∥G(t)u-GN(t)u∥2≤∑k=N+1∞e-2tλk|uk|2≤e-2tλN+1∥u∥2 if we take an increasing sequence (λk), then one gets
(3.39)‖G(t)-GN(t)‖≤e-tλN+1.
Thus, we have
(3.40)∫0T‖G(t)-GN(t)‖dt≤1λN+1,
which tends to 0 if N→∞.
Thus,𝒢i is compact for all i. We have
(3.41)uin(t)=Gi(t)ui0+Gi(gin)(t),
where (gin)n≥0 is bounded in C0([0,T],H), then (uin)n≥0 belongs to a relatively compact set of C0([0,T],H).
Therefore, from the sequence(uin)n≥0, we can extract a subsequence (uim)m≥0 which converges uniformly to ui∈𝒞0([0,T],H) for each i.
Moreover, we can assume thatuim converges weakly to ui in L2(]0,T[,V).
ButAi∈ℒ(V,V′) and (uin) bounded in L2(]0,T[,V) imply
(3.42)∫0T‖Aiuin‖V′2dt≤‖Ai‖2∫0T‖uin‖V2dt.
Thus, the sequence(Aiuin)n≥0 is bounded in L2(]0,T[,V′). We can assume that Aiuim converges weakly to Aiui in L2(]0,T[,V′).
Remark that for allT<∞, we have
(3.43)C0([0,T],H)↪L2(]0,T[,V′).Conclusion 2.
One has(3.44)uim⟶uiinC0([0,T],H).
Thus, ui≥0 and ui(0)=u0i.
We check that(3.45)qi(um-1)⟶qi(u),qi(um-1)uim⟶qi(u)uiinC0([0,T],H)∀i.
We have also(3.46)fi(um-1)⟶fi(u)inC0([0,T],H)∩L2(]0,T[,V′)∀i.
For alli, uim is solution of
(3.47)〈∂uim∂t,v〉+〈Aiuim,v〉+(qi(um-1)uim,v)=〈fi(um-1),v〉∀v∈V.
We takeϕ∈𝒟(]0,T[), and therefore ϕv∈L2(]0,T[,V),
(3.48)∫0T〈∂uim∂t,ϕv〉dt+∫0T〈Aiuim,ϕv〉dt+∫0T(qi(um-1)uim,ϕv)dt=∫0T〈fi(um-1),ϕv〉dt.
The second term in the left side and the right side of the egality converges due to the weak convergence inL2(]0,T[,V′). The third term in the left-hand side also passes to the limit, due to the convergence in 𝒞0([0,T],H).
We deduce that∂uim/∂t converges weakly in L2(]0,T[,V′).
But we have(3.49)uim⟶uiinC0([0,T],H),
then
(3.50)∂uim∂t⟶∂ui∂tinD′(]0,T[,H).
Therefore, we obtain
(3.51)∂uim∂t⟶∂ui∂tweaklyinL2(]0,T[,V′),∫0T〈∂ui∂t,ϕv〉dt+∫0T〈Aiui,ϕv〉dt+∫0T(qi(u)ui,ϕv)Hdt=∫0T〈fi(u),ϕv〉dt.
This formulation being true for all ϕ, we have
(3.52)〈∂ui∂t,v〉+〈Aiui,v〉+(qi(u)ui,v)H=〈fi(u),v〉∀v∈V,
that is to say,
(3.53)ddt(ui,v)H+a(ui,v)+(qi(u)ui,v)H=〈fi(u),v〉∀v∈V,i,∂ui∂t=fi(u)-Aiui-qi(u)ui∈L2(]0,T[,V′)∀i.
For alli, one has
(3.54)uim(t)=Gi(t)u0i+∫0tGi(t-s)(-qi(um-1)uim+fi(uim-1))(s)ds,
and as qi(um-1)uim and fi(um-1) converge in C0([0,T],H), we have
(R)ui(t)=Gi(t)u0i+∫0tGi(t-s)(-qi(u)ui+fi(u))(s)ds.


## 3.4. Main Result

Theorem 3.3.
If the initial condition satisfies (3.1), then system (𝒮) admits a unique nonnegative solution ui∈W(0,T,V,V′) for all i.
Moreover, for alli, ui satisfies relation (3.62).Proof.
We have already shown existence of solution; thus, it remains to show uniqueness.
Letv be another solution of system (𝒮)(3.55)vi∈W(0,T,V,V′)⟹vi∈C0([0,T],H),vi≥0.
Consequently, we obtain
(3.56)qi(v)vi+fi(v)∈L2(]0,T[,V′).
Thus, by Proposition 2.11, we have
(3.57)vi(t)=Gi(t)u0i+∫0tGi(t-s)(-qi(v)vi+fi(v))(s)ds.
By subtraction, we have(3.58)ui(t)-vi(t)=∫0tGi(t-s)(-qi(u)ui+qi(v)vi+fi(u)-fi(v))(s)ds,
with
(3.59)qi(u)ui-qi(v)vi=qi(u)(ui-vi)+(qi(u)-qi(v))vi.
ui being positive, we have
(3.60)‖ujK+ui‖≤1K‖uj‖∞,
where we have defined
(3.61)‖uj‖∞=‖uj‖L∞(]0,T[,H).
If we define(3.62)‖u‖∞=∑j=16‖uj‖∞,
there is M1>0 such that
(3.63)‖qi(u)‖∞≤M1‖u‖∞∀u.
So the numerator ofqi(u)-qi(v) is the sum of terms of form (uk-vk)vj or (uj-vj)uk, and we can find M2>0 such that
(3.64)|qi(u)-qi(v)|H(s)≤M2(∑j=16|uj(s)-vj(s)|H).
Also we can findM3>0 such that
(3.65)|fi(u)-fi(v)|H(s)≤M3(∑j=16|uj(s)-vj(s)|H).
Summing and noting that∥Gi(t-s)∥≤NieΩiT with Ni, Ωi>0, we can find M such that
(3.66)∑i=16|ui(t)-vi(t)|H≤M‖u-v‖∞.
Replacing in (3.58), we obtain
(3.67)∑i=16|ui(t)-vi(t)|H≤M2‖u-v‖∞∫0tsds=M2t22‖u-v‖∞.
By induction, we have(3.68)∑j=16|uj(t)-vj(t)|H≤Mnn!Tn‖u-v‖∞,lim⁡n→+∞⁡Mnn!Tn‖u-v‖∞=0.



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*Source: 831436-2011-08-16.xml*