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

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.

1. Introduction

Modelling biological dynamic in the soil is of great interest during these last years. Several attempts are made in 1𝐷, 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].

Let 𝑇>0 be a fixed time, Ω⊂ℝ³ an open smooth bounded domain, 𝑄_𝑇=]0,𝑇[×Ω, and Γ_𝑇=]0,𝑇[×𝜕Ω.

The set of equations describing the organic matter cycle of decomposition in the soil is given by the following system:(𝒮)𝜕𝑢_𝑖𝜕𝑡−𝐷_𝑖Δ𝑢_𝑖+𝑞_𝑖(𝑢)𝑢_𝑖=𝑓_𝑖(𝑢)in𝑄_𝑇,𝜕𝑢_i𝜕𝑛=0overΓ_𝑇,𝑢_𝑖(0,𝑥)=𝑢_0𝑖(𝑥)inΩ, (1.1) for 𝑖=1,…,6.

We have noticed 𝑢=(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,𝑞₁(𝑢)=−𝑘𝑢₂𝐾_𝑠+𝑢₂+𝜇+𝑟+𝜈,𝑞₂(𝑢)=𝑘𝑢₁𝐾_𝑠+𝑢₂,𝑞₃(𝑢)=𝑐₁𝑢₅𝐾_𝑚+𝑢₅,𝑞₄(𝑢)=𝑐₂𝑢₅𝐾_𝑚+𝑢₅,𝑞₅(𝑢)=𝜁,𝑞₆(𝑢)𝑓=0,₁(𝑢)=0,𝑓₂(𝑢)=𝑢₅𝐾_𝑚+𝑢₅𝑐₁𝑢₃+𝑐₂𝑢₄+𝜁𝑢₅+𝜇𝑢₁2,𝑓₃(𝑢)=𝜁𝑢₅+𝜇𝑢₁2,𝑓₄(𝑢)=0,𝑓₅(𝑢)=𝜈𝑢₁,𝑓₆(𝑢)=𝑟𝑢₁, (1.2) 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, 𝐾_𝑚 and 𝐾_𝑠 represent half-saturation constants, and 𝐷_𝑖, 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, 𝑢_𝑖 denoting the densities of species present and 𝑞_𝑖 growth rate. This system is also involved in biochemical reactions. In this case, the 𝑢_𝑖 are the concentrations of various molecules, 𝑞_𝑖 is the rate of loss, and 𝑓_𝑖 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, 𝑢_𝑡=𝑢_𝑥𝑥()𝑓(𝑢)(),𝑢−𝑎𝑥,𝑡0<𝑥<1,𝑡∈0,𝑇_𝑥()(),𝑢0,𝑡=0𝑡∈0,𝑇_𝑥(1,𝑡)=𝑏(𝑡)𝑔(𝑢(1,𝑡))𝑡∈(0,𝑇). (1.3)

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]:𝜕_𝑡+ℒ𝑢+𝐹(𝑡,𝑥,𝑢,∇𝑢)=0∀(𝑡,𝑥)∈[0,𝑇[×ℝ^𝑑,𝑢(𝑥,𝑇)=𝑔(𝑥)∀𝑥∈ℝ^𝑑, (1.4) with𝐹∶[0,𝑇]×ℝ^𝑑×ℝ^𝑚×ℝ^𝑚×𝑑⟶ℝ^𝑚,𝑏∈𝐶²_𝑏ℝ^𝑑;ℝ^𝑑,𝜎∈𝐶³_𝑏ℝ^𝑑;ℝ^𝑑×𝑑,𝑎=𝜎𝜎^𝑡,ℒ=^𝑑_𝑖=1𝑏_𝑖𝜕𝜕𝑥_𝑖+12^𝑑_𝑖,𝑗=1𝑎_𝑖,𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗, (1.5) 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𝜕𝑢()𝑢()𝜕𝑡=𝒟Δ𝑢+𝑢𝑓𝑥,𝑢𝑥∈Ω,𝑡>0,𝑥,0=𝑢₀(𝑥)≥0𝑥∈Ω. (1.6)

A Cauchy problem for parabolic semilinear equations with initial data in 𝐻^𝑠_𝑝(ℝ^𝑛) 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:𝜕∀𝑖=1,…,𝑚,_𝑡𝑢_𝑖−𝑑_𝑖Δ𝑢_𝑖=𝑓_𝑖𝑢₁,…,𝑢_𝑚()𝛼in0,𝑇×Ω,_𝑖𝜕𝑢_𝑖+𝜕𝑛1−𝛼_𝑖𝑢_𝑖=𝛽_𝑖()𝑢on0,𝑇×𝜕Ω,_𝑖(0)=𝑢_𝑖0inΩ, (1.7) where the 𝑓_𝑖∶ℝ^𝑚→ℝ are 𝐶¹ functions of 𝑢=(𝑢₁,…,𝑢_𝑚), and 𝑑_𝑖∈(0,∞), 𝛼_𝑖∈[0,1], 𝛽_𝑖∈𝐶²([0,𝑇]×Ω), 𝛽_𝑖≥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”∀𝑟=𝑟₁,…,𝑟_𝑛,^𝑚_𝑖=1𝑓(𝑟)≤𝐶1+^𝑚_𝑖=1𝑟_𝑖,𝐶>0. (1.8) 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 in 𝐿¹((0,𝑇)×Ω) are also considered, for instance, for 𝑓_𝑖 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:𝜕_𝑡𝑢−𝑑₁(),𝜕Δ𝑢=𝑓𝑢,𝑣_𝑡𝑣−𝑑₂(),𝑢()Δ𝑣=𝑔𝑢,𝑣0,⋅=𝑢₀(⋅)≥0,𝑣(0,⋅)=𝑣₀(⋅)≥0witheither:𝜕𝑢𝜕𝑛=𝛽₁,𝜕𝑣𝜕𝑛=𝛽₂on(0,+∞)×𝜕Ω,or:𝑢=𝛽₁,𝑣=𝛽₂on(0,+∞)×𝜕Ω, (1.9) where 𝑑₁,𝑑₂∈(0,+∞),𝛽₁,𝛽₂∈[0,+∞), and 𝑓,𝑔∶[0,+∞)²→ℝ are 𝐶¹, holds for the additional following hypothesis:()[]𝑔(∀𝑢≥𝑈,∀𝑣≥0,𝑓𝑢,𝑣≤𝐶1+𝑢+𝑣,𝑈,𝐶≥0,∃𝑟≥1,∀𝑢,𝑣≥0,𝑢,𝑣)≤𝐶1+𝑢^𝑟+𝑣^𝑟. (1.10)

This approach has been extended to 𝑚×𝑚 systems for which 𝑓₁,𝑓₁+𝑓₂,𝑓₁+𝑓₂+𝑓₃,… are all bounded by a linear of the 𝑢_𝑖 (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 in 𝐿¹ may be found in [18]. It is assumed that, for all 𝑖=1,…,𝑚,𝑓_𝑖∶]0,𝑇[×Ω×[0,+∞)^𝑚⟶ℝismeasurable,𝑓_𝑖(⋅,0)∈𝐿¹[)][[)⟶[)()(]0,𝑇×Ω∃𝐾∶0,𝑇×Ω×0,+∞0,+∞with∀𝑀>0,𝐾⋅,𝑀∈𝐿¹[)()∈][̃[)(]0,𝑇×Ωanda.e.𝑡,𝑥0,𝑇×Ω,∀𝑟,𝑟∈0,+∞^𝑚|𝑟|,|𝑟|̃𝑓with≤𝑀,_𝑖(𝑡,𝑥,𝑟)−𝑓_𝑖(𝑡,𝑥,𝑟)̃≤𝐾(𝑡,𝑥,𝑀)|𝑟−𝑟|̃,∀𝑟∈[0,+∞)^𝑚,𝑓_𝑖𝑡,𝑥,𝑟₁,…,𝑟_𝑖−1,0,𝑟_𝑖+1,…,𝑟_𝑚≥0, (1.11) 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 with 𝑢₀∈𝐿²(Ω)^𝑚 can be obtained.

Results are also obtained in the weak sense for systems satisfying 𝑊_𝑡−Δ𝑍≤𝐻, where 𝑊=∑_𝑖𝑢_𝑖, 𝑍=∑_𝑖𝑑_𝑖𝑢_𝑖, 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 that 𝑢 being in the denominator of some 𝑞_𝑖(𝑢) and 𝑓_𝑖(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 of 𝑞_𝑖(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 spaces 𝐻 and 𝑉 such that 𝑉 is embedded continuously and densely in 𝐻.

Then, we have duality 𝐻^′↪𝑉^′. Using Riesz theorem, we identify 𝐻 and 𝐻^′. So we get 𝑉↪𝐻↪𝑉^′.

Definition 2.1. We define the Hilbert space 𝑊0,𝑇,𝑉,𝑉^′=𝑢∈𝐿²(]0,𝑇[,𝑉)suchthat𝜕𝑢𝜕𝑡∈𝐿²]0,𝑇[,𝑉^′, (2.1) equipped with the norm ‖𝑢‖²_𝑊=‖𝑢‖²_{𝐿2(]0,𝑇[,𝑉)}+𝜕𝑢𝜕𝑡²_𝐿²_{(]0,𝑇[,𝑉}^′₎. (2.2)

We assume the two following lemmas, see [19].

Lemma 2.2. There exists a continuous prolongation operator 𝑃 from 𝑊(0,𝑇,𝑉,𝑉^′) to 𝑊(−∞,+∞,𝑉,𝑉^′) such that 𝑃𝑢_∣]0,𝑇[=𝑢∀𝑢∈𝑊0,𝑇,𝑉,𝑉^′. (2.3)

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. If 𝑢∈𝑊(0,𝑇,𝑉,𝑉^′), one takes a sequence 𝑢_𝑛 of 𝒟(ℝ,𝑉) which converges in 𝑊(−∞,+∞,𝑉,𝑉^′) toward 𝑃𝑢, and then (𝑢_𝑛)_∣]0,𝑇[ converges toward 𝑢 and (𝑢_𝑛)_∣]0,𝑇[∈𝐶^∞([0,𝑇],𝑉) for all 𝑛∈ℕ.

Proposition 2.5. Every element 𝑢∈𝑊(0,𝑇,𝑉,𝑉^′) is almost everywhere equal to a function in 𝐶⁰([0,𝑇],𝐻).
Furthermore, the injection of 𝑊(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 all 𝑡∈[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 ∃𝑀>0,suchthat|𝑎(𝑡;𝑢,𝑣)|≤𝑀‖𝑢‖_𝑉‖𝑣‖_𝑉∀𝑢,𝑣∈𝑉. (2.4)
For each fixed 𝑡, one defines a continuous linear application 𝐴(t)∈ℒ(𝑉,𝑉^′) by ⟨𝐴(𝑡)𝑢,𝑣⟩=𝑎(𝑡;𝑢,𝑣). (2.5)
Then, we have ‖𝐴(𝑡)‖_{ℒ(𝑉,𝑉′)}≤𝑀. (2.6)
Also we associate, for all fixed 𝑡, 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 𝑎(𝑡;𝑢,𝑣)=(𝐴(𝑡)𝑢,𝑣)_𝐻. (2.7)
To simplify the writting the unbounded operator is noted 𝐴(𝑡).
Let 𝜓∈𝐶^∞([0,𝑇],𝑉), we have, for 𝑣∈𝑉, 𝑑𝜓(𝑡)𝑑𝑡,𝑣_𝐻=𝜕𝜓𝜕𝑡,𝑣_𝐻=𝜓^′(𝑡),𝑣, (2.8) 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, 𝑑(𝑢(𝑡))𝑑𝑡,𝑣_𝐻=𝑢^′(𝑡),𝑣fora.e.𝑡. (2.9)
The variational parabolic problem associated to the triple (𝐻,𝑉,𝑎(𝑡;⋅,⋅)) is the following.
Given 𝑓(𝑡)∈𝐿²(]0,𝑇[;𝑉^′) and 𝑢₀∈𝐻, find 𝑢∈𝑊(0,𝑇,𝑉,𝑉^′) such that (𝒫)𝑑(𝑢(𝑡))𝑑𝑡,𝑣_𝐻((𝑡))=𝑓(𝑡)𝑢(0)+𝑎𝑡;𝑢,𝑣,𝑣∀𝑣∈𝑉,=𝑢₀. (2.10)
This problem is equivalent to 𝑑𝑢(𝑡)(𝑡)𝑢(𝑡)(𝑡),𝑢(0)𝑑𝑡+𝐴=𝑓=𝑢₀. (2.11)

Definition 2.6. The form 𝑎 is coercive or 𝑉 coercive if 𝛼>0 exists such that 𝑎(𝑡;𝑢,𝑢)≥𝛼‖𝑢‖²_𝑉∀(𝑡,𝑢)∈[0,𝑇]×𝑉. (2.12)

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 is 𝐻 coercive if there exist two constants 𝜆 and 𝛼>0 such that 𝑎(𝑡;𝑢,𝑢)+𝜆‖𝑢‖²_𝐻≥𝛼‖𝑢‖²_𝑉,∀𝑡∈[0,𝑇],∀𝑢∈𝑉. (2.13)

If we set 𝑢(𝑡)=𝑒^𝜆𝑡𝑤(𝑡), then 𝑢 is solution of (𝒫) if and only if 𝑤 is solution of𝒫^′𝑑(𝑤(𝑡))𝑑𝑡,𝑣_𝐻+𝑎(𝑡;𝑤,𝑣)+𝜆(𝑤,𝑣)_𝐻=𝑒^−𝜆𝑡𝑓(𝑡)𝑤(0),𝑣∀𝑣∈𝑉,=𝑢₀. (2.14)

Writing 𝑏(𝑡;𝑢,𝑣)=𝑎(𝑡;𝑢,𝑣)+𝜆(𝑢,𝑣)_𝐻, (2.15)

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: 𝐻=𝐿²(Ω),𝑉=𝐻¹(Ω), (2.16) and defining𝑎(𝑡;𝑢,𝑣)=³_{𝑖,𝑗=1Ω}𝑎_𝑖𝑗(𝑡,𝑥)𝜕𝑢𝜕𝑥_𝑖𝜕𝑣𝜕𝑥_𝑗𝑑𝑥, (2.17) we assume that𝑎_𝑖𝑗(𝑡,𝑥)∈𝐿^∞(]0,𝑇[×Ω)∀𝑖,𝑗, (2.18) and there exists 𝛼>0 such that, for all 𝜁=(𝜁_𝑖)_𝑖=1,2,3, we have³_𝑖,𝑗=1𝑎_𝑖𝑗(𝑡,𝑥)𝜁_𝑖𝜁_𝑗≥𝛼𝜁²_ℝ3a.e.in]0,𝑇[×Ω. (2.19)

Then, we deduce that𝑎(𝑡;𝑢,𝑢)≥𝛼³_𝑖=1𝜕𝑢𝜕𝑥_𝑖²_𝐿2(Ω). (2.20)

The form is then 𝐻 coercive, and it suffices to take 𝜆=𝛼.

In addition, let us take 𝑎₀∈𝐿^∞(]0,𝑇[×Ω) with 𝑎₀(𝑡,𝑥)≥0 for all 𝑡, 𝑥.

The form𝑏(𝑡;𝑢,𝑣)=𝑎(𝑡;𝑢,𝑣)+𝑎₀(𝑡)𝑢,𝑣_𝐻 (2.21) 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, if 𝑢₀≥0 and 𝑓(𝑡),𝑣≥0∀𝑣∈𝑉𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑣≥0,𝑡∈]0,𝑇[a.e., (2.22) one has 𝑢(𝑡)≥0 for all 𝑡∈[0,𝑇].

Proof It remains to show that the solution is nonnegative.
Given 𝑢∈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 replacing 𝑣 by 𝑢⁻(𝑡) in (𝒫), we obtain 𝑑𝑢𝑑𝑡,𝑢⁻(𝑡)+𝑎(𝑡;𝑢(𝑡),𝑢⁻(𝑡))+𝑎₀(𝑡)𝑢,𝑢⁻_𝐻=𝑓(𝑡),𝑢⁻. (2.22)
One gets 𝑢=𝑢⁺−𝑢⁻,𝑢⁺𝑢⁻=0,𝑑𝑢⁺𝑢𝑑𝑡⁻=0, (2.24) and by linearity, we obtain −𝑑𝑢⁻𝑑𝑡,𝑢⁻−𝑎(𝑡;𝑢⁻(𝑡),𝑢⁻(𝑡))−𝑎₀(𝑡)𝑢⁻,𝑢⁻_𝐻=𝑓(𝑡),𝑢⁻, (2.25)
but 𝑑𝑢⁻𝑑𝑡,𝑢⁻=12𝑑(𝑢𝑑𝑡⁻(𝑡),𝑢⁻(𝑡))_𝐻. (2.26)
Since (𝑎(𝑡)𝑢⁻,𝑢⁻)≥0,𝑎₀(𝑡)𝑢⁻,𝑢⁻_𝐻≥0,𝑓(𝑡),𝑢⁻(𝑡)≥0, (2.27) it comes that 𝑑‖𝑢𝑑𝑡⁻(𝑡)‖²_𝐻≤0. (2.28)
By integration over ]0,𝑡[, we deduce ‖𝑢⁻(𝑡)‖²_𝐻≤‖𝑢⁻(0)‖²_𝐻,∀𝑡. (2.29)
But 𝑢₀≥0, then 𝑢⁻₀=0, so 𝑢⁻(0)=0.
Hence, we conclude that 𝑢⁻(𝑡)=0 for all 𝑡.
Instead of 𝑎₀(𝑡,𝑥)≥0, assume that ∃𝐶>0suchthat𝑎₀(𝑡,𝑥)≥−𝐶∀𝑡,𝑥. (2.30)
As previously mentioned, if we set 𝑢(𝑡)=𝑒^𝜆𝑡𝑤(𝑡), 𝑤(𝑡) is solution of 𝑑(𝑤(𝑡))𝑑𝑡,𝑣_𝐻+𝑎(𝑡;𝑤(𝑡),𝑣)+𝑎₀(𝑡)+𝜆𝑤,𝑣_𝐻=𝑒^−𝜆𝑡𝑓(𝑡),𝑣, (2.31) with 𝑎₀(𝑡,𝑥)+𝜆≥𝜆−𝐶. (2.32)
It suffices to take 𝜆≥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.
If 𝑎₀∈𝐿^∞(]0,𝑇[×Ω) and ∃𝐶>0suchthat𝑎₀(𝑡,𝑥)≥−𝐶for𝑎.𝑒.(𝑡,𝑥)∈]0,𝑇[×Ω, (2.33) then the variational problem 𝑑(𝑢(𝑡))()+𝑎𝑑𝑡,𝑣+𝑎𝑡;𝑢,𝑣₀(𝑡)𝑢,𝑣_𝐻=𝑓(𝑡)𝑢(0),𝑣∀𝑣∈𝑉,=𝑢₀ (2.34) admits a unique solution in 𝑊(0,𝑇,𝑉,𝑉^′) for all 𝑢₀∈𝐻 and 𝑓∈𝐿²(]0,𝑇[,𝑉^′).
Moreover, if 𝑢₀≥0 and 𝑓(𝑡),𝑣≥0∀𝑣∈𝑉,𝑡∈]0,𝑇[a.e., (2.35) then one has 𝑢(𝑡)≥0 for all 𝑡∈[0,𝑇].

Equivalence of the Variational Solution with the Initial Problem
We have 𝜕𝑄_𝑇=({0}×Ω)∪({𝑇}×Ω)∪(]0,𝑇[×𝜕Ω). (2.35)
For the sake of simplicity, we set 𝑎_𝑖𝑗=𝛿_𝑖𝑗 which is the Kronecker symbol.
Then, 𝐴=−Δ over 𝒟(Ω), and we have 𝜕𝑢𝜕𝑡−Δ𝑢+𝑎₀𝑢=𝑓in]0,𝑇[×Ω. (2.37)
We assume that 𝑓∈𝐿²(]0,𝑇[,𝐻), then 𝜕𝑢𝜕𝑡−Δ𝑢∈𝐿²(]0,𝑇[,𝐻) (2.38) if 𝑢∈𝐿²(]0,𝑇[,𝑉). We set 𝑣=𝑢,−grad(𝑢)^𝑇=𝑢,−𝜕𝑢𝜕𝑥₁,−𝜕𝑢𝜕𝑥₂,−𝜕𝑢𝜕𝑥₃^𝑇. (2.39)
Consequently, we have 𝑣∈(𝐿²(]0,𝑇[×Ω))⁴ and div(𝑣)=𝜕𝑢𝜕𝑡−Δ𝑢∈𝐿²(]0,𝑇[×Ω). (2.40)
Then 𝑢(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.
Let 𝜙∈𝒟(]0,𝑇[), and we multiply (2.34) ₁ by 𝜙, and by integration over ]0,𝑇[, one gets ^𝑇_0Ω𝜙𝑣𝜕𝑢𝜕𝑡𝑑𝑡𝑑𝑥+³_𝑖=1^𝑇_0Ω𝜙𝜕𝑢𝜕𝑥_𝑖𝜕𝑣𝜕𝑥_𝑖𝑑𝑡𝑑𝑥+^𝑇_0Ω𝜙𝑎₀=𝑢𝑣𝑑𝑡𝑑𝑥_𝑇0Ω𝑓𝑣𝜙𝑑𝑡𝑑𝑥∀𝑣∈𝐻¹(Ω). (2.41)
Using Green formula with (𝑢,−grad(𝑢))^𝑇, we have ^𝑇_0Ω𝜙𝑣𝜕𝑢𝜕𝑡−Δ𝑢+𝑎₀𝑢𝑑𝑥𝑑𝑡=^𝑇_0Ω𝑓𝜙𝑣𝑑𝑥𝑑𝑡−^𝑇_0𝜕Ω𝜙𝑣𝜕𝑢𝜕𝑛𝑑𝑠𝑑𝑡. (2.42)
As 𝜕𝑢𝜕𝑡−Δ𝑢+𝑎₀𝑢=𝑓, (2.43) we can conclude the following statement: ^𝑇_0𝜕Ω𝜙𝑣𝜕𝑢𝜕𝑛𝑑𝑡𝑑𝑠=0∀𝜙,𝑣∈𝒟(]0,𝑇[)×𝐻¹(Ω). (2.44)
We deduce that 𝑣,𝜕𝑢𝜕𝑛_𝐻^1/2_𝐻^−1/2=0∀𝑣∈𝐻¹(Ω)and𝑡a.e. (2.45)
Function 𝑣→𝑣_|𝜕Ω from 𝐻¹(Ω) into 𝐻^1/2(𝜕Ω) being surjective, we deduce that 𝜕𝑢𝜕𝑛=0in𝐻^−1/2(Γ)for𝑡a.e. (2.46)

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 𝐴_𝐻 in 𝐻 with𝐷𝐴_𝐻={𝑢∈𝑉suchthat𝐴𝑢∈𝐻}. (2.47)

Assume that 𝑎 is 𝐻 coercive, then 𝐴_𝐻 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 𝐴_𝐻.

Proposition 2.11. For 𝑢₀∈𝐻 and 𝑓∈𝐿²(]0,𝑇[,𝑉^′), problem (𝒫) which consists in finding 𝑢∈𝑊(0,𝑇,𝑉,𝑉^′) such that 𝑑𝑢(0)𝑑𝑡+𝐴𝑢=𝑓𝑤𝑖𝑡ℎ𝑢=𝑢₀ (2.48) admits a unique solution given by 𝑢(𝑡)=𝐺(𝑡)𝑢₀+^t₀𝐺(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (2.49)

Proof. Note 𝑢 and 𝑓 the extensions by 0 of 𝑢 and 𝑓 outside ]0,𝑇[, then we have 𝑑𝑢=𝑑𝑡𝑑𝑢(0)𝛿(𝑡)(𝑇)𝛿()𝑑𝑡+𝑢−𝑢𝑇−𝑡, (2.50) with 𝛿(𝑡) the Dirac measure on ℝ.
Thus, 𝑑𝑢𝑑𝑡+𝐴𝑢=𝑑𝑢+(0)𝛿(𝑡)(𝑇)𝛿()=(0)𝛿(𝑡)(𝑇)𝛿().𝑑𝑡𝐴𝑢+𝑢−𝑢𝑇−𝑡𝑓+𝑢−𝑢𝑇−𝑡 (2.51)
Hence, an equation of the form 𝑑𝑈𝑑𝑡+𝐴𝑈=𝐹in𝒟^′₊𝑉^′, (2.52) where 𝒟^′₊(𝑉^′) is the space of distributions over ℝ into 𝑉^′whose support is in [0,+∞[. By Laplace transform, one is reduced to (𝐴+𝑃𝐼)ℒ(𝑈)=ℒ(𝐹), (2.53) where ℒ(𝑈)=(𝐴+𝑃𝐼)⁻¹ℒ(𝐹), (2.54) and therefore, we have 𝑈=𝐺∗𝐹.
But since supp(𝛿(𝑇−𝑡))=𝑇,supp𝐺∗(𝑈(𝑇)𝛿(𝑇−𝑡))⊂[𝑇,+∞[, (2.55) we have 𝑈_∣]0,𝑇[=𝑢=𝐺∗𝐹_∣]0,𝑇[. (2.56)
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 define 𝐻=𝐿²(Ω) and 𝑉=𝐻¹(Ω) and the following hypothesis for initial conditions:𝑢₀₁∈𝐿^∞(Ω),𝑢_0𝑖∈𝐻for𝑖∈{2,…,6},𝑢_0𝑖≥0for𝑖∈{1,…,6}. (3.1)

We will make a resolution component by applying Theorem 2.7 with, for each 𝑖, the form a(𝑡;𝑢,𝑣)=³_𝑗=1𝐷_𝑖Ω𝜕𝑢𝜕𝑥_𝑗𝜕𝑣𝜕𝑥_𝑗𝑑𝑥. (3.2)

One approaches the solution by a sequence of solutions of linear equations.

3.1. Recursive Sequence of Solutions

For 𝑛=0, we note that 𝑢⁰_𝑖 is the solution of𝜕𝑢⁰_𝑖𝜕𝑡−𝐷_𝑖Δ𝑢⁰_𝑖=0in𝑄_𝑇,𝑢⁰_𝑖(0)=𝑢_0𝑖inΩ,𝜕𝑢⁰_𝑖𝜕𝑛_∣𝜕Ω=0. (3.3)

This equation admits strong solution and 𝑢⁰_𝑖≥0.

By induction, we note that 𝑢^𝑛_𝑖 is solution of equation𝜕𝑢^𝑛_𝑖𝜕𝑡−𝐷_𝑖Δ𝑢^𝑛_𝑖+𝑞_𝑖𝑢^𝑛−1𝑢^𝑛_𝑖=𝑓_𝑖𝑢^𝑛−1in𝑄_𝑇,𝑢^𝑛_𝑖(0)=𝑢_0𝑖inΩ,𝜕𝑢^𝑛_𝑖𝜕𝑛_∣𝜕Ω=0. (3.4)

It is a linear equation within the framework of Corollary 2.10 with 𝑎₀=𝑞_𝑖(𝑢^𝑛−1) and 𝑓(𝑡)=𝑓_𝑖(𝑢^𝑛−1(𝑡)). Let us suppose that there exists a unique nonnegative solution 𝑢^𝑛−1. Assuming by induction that 𝑢^𝑗_𝑖≥0 for 0≤𝑗≤𝑛−1, we have 0≤𝑘𝑢₂^𝑛−1𝐾_𝑠+𝑢₂^𝑛−1≤𝑘, (3.5) which implies that 𝜇+𝑟+𝜈−𝑘≤𝑞₁𝑢^𝑛−1≤𝜇+𝑟+𝜈. (3.6)𝑢^𝑛−1 is nonnegative also that implies that there are two positive constants 𝐶₁, 𝐶₂ such that 0≤𝑞₃𝑢^𝑛−1≤𝐶₁,0≤𝑞₄𝑢^𝑛−1≤𝐶₂. (3.7) For the rest, we notice that 𝑞₅ and 𝑞₆ are constant.

We have shown that 𝑞_𝑖(𝑢^𝑛−1)∈𝐿^∞(]0,𝑇[×Ω) for 𝑖≠2. It remains to prove that the same property is satisfied by 𝑞₂(𝑢^𝑛−1).

To prove that 𝑞₂(𝑢^𝑛−1) is bounded, we need to show that 𝑢^𝑛₁∈𝐿^∞(0,𝑇;𝐿^∞(Ω)).

Case of u⁰₁
Let 𝑘∈ℕ^∗, we multiply (3.3) ₁ by (𝑢⁰₁)^2𝑘−1 and integrate it over Ω, and it comes that 1𝑑2𝑘𝑑𝑡_Ω𝑢⁰₁^2𝑘𝑑𝑥+(2𝑘−1)𝐷_1Ω𝑢⁰₁^2(𝑘−1)∇𝑢⁰₁²𝑑𝑥=0. (3.8) The second term is nonnegative, then we have 1𝑑2𝑘𝑑𝑡_Ω𝑢⁰₁^2𝑘𝑑𝑥≤0. (3.9) By integrating over ]0,𝑡[, we obtain 𝑢⁰₁(𝑡)_{𝐿2𝑘(Ω)}≤𝑢_{01𝐿2𝑘(Ω)}. (3.10) When 𝑘 tends to +∞, it comes that, 𝑢⁰₁(𝑡)_𝐿∞(Ω)≤𝑢_{01𝐿∞(Ω)}, (3.11) which implies that 𝑢⁰₁∈𝐿^∞(0,𝑇;𝐿^∞(Ω)).

Case of 𝑢^𝑛₁ with 𝑛∈ℕ^∗
By induction, we suppose that 𝑢⁰₁,𝑢¹₁,…,𝑢₁^𝑛−1 are in 𝐿^∞(0,𝑇;𝐿^∞(Ω)).

Remark 3.1. We make the following change: 𝑤^𝑛_𝑖=𝑒^−𝜆𝑡𝑢^𝑛_𝑖,𝑖=1,…,6. We obtain 𝜕𝑤^𝑛₁𝜕𝑡−𝐷₁Δ𝑤^𝑛₁+𝜆+𝑞₁𝑒^𝜆𝑡𝑤_𝑖^𝑛−1_𝑖𝑤^𝑛₁=0. (3.12)
The function 𝑞₁ being undervalued, we can choose 𝜆≥0 such that 𝜆+𝑞₁𝑒^𝜆𝑡𝑤_𝑖^𝑛−1_𝑖≥0. (3.13) We multiply (3.12) by (𝑤^𝑛₁)^2𝑘−1 and integrate it over Ω. We obtain 1𝑑2𝑘𝑑𝑡_Ω𝑤^𝑛₁^2𝑘𝑑𝑥+(2𝑘−1)𝐷_1Ω𝑤^𝑛₁^2(𝑘−1)∇𝑤^𝑛₁²+𝑑𝑥_Ω𝜆+𝑞₁𝑒^𝜆𝑡𝑤_𝑖^𝑛−1_𝑖𝑤^𝑛₁^2𝑘𝑑𝑥=0. (3.14) The second and third term being nonnegative, we can conclude as in the previous case that 𝑤^𝑛₁(𝑡)_{𝐿2𝑘(Ω)}≤𝑢_{01𝐿2𝑘(Ω)}. (3.15) Since 𝑘 tends to ∞, it follows that 𝑤^𝑛₁(𝑡)_𝐿∞(Ω)≤𝑢_{01𝐿∞(Ω)}. (3.16) As a result, we have proved that 𝑤^𝑛₁∈𝐿^∞(0,∞;𝐿^∞(Ω)), and since 𝑢^𝑛₁=𝑒^𝜆𝑡𝑤^𝑛₁, we have 𝑢^𝑛₁∈𝐿^∞(0,𝑇;𝐿^∞(Ω)).

Conclusion 1. With the previous demonstration, we obtain by induction that if 𝑢₀₁∈𝐿^∞(Ω) with 𝑢₀₁≥0, then 𝑞_𝑖(𝑢^𝑛)∈𝐿^∞(]0,𝑇[×Ω) for all 𝑛 and 𝑖=1,…,6.
We also have 𝑓_𝑖(𝑢^𝑛−1)≥0 and 𝑓_𝑖(𝑢^𝑛−1)∈𝐿²(]0,𝑇[;𝑉^′). Then by means of Corollary 2.10, there exists a unique solution 𝑢^𝑛_𝑖∈𝑊(0,𝑇,𝑉,𝑉^′) with 𝑢^𝑛_𝑖≥0.

3.2. Boundedness of the Solution

Let us show that the sequence is bounded. 𝑢^𝑛_𝑖 satisfies (2.34), so𝜕𝑢𝜕𝑡^𝑛𝑖,𝑣_𝐻+𝐷_𝑖𝑎𝑢^𝑛_𝑖,𝑣+𝑞_𝑖𝑢^𝑛−1𝑢^𝑛_𝑖,𝑣_𝐻=𝑓_𝑖𝑢^𝑛−1,𝑣∀𝑣∈𝑉. (3.17)

We remark that 𝜕𝑢𝜕𝑡^𝑛𝑖,𝑣_𝐻=𝜕𝑢^𝑛_𝑖𝜕𝑡,𝑣. (3.18)

For 𝜓∈𝐶^∞([0,𝑇],𝑉), 𝜕𝜓(𝑡)(𝑡=1𝜕𝑡,𝜓)2𝑑𝜓(𝑡)(𝑡)𝑑𝑡,𝜓_𝐻. (3.19) By density and choosing 𝑣=𝑢^𝑛_𝑖(𝑡), we have 𝜕𝑢^𝑛_𝑖(𝑡)𝜕𝑡,𝑢^𝑛_𝑖(𝑡=1)2𝑑𝑢𝑑𝑡^𝑛𝑖(𝑡),𝑢^𝑛_𝑖(𝑡)_𝐻. (3.20)

Hence, 12𝑑𝑢𝑑𝑡^{𝑛𝑖2𝐻}+𝐷_𝑖𝑎𝑢^𝑛_𝑖,𝑢^𝑛_𝑖+𝑞_𝑖𝑢^𝑛−1𝑢^𝑛_𝑖,𝑢^𝑛_𝑖𝐻=𝑓_𝑖𝑢^𝑛−1,𝑢^𝑛_𝑖. (3.21)

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 then 𝜆=𝛽+𝛿, and one is reduced to 12𝑑𝑢𝑑𝑡^{𝑛𝑖2𝐻}+𝐷_𝑖𝑎𝑢^𝑛_𝑖,𝑢^𝑛_𝑖+𝛽𝑢^𝑛_𝑖²_𝐻+𝛿+𝑞_𝑖𝑢^𝑛−1𝑢^𝑛_𝑖,𝑢^𝑛_𝑖𝐻=𝑓_𝑖𝑢^𝑛−1,𝑢^𝑛_𝑖. (3.22)

The form 𝐷_𝑖𝑎 is 𝐻 coercive, so we take 𝛽 such that 𝐷_𝑖𝑎(𝑢,𝑢)+𝛽‖𝑢‖²_𝐻≥𝛼‖𝑢‖²_𝑉∀𝑢∈𝑉. (3.23)

The 𝑞_𝑖 are bounded, so∃𝛿>0suchthat𝛿+𝑞_𝑖(𝑢)≥𝑙∀𝑖,𝑢≥0. (3.24) Therefore, 12𝑑𝑢𝑑𝑡^{𝑛𝑖2𝐻}+𝛼𝑢^𝑛_𝑖²_𝑉+𝑙𝑢^𝑛_𝑖²_𝐻≤𝑓_𝑖𝑢^𝑛−1_𝑉′𝑢^𝑛_𝑖𝑉≤𝜖2𝑢^𝑛_𝑖²_𝑉+1𝑓2𝜖^𝑖𝑢^𝑛−12_𝑉′𝜖>0. (3.25)

We take 𝜖 small enough such that 𝛼−𝜖/2=𝛾>0. Hence, 12𝑢^𝑛_𝑖²_𝐻+𝛾𝑢^𝑛_𝑖²_𝑉+𝑙𝑢^𝑛_𝑖²_𝐻≤1𝑓2𝜖^𝑖𝑢^𝑛−12_𝑉′. (3.26)

For 𝑖=1 and 𝑖=4, 𝑓_𝑖=0. Therefore, by integration, 12𝑢^𝑛_𝑖(𝑡)²_𝐻+𝛾^𝑡₀𝑢^𝑛_𝑖(𝑠)²_𝑉𝑑𝑠+𝑙^𝑡₀𝑢^𝑛_𝑖(𝑠)²_𝐻≤12𝑢^𝑛_𝑖(0)²_𝐻. (3.27)

We deduce that (𝑢^𝑛₁) and (𝑢^𝑛₄) remain bounded in 𝐶⁰([0,𝑇],𝐻) and 𝐿²(]0,𝑇[,𝑉).

𝑓₅(𝑢)=𝜈𝑢₁, thus, 𝑓₅(𝑢^𝑛−1)=𝜈𝑢₁^𝑛−1 remains bounded in 𝐿²(]0,𝑇[,𝑉); therefore, 𝑢^𝑛₅ has the same property as 𝑢^𝑛₁, 𝑢^𝑛₄. It is the same for 𝑢^𝑛₆ because 𝑓₆(𝑢)=𝑟𝑢₁.

We have 𝑓₃(𝑢)=(𝜁𝑢₅+𝜇𝑢₁)/2; therefore, we have the same conclusion for 𝑢^𝑛₃ and finally for 𝑢^𝑛₂ because 𝑓₂(𝑢) depends on 𝑢₁, 𝑢₂, 𝑢₄, and 𝑢₅.

3.3. Convergence of the Sequence

We deduce at this stage that the sequence (𝑢^𝑛_𝑖)_𝑛≥0 (one can extract subsequence (𝑢^𝑚_𝑖)_𝑚≥0) converges weakly in 𝐿²(]0,𝑇[,𝑉) to 𝑢_𝑖 and weakly star in 𝐿^∞(]0,𝑇[,𝐻) to 𝑢_𝑖.

But it is not enough to pass to the limit in the equation, we need the pouctual convergence for almost all 𝑡 to deduce that 𝑢_𝑖≥0 for all 𝑖 and to pass to the limit in 𝑞_𝑖(𝑢^𝑛−1) and 𝑓_𝑖(𝑢^𝑛−1). To pass to the limit, we need strong compactness. Using Proposition 2.11, for all 𝑛, we have 𝑢^𝑛_𝑖(𝑡)=^𝑡₀𝐺_𝑖(𝑡−𝑠)𝑔ⁿ_𝑖(𝑠)𝑑𝑠+𝐺_𝑖(𝑡)𝑢_0𝑖, (3.28) where 𝐺_𝑖(𝑡) is the semigroup generated by the unbounded operator −𝐷_𝑖𝐴_𝐻. Let us denote 𝑔^𝑛_𝑖(𝑠)=−𝑞_𝑖𝑢^𝑛−1(𝑠)𝑢^𝑛_𝑖(𝑠)+𝑓_𝑖𝑢^𝑛−1(𝑠), (3.29) and we deduce 𝑔^𝑛_𝑖∈𝐿²(]0,𝑇[,𝑉).

Moreover, the sequence (𝑢^𝑛_𝑖)_𝑛≥0 is bounded in 𝐶⁰([0,𝑇],𝐻) which implies that the sequence (𝑔^𝑛_𝑖)_𝑛≥0 is bounded in 𝐶⁰([0,𝑇],𝐻) for all 𝑖.

Then, we can conclude showing that operator 𝒢_𝑖 from 𝐶⁰([0,𝑇],𝐻) into 𝐶⁰([0,𝑇],𝐻) defined by 𝒢_𝑖𝑓(𝑡)=^𝑡₀𝐺_𝑖(𝑡−𝑠)𝑓(𝑠)𝑑𝑠 (3.30) is compact.

One takes the triple (𝐿²(Ω),𝐻¹(Ω),𝑎) with𝑎(𝑢,𝑣)=^𝑛_𝑗=1Ω𝜕𝑢𝜕𝑥_𝑗𝜕𝑣𝜕𝑥_𝑗𝑑𝑥, (3.31) where Ω is regular and bounded. The unbounded variational operator 𝐴_𝐻 associated to 𝑎 is a positive symmetric operator with compact resolvent. It admits a sequence (𝜆_𝑘)_𝑘 of positive eigenvalues with lim_𝑘→+∞𝜆_𝑘=+∞ and a Hilbert basis (𝑒_𝑘)_𝑘 of 𝐻 consisting of eigenvectors of 𝐴_𝐻. If (𝐺(𝑡))_𝑡>0 is the semigroup generated by −𝐴_𝐻, then for all 𝑢₀∈𝐻,𝐺(𝑡)𝑢₀=^+∞_𝑘=0𝑒^{−𝑡𝜆𝑘}𝑢₀,𝑒_𝑘𝑒_𝑘, (3.32)

which proves that the operator is compact for all 𝑡>0 because lim_𝑘→+∞𝑒^{−𝑡𝜆𝑘}=0. (3.33) We have the same formula for 𝐺_𝑖(𝑡), and it suffices to replace 𝜆_𝑘 by 𝐷_𝑖𝜆_𝑘.

If we set 𝐺_𝑁(𝑡)𝑢=^𝑁_𝑘=0𝑒^{−𝑡𝜆𝑘}𝑢,𝑒_𝑘𝑒_𝑘, (3.34) then 𝐺_𝑁(𝑡) is an operator with finite rank which converges to 𝐺(𝑡).

Theorem 3.2. Let 𝑡→𝐺(𝑡) be an application from [0,∞[ into ℒ(𝐻). One assumes that there exists a sequence of operators (𝐺_𝑁(𝑡))_𝑁≥0 of 𝐻 with the following properties: (1)for all 𝑁 and all 𝑡>0, 𝐺_𝑁(𝑡) is finite rank independent of 𝑡, (2)𝑡→𝐺_𝑁(𝑡) is continuous from [0,∞[ into ℒ(𝐻) for all 𝑁, (3)for 𝑁→∞, 𝐺_𝑁(𝑡) converges to 𝐺(𝑡) in 𝐿¹(]0,𝑇[,ℒ(𝐻)) for all 𝑇>0,
then the operator 𝒢 is compact from 𝒞⁰([0,𝑇],𝐻) into 𝒞⁰([0,𝑇],𝐻) for all 𝑇>0.

Proof. Regarding property (3) of Theorem 3.2, 𝒢 is well defined on 𝒞⁰([0,𝑇],𝐻), and we have ^𝑡₀𝐺(𝑠)𝑓(𝑡−𝑠)𝑑𝑠−^𝑡₀𝐺_𝑁(𝑠)𝑓(𝑡−𝑠)𝑑𝑠≤^𝑇₀𝐺(𝑠)−𝐺_𝑁(𝑠)𝑑𝑠𝑓_∞. (3.35) This proves that 𝒢_𝑁 converges to 𝒢 in ℒ([0,𝑇],𝐻) using property (3) of Theorem 3.2.
To show that 𝒢 is compact, it suffices to show that for all 𝑁, 𝒢_𝑁 is compact.
Let ℬ be a bounded set of 𝒞⁰([0,𝑇],𝐻), 𝒢_𝑁(ℬ) is bounded in 𝒞⁰([0,𝑇],𝐻), using Ascoli result, it will be relatively compact if 𝒢_𝑁(ℬ) is equicontinuous and if for all 𝑡₀ in [0,𝑇], 𝒢_𝑁(ℬ)(𝑡₀) is relatively compact in 𝐻.
But 𝒢_𝑁(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.
Let 𝑀 and 𝐶_𝑁 such that 𝑓_∞≤𝑀,∀𝑓∈ℬ,𝒢_𝑁(𝑡)≤𝐶_𝑁∀𝑡∈[0,𝑇]. (3.36) For 0≤𝑡₀≤𝑡≤𝑇, one has 𝒢_𝑁𝑓(𝑡)−𝒢_𝑁𝑓𝑡₀=^𝑡0₀𝐺_𝑁(𝑡−𝑠)−𝐺_𝑁𝑡₀−𝑠𝑓(𝑠)𝑑𝑠+^𝑡_𝑡0𝐺_𝑁(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (3.37) We have ^𝑡_𝑡0𝐺_𝑁(𝑡−𝑠)𝑓(𝑠)𝑑𝑠≤𝑀𝐶_𝑁𝑡−𝑡₀,_𝑡00𝐺_𝑁(𝑡−𝑠)−𝐺_𝑁𝑡₀−𝑠𝑓(𝑠)𝑑𝑠≤𝑀^𝑡0₀𝐺_𝑁(𝑡−𝑠)−𝐺_𝑁𝑡₀−𝑠𝑑𝑠, (3.38) which tend to 0 when 𝑡→𝑡₀ using property (2) of Theorem 3.2 and the continuity under the integral.
We apply Theorem 3.2 to the semigroup generated by the Laplacien, and we obtain(1)𝐺_𝑁(𝑡) is of rank 𝑁+1 for all 𝑡≥0, (2)‖𝐺_𝑁(𝑡)−𝐺_𝑁(𝑡₀)‖²≤∑^𝑁_𝑘=0|𝑒^{−𝑡𝜆𝑘}−𝑒^{−𝑡0𝜆𝑘}|² which tends to 0 if 𝑡→𝑡₀, (3)‖𝐺(𝑡)𝑢−𝐺_𝑁(𝑡)𝑢‖²≤∑^∞_𝑘=𝑁+1𝑒^{−2𝑡𝜆𝑘}|𝑢_𝑘|²≤𝑒^{−2𝑡𝜆𝑁+1}‖𝑢‖² if we take an increasing sequence (𝜆_𝑘), then one gets 𝐺(𝑡)−𝐺_𝑁(𝑡)≤𝑒^{−𝑡𝜆𝑁+1}. (3.39) Thus, we have ^𝑇₀𝐺(𝑡)−𝐺_𝑁(𝑡1)𝑑𝑡≤𝜆_𝑁+1, (3.40) which tends to 0 if 𝑁→∞.
Thus, 𝒢_𝑖 is compact for all 𝑖. We have 𝑢^𝑛_𝑖(𝑡)=𝐺_𝑖(𝑡)𝑢⁰_𝑖+𝒢_𝑖𝑔^𝑛_𝑖(𝑡), (3.41) where (𝑔^𝑛_𝑖)_𝑛≥0 is bounded in 𝐶⁰([0,𝑇],𝐻), then (𝑢^𝑛_𝑖)_𝑛≥0 belongs to a relatively compact set of 𝐶⁰([0,𝑇],𝐻).
Therefore, from the sequence (𝑢^𝑛_𝑖)_𝑛≥0, we can extract a subsequence (𝑢^𝑚_𝑖)_𝑚≥0 which converges uniformly to 𝑢_𝑖∈𝒞⁰([0,𝑇],𝐻) for each 𝑖.
Moreover, we can assume that 𝑢^𝑚_𝑖 converges weakly to 𝑢_𝑖 in 𝐿²(]0,𝑇[,𝑉).
But 𝐴_𝑖∈ℒ(𝑉,𝑉^′) and (𝑢^𝑛_𝑖) bounded in 𝐿²(]0,𝑇[,𝑉) imply ^𝑇₀𝐴_𝑖𝑢^𝑛_𝑖²_𝑉′𝑑𝑡≤𝐴_𝑖^2𝑇₀𝑢^𝑛_𝑖²_𝑉𝑑𝑡. (3.42)
Thus, the sequence (𝐴_𝑖𝑢^𝑛_𝑖)_𝑛≥0 is bounded in 𝐿²(]0,𝑇[,𝑉^′). We can assume that 𝐴_𝑖𝑢^𝑚_𝑖 converges weakly to 𝐴_𝑖𝑢_𝑖 in 𝐿²(]0,𝑇[,𝑉^′).
Remark that for all 𝑇<∞, we have 𝒞⁰([0,𝑇],𝐻)↪𝐿²]0,𝑇[,𝑉^′. (3.43)

Conclusion 2. One has 𝑢^𝑚_𝑖⟶𝑢_𝑖in𝒞⁰([0,𝑇],𝐻). (3.44) Thus, 𝑢_𝑖≥0 and 𝑢_𝑖(0)=𝑢_0𝑖.
We check that 𝑞_𝑖𝑢^𝑚−1⟶𝑞_𝑖(𝑢),𝑞_𝑖𝑢^𝑚−1𝑢^𝑚_𝑖⟶𝑞_𝑖(𝑢)𝑢_𝑖in𝒞⁰([0,𝑇],𝐻)∀𝑖. (3.45)
We have also 𝑓_𝑖𝑢^𝑚−1⟶𝑓_𝑖(𝑢)in𝒞⁰([0,𝑇],𝐻)∩𝐿²]0,𝑇[,𝑉^′∀𝑖. (3.46)
For all 𝑖, 𝑢^𝑚_𝑖 is solution of 𝜕𝑢^𝑚_𝑖+𝐴𝜕𝑡,𝑣_𝑖𝑢^𝑚_𝑖,𝑣+𝑞_𝑖𝑢^𝑚−1𝑢^𝑚_𝑖,𝑣=𝑓_𝑖𝑢^𝑚−1,𝑣∀𝑣∈𝑉. (3.47)
We take 𝜙∈𝒟(]0,𝑇[), and therefore 𝜙𝑣∈𝐿²(]0,𝑇[,𝑉), ^𝑇₀𝜕𝑢^𝑚_𝑖𝜕𝑡,𝜙𝑣𝑑𝑡+^𝑇₀𝐴_𝑖𝑢^𝑚_𝑖,𝜙𝑣𝑑𝑡+^𝑇₀𝑞_𝑖𝑢^𝑚−1𝑢^𝑚_𝑖=,𝜙𝑣𝑑𝑡_𝑇0𝑓_𝑖𝑢^𝑚−1,𝜙𝑣𝑑𝑡. (3.48)
The second term in the left side and the right side of the egality converges due to the weak convergence in 𝐿²(]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 𝜕𝑢^𝑚_𝑖/𝜕𝑡 converges weakly in 𝐿²(]0,𝑇[,𝑉^′).
But we have 𝑢^𝑚_𝑖⟶𝑢_𝑖in𝒞⁰([0,𝑇],𝐻), (3.49) then 𝜕𝑢^𝑚_𝑖⟶𝜕𝑡𝜕𝑢_𝑖𝜕𝑡in𝒟^′(]0,𝑇[,𝐻). (3.50) Therefore, we obtain 𝜕𝑢^𝑚_𝑖⟶𝜕𝑡𝜕𝑢_𝑖𝜕𝑡weaklyin𝐿²]0,𝑇[,𝑉^′,_𝑇0𝜕𝑢_𝑖𝜕𝑡,𝜙𝑣𝑑𝑡+^𝑇₀𝐴_𝑖𝑢_𝑖,𝜙𝑣𝑑𝑡+^𝑇₀𝑞_𝑖(𝑢)𝑢_𝑖,𝜙𝑣_𝐻𝑑𝑡=^𝑇₀𝑓_𝑖(𝑢),𝜙𝑣𝑑𝑡. (3.51) This formulation being true for all 𝜙, we have 𝜕𝑢_𝑖+𝐴𝜕𝑡,𝑣_𝑖𝑢_𝑖,𝑣+𝑞_𝑖(𝑢)𝑢_𝑖,𝑣_𝐻=𝑓_𝑖(𝑢),𝑣∀𝑣∈𝑉, (3.52) that is to say, 𝑑𝑢𝑑𝑡^𝑖,𝑣_𝐻+𝑎𝑢_𝑖,𝑣+𝑞_𝑖(𝑢)𝑢_𝑖,𝑣_𝐻=𝑓_𝑖(𝑢),𝑣∀𝑣∈𝑉,𝑖,𝜕𝑢_𝑖𝜕𝑡=𝑓_𝑖(𝑢)−𝐴_𝑖𝑢_𝑖−𝑞_𝑖(𝑢)𝑢_𝑖∈𝐿²]0,𝑇[,𝑉^′∀𝑖. (3.53)
For all 𝑖, one has 𝑢^𝑚_𝑖(𝑡)=𝐺_𝑖(𝑡)𝑢_0𝑖+^𝑡₀𝐺_𝑖(𝑡−𝑠)−𝑞_𝑖𝑢^𝑚−1𝑢^𝑚_𝑖+𝑓_𝑖𝑢_𝑖^𝑚−1(𝑠)𝑑𝑠, (3.54) and as 𝑞_𝑖(𝑢^𝑚−1)𝑢^𝑚_𝑖 and 𝑓_𝑖(𝑢^𝑚−1) converge in 𝐶⁰([0,𝑇],𝐻), we have 𝑢_𝑖(𝑡)=𝐺_𝑖(𝑡)𝑢_0𝑖+^𝑡₀𝐺_𝑖(𝑡−𝑠)−𝑞_𝑖(𝑢)𝑢_𝑖+𝑓_𝑖(𝑢)(𝑠)𝑑𝑠. (R)

3.4. Main Result

Theorem 3.3. If the initial condition satisfies (3.1), then system (𝒮) admits a unique nonnegative solution 𝑢_𝑖∈𝑊(0,𝑇,𝑉,𝑉^′) for all 𝑖.
Moreover, for all 𝑖, 𝑢_𝑖 satisfies relation (3.62).

Proof. We have already shown existence of solution; thus, it remains to show uniqueness.
Let 𝑣 be another solution of system (𝒮)𝑣_𝑖∈𝑊0,𝑇,𝑉,𝑉^′⟹𝑣_𝑖∈𝒞⁰([0,𝑇],𝐻),𝑣_𝑖≥0. (3.55) Consequently, we obtain 𝑞_𝑖(𝑣)𝑣_𝑖+𝑓_𝑖(𝑣)∈𝐿²]0,𝑇[,𝑉^′. (3.56) Thus, by Proposition 2.11, we have 𝑣_𝑖(𝑡)=𝐺_𝑖(𝑡)𝑢_0𝑖+^𝑡₀𝐺_𝑖(𝑡−𝑠)−𝑞_𝑖(𝑣)𝑣_𝑖+𝑓_𝑖(𝑣)(𝑠)𝑑𝑠. (3.57)
By subtraction, we have 𝑢_𝑖(𝑡)−𝑣_𝑖(𝑡)=^𝑡₀𝐺_𝑖(𝑡−𝑠)−𝑞_𝑖(𝑢)𝑢_𝑖+𝑞_𝑖(𝑣)𝑣_𝑖+𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)(𝑠)𝑑𝑠, (3.58) with 𝑞_𝑖(𝑢)𝑢_𝑖−𝑞_𝑖(𝑣)𝑣_𝑖=𝑞_𝑖(𝑢)𝑢_𝑖−𝑣_𝑖+𝑞_𝑖(𝑢)−𝑞_𝑖(𝑣)𝑣_𝑖. (3.59)
𝑢_𝑖 being positive, we have 𝑢_𝑗𝐾+𝑢_𝑖≤1𝐾𝑢_𝑗∞, (3.60) where we have defined 𝑢_𝑗∞=𝑢_{𝑗𝐿∞(]0,𝑇[,𝐻)}. (3.61)
If we define ‖𝑢‖_∞=⁶_𝑗=1𝑢_𝑗∞, (3.62) there is 𝑀₁>0 such that 𝑞_𝑖(𝑢)_∞≤𝑀₁‖𝑢‖_∞∀𝑢. (3.63)
So the numerator of 𝑞_𝑖(𝑢)−𝑞_𝑖(𝑣) is the sum of terms of form (𝑢_𝑘−𝑣_𝑘)𝑣_𝑗 or (𝑢_𝑗−𝑣_𝑗)𝑢_𝑘, and we can find 𝑀₂>0 such that 𝑞_𝑖(𝑢)−𝑞_𝑖(𝑣)_𝐻(𝑠)≤𝑀₂⁶_𝑗=1𝑢_𝑗(𝑠)−𝑣_𝑗(𝑠)_𝐻. (3.64)
Also we can find 𝑀₃>0 such that 𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)_𝐻(𝑠)≤𝑀₃⁶_𝑗=1𝑢_𝑗(𝑠)−𝑣_𝑗(𝑠)_𝐻. (3.65)
Summing and noting that ‖𝐺_𝑖(𝑡−𝑠)‖≤𝑁_𝑖𝑒^Ω𝑖𝑇 with 𝑁_𝑖, Ω_𝑖>0, we can find 𝑀 such that ⁶_𝑖=1𝑢_𝑖(𝑡)−𝑣_𝑖(𝑡)_𝐻≤𝑀‖𝑢−𝑣‖_∞. (3.66)
Replacing in (3.58), we obtain ⁶_𝑖=1𝑢_𝑖(𝑡)−𝑣_𝑖(𝑡)_𝐻≤𝑀²‖𝑢−𝑣‖_∞^𝑡₀𝑠𝑑𝑠=𝑀²𝑡²2‖𝑢−𝑣‖_∞. (3.67)
By induction, we have ⁶_𝑗=1𝑢_𝑗(𝑡)−𝑣_𝑗(𝑡)_𝐻≤𝑀^𝑛𝑇𝑛!^𝑛‖𝑢−𝑣‖_∞,lim_𝑛→+∞𝑀^𝑛𝑇𝑛!^𝑛‖𝑢−𝑣‖_∞=0. (3.68)

Acknowledgment

The third author thanks Associate and Federation Schemes of ICTP for its support.