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

We study a nonlinear parabolic system governing the biological dynamic in the soil. We prove global existence (cid:3) in time (cid:4) and uniqueness of weak and positive solution for this reaction-di ﬀ usion semilinear system in a bounded domain, completed with homogeneous Neumann boundary conditions and positive initial conditions.


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 ⊂ R 3 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: for i 1, . . . , 6.

International Journal of Differential Equations
We have noticed u u 1 , u 2 , . . . , u 6 T with u 1 is the density of microorganisms MB , u 2 is the density of DOM, u 3 is the density of SOM, u 4 is the density of FOM, u 5 is the density of enzymes, and u 6 is the density of CO 2 , q 1 u − ku 2 K s u 2 μ r ν, q 2 u ku 1 K s u 2 , q 3 u c 1 u 5 K m u 5 , q 4 u c 2 u 5 K m u 5 , q 5 u ζ, q 6 u 0, with μ mortality rate, r is the breathing rate, ν is the enzymes production rate, ζ is the transformation rate of deteriorated enzymes, c 1 is the maximal transformation rate of SOM, c 2 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,

1.3
A sufficient condition for the blowup of the solution of parabolic semilinear secondorder equation is obtained in 6 with nonlinear boundary conditions, and so the set in which International Journal of Differential Equations 3 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 : 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 ∂t DΔu uf x, u x ∈ Ω, t > 0,

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 reactiondiffusion systems similar to the following one: 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, 4 International Journal of Differential Equations ii they satisfy a "mass-control structure" ∀r r 1 , . . . , r n , 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 L 1 0, T × Ω are also considered, for instance, for f i in L 1 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: where d 1 , d 2 ∈ 0, ∞ , β 1 , β 2 ∈ 0, ∞ , and f, g : 0, ∞ 2 → R are C 1 , holds for the additional following hypothesis: This approach has been extended to m × m systems for which . . 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 L 1 may be found in 18 . It is assumed that, for all i 1, . . . , m, there is a sequence which converges in L 1 0, T ×Ω to a supersolution of 1.7 .
International Journal of Differential Equations 5 One consequence is that global existence of weak solutions for systems whose nonlinearities are at most quadratic with u 0 ∈ L 2 Ω m can be obtained.
Results are also obtained in the weak sense for systems satisfying and H ∈ L 2 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.

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 .
We assume the two following lemmas, see 19 .

Lemma 2.2. There exists a continuous prolongation operator
International Journal of Differential Equations 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.

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 For each fixed t, one defines a continuous linear application A t ∈ L V, V by Then, we have 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 .
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 dt u t , v H u t , v for a.e. t.

2.9
International Journal of Differential Equations 7 The variational parabolic problem associated to the triple H, V, a t; ·, · is the following.
Given f t ∈ L 2 0, T ; V and u 0 ∈ H, find u ∈ W 0, T, V, V such that

2.10
This problem is equivalent to u 0 u 0 .

2.11
Definition 2.6. The form a is coercive or V coercive if α > 0 exists such that Theorem 2.7. If the form is coercive, then the problem P admits a unique solution.
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 λt w t , then u is solution of P if and only if w is solution of 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: a ij t, x ζ i ζ j ≥ α ζ 2 R 3 a.e. in 0, T × Ω.

2.19
Then, we deduce that

2.20
The form is then H coercive, and it suffices to take λ α.
In addition, let us take a 0 ∈ L ∞ 0, T ×Ω with a 0 t, x ≥ 0 for all t, x.

2.22
one has u t ≥ 0 for all t ∈ 0, T .
Proof. It remains to show that the solution is nonnegative. Given u ∈ L 2 Ω , we set u max 0, u x and u − max 0, −u x . If u ∈ H 1 Ω , then we have u and u − ∈ H 1 Ω .
By replacing v by u − t in P , we obtain

2.23
International Journal of Differential Equations 9 One gets and by linearity, we obtain By integration over 0, t , we deduce

2.30
As previously mentioned, if we set u t e λt w t , w t is solution of It suffices to take λ ≥ C to reduce to the previous case, and w t ≥ 0 implies u t ≥ 0. Then, we get.

Equivalence of the Variational Solution with the Initial Problem
We have 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 a 0 u f in 0, T × Ω.

2.37
We assume that f ∈ L 2 0, T , H , then

2.40
International Journal of Differential Equations

2.41
Using Green formula with u, −grad u T , we have

Semigroup Method
Consider the variational triple H, V, a where a is independent of t. We associate operators Assume that a is H coercive, then A H is the infinitesimal generator of semigroup t → G t of class C 0 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 .

International Journal of Differential Equations
Proposition 2.11. For u 0 ∈ H and f ∈ L 2 0, T , V , problem P which consists in finding u ∈ W 0, T, V, V such that admits a unique solution given by Proof. Note u and f the extensions by 0 of u and f outside 0, T , then we have with δ t the Dirac measure on R. Thus,

2.51
Hence, an equation of the form where D V is the space of distributions over R into V whose support is in 0, ∞ . By Laplace transform, one is reduced to and therefore, we have U G * F. But since International Journal of Differential Equations 13 we have

2.56
Hence, we get the result.

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 L 2 Ω and V H 1 Ω and the following hypothesis for initial conditions: We will make a resolution component by applying Theorem 2.7 with, for each i, the form One approaches the solution by a sequence of solutions of linear equations.

Recursive Sequence of Solutions
For n 0, we note that u 0 i is the solution of

3.3
This equation admits strong solution and u 0 i ≥ 0. By induction, we note that u n i is solution of equation

3.4
It is a linear equation within the framework of Corollary 2.10 with a 0 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 u n−1 is nonnegative also that implies that there are two positive constants C 1 , C 2 such that For the rest, we notice that q 5 and q 6 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 q 2 u n−1 .
To prove that q 2 u n−1 is bounded, we need to show that u n 1 ∈ L ∞ 0, T; L ∞ Ω .
Case of u 0 1 Let k ∈ N * , we multiply 3.3 1 by u 0 1 2k−1 and integrate it over Ω, and it comes that The second term is nonnegative, then we have 1 2k By integrating over 0, t , we obtain When k tends to ∞, it comes that, The function q 1 being undervalued, we can choose λ ≥ 0 such that λ q 1 e λt w n−1 We multiply 3.12 by w n 1 2k−1 and integrate it over Ω. We obtain 1 2k

3.14
The second and third term being nonnegative, we can conclude as in the previous case that Since k tends to ∞, it follows that As a result, we have proved that w n 1 ∈ L ∞ 0, ∞; L ∞ Ω , and since u n 1 e λt w n 1 , we have u n 1 ∈ L ∞ 0, T; L ∞ Ω .

Conclusion 1.
With the previous demonstration, we obtain by induction that if u 01 ∈ L ∞ Ω with u 01 ≥ 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 ∈ L 2 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.

Boundedness of the Solution
where G i t is the semigroup generated by the unbounded operator −D i A H . Let us denote g n i s −q i u n−1 s u n i s f i u n−1 s , 3.29 and we deduce g n i ∈ L 2 0, T , V . Moreover, the sequence u n i n≥0 is bounded in C 0 0, T , H which implies that the sequence g n i n≥0 is bounded in C 0 0, T , H for all i. Then, we can conclude showing that operator G i from C 0 0, T , H into C 0 0, T , H defined by 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 18 International Journal of Differential Equations 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 u 0 ∈ H, G t u 0 ∞ k 0 e −tλ k u 0 , e k e k , 3.32 which proves that the operator is compact for all t > 0 because lim k → ∞ e −tλ k 0.

3.33
We have the same formula for G i t , and it suffices to replace λ k by D i λ k . If we set G N t u N k 0 e −tλ k u, e k e k , 3.34 then G N t is an operator with finite rank which converges to G t .