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 (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.
Modelling biological dynamic in the soil is of great interest during these last years. Several attempts are made in , , and rarely in . For more details, readers are referred to [1–3]. We deal here with the mathematical study of the model described in .
Let be a fixed time, an open smooth bounded domain, , and .
The set of equations describing the organic matter cycle of decomposition in the soil is given by the following system: for .
We have noticed with is the density of microorganisms (MB), is the density of DOM, is the density of SOM, is the density of FOM, density of enzymes, and is the density of CO2, with mortality rate, is the breathing rate, is the enzymes production rate, is the transformation rate of deteriorated enzymes, is the maximal transformation rate of SOM, is the maximal transformation rate of FOM, maximal growth rate, and represent half-saturation constants, and , to 6, are strictly positive constants.
System is introduced in . 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 operate in other situations. It comes in population dynamics as Lotka-Voltera equation which corresponds to the case , denoting the densities of species present and growth rate. This system is also involved in biochemical reactions. In this case, the are the concentrations of various molecules, is the rate of loss, and represents the gains.
For models in biology, interested reader can consult with profit  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 , the authors prove the blowup in finite time for the system in ,
A sufficient condition for the blowup of the solution of parabolic semilinear second-order equation is obtained in  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 : with with obstacles, giving a probabilistic interpretation of solution. This problem is solved using a probabilistic method under monotony assumptions.
By using bifurcation theory, in , authors determine the overall behavior of the dynamic system
A Cauchy problem for parabolic semilinear equations with initial data in is studied in . Particularly the author solves local existence using distributions data.
Michel Pierre’s paper, see , presents few results and open problems on reaction-diffusion systems similar to the following one: where the are functions of , and , , , .
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.
It is assumed throughout the paper that (i)all nonlinearities are quasipositives, (ii)they satisfy a “mass-control structure” 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 are also considered, for instance, for in whose growth rate is less than when tends to .
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 , and are , holds for the additional following hypothesis:
This approach has been extended to systems for which ,, are all bounded by a linear of the (see ).
A very general result for systems which preserves positivity and for which the nonlinearities are bounded in may be found in . It is assumed that, for all , there is a sequence which converges in to a supersolution of (1.7).
One consequence is that global existence of weak solutions for systems whose nonlinearities are at most quadratic with can be obtained.
Results are also obtained in the weak sense for systems satisfying , where , , and .
The aim of our paper is to study the global existence in time of solution for the system . 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 being in the denominator of some and , it is necessary to guarantee that 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 by a function of .
In our work, we show existence and unicity of a global positive weak solution of System 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 )
We consider two Hilbert spaces and such that is embedded continuously and densely in .
Then, we have duality . Using Riesz theorem, we identify and . So we get .
Definition 2.1. We define the Hilbert space equipped with the norm
We assume the two following lemmas, see .
Lemma 2.2. There exists a continuous prolongation operator from to such that
Lemma 2.3. is dense in .
Corollary 2.4. is dense in .
Proof. If , one takes a sequence of which converges in toward , and then converges toward and for all .
Proposition 2.5. Every element is almost everywhere equal to a function in .
Furthermore, the injection of into is continuous when is equipped with the supnorm.
Proof. See .
For all , a bilinear form is given on such that for and fixed, is measurable and
For each fixed , one defines a continuous linear application by
Then, we have
Also we associate, for all fixed , an unbounded operator in whose domain is the set of such that is continuous on for the induced norm by . It is exactly the set of such that and then
To simplify the writting the unbounded operator is noted .
Let , we have, for , where the bracket is the duality between and because . By density, if , one has, for all ,
The variational parabolic problem associated to the triple is the following.
Given and , find such that
This problem is equivalent to
Definition 2.6. The form is coercive or coercive if exists such that
Theorem 2.7. If the form is coercive, then the problem admits a unique solution.
Proof. See Dautray-Lions .
Definition 2.8. The form is coercive if there exist two constants and such that
If we set , then is solution of if and only if is solution of
is a coercive form, and then admits a unique solution, and therefore too. We apply Theorem 2.7 in the following case: and defining we assume that and there exists such that, for all , we have
Then, we deduce that
The form is then coercive, and it suffices to take .
In addition, let us take with for all , .
The form is still coercive. We have the following theorem.
Theorem 2.9. Under the previous hypothesis, problem associated to the triple admits a unique solution for all and .
Moreover, if and one has for all .
Proof It remains to show that the solution is nonnegative.
Given , we set and . If , then we have and .
By replacing by in , we obtain
One gets and by linearity, we obtain
Since it comes that
By integration over , we deduce
But , then , so .
Hence, we conclude that for all .
Instead of , assume that
As previously mentioned, if we set , is solution of with
It suffices to take to reduce to the previous case, and implies .
Then, we get.
Corollary 2.10. Consider the triple satisfying assumptions of Theorem 2.9.
If and then the variational problem admits a unique solution in for all and .
Moreover, if and then one has for all .
Equivalence of the Variational Solution with the Initial Problem
For the sake of simplicity, we set which is the Kronecker symbol.
Then, over , and we have
We assume that , then if . We set
Consequently, we have and
Then , , and are well defined.
It remains to show that .
Let , and we multiply (2.34) 1 by , and by integration over , one gets
Using Green formula with , we have
As we can conclude the following statement:
We deduce that
Function from into being surjective, we deduce that
2.2. Semigroup Method
Consider the variational triple where is independent of . We associate operators and in with
Assume that is coercive, then is the infinitesimal generator of semigroup of class over , and operates over and . If we note the extension of by 0 for , then the Laplace transform of is the resolvent of .
Proposition 2.11. For and , problem which consists in finding such that admits a unique solution given by
Proof. Note and the extensions by 0 of and outside , then we have
with the Dirac measure on .
Hence, an equation of the form where is the space of distributions over into whose support is in . By Laplace transform, one is reduced to where and therefore, we have .
But since we have
Hence, we get the result.
3. System Resolution
In this part, we go back to system with assumptions and will analyze this problem by using the framework described in the previous section.
We define and and the following hypothesis for initial conditions:
We will make a resolution component by applying Theorem 2.7 with, for each , the form
One approaches the solution by a sequence of solutions of linear equations.
3.1. Recursive Sequence of Solutions
For , we note that is the solution of
This equation admits strong solution and .
By induction, we note that is solution of equation
It is a linear equation within the framework of Corollary 2.10 with and . Let us suppose that there exists a unique nonnegative solution . Assuming by induction that for , we have which implies that is nonnegative also that implies that there are two positive constants , such that For the rest, we notice that and are constant.
We have shown that for . It remains to prove that the same property is satisfied by .
To prove that is bounded, we need to show that .
Let , we multiply (3.3) 1 by and integrate it over , and it comes that The second term is nonnegative, then we have By integrating over , we obtain When tends to , it comes that, which implies that .
Case of with
By induction, we suppose that are in .
Remark 3.1. We make the following change: . We obtain
The function being undervalued, we can choose such that We multiply (3.12) by and integrate it over . We obtain The second and third term being nonnegative, we can conclude as in the previous case that Since tends to , it follows that As a result, we have proved that , and since , we have .
Conclusion 1. With the previous demonstration, we obtain by induction that if with , then for all and .
We also have and . Then by means of Corollary 2.10, there exists a unique solution with .
3.2. Boundedness of the Solution
Let us show that the sequence is bounded. satisfies (2.34), so
We remark that
For , By density and choosing , we have
We have seen that we can obtain problem replacing by , and since , if is bounded, is also bounded.
We take then , and one is reduced to
The form is coercive, so we take such that
The are bounded, so Therefore,
We take small enough such that . Hence,
For and , . Therefore, by integration,
We deduce that and remain bounded in and .
, thus, remains bounded in ; therefore, has the same property as , . It is the same for because .
We have ; therefore, we have the same conclusion for and finally for because depends on , , , and .
3.3. Convergence of the Sequence
We deduce at this stage that the sequence (one can extract subsequence ) converges weakly in to and weakly star in to .
But it is not enough to pass to the limit in the equation, we need the pouctual convergence for almost all to deduce that for all and to pass to the limit in and . To pass to the limit, we need strong compactness. Using Proposition 2.11, for all , we have where is the semigroup generated by the unbounded operator . Let us denote and we deduce .
Moreover, the sequence is bounded in which implies that the sequence is bounded in for all .
Then, we can conclude showing that operator from into defined by is compact.
One takes the triple with where is regular and bounded. The unbounded variational operator associated to is a positive symmetric operator with compact resolvent. It admits a sequence of positive eigenvalues with and a Hilbert basis of consisting of eigenvectors of . If is the semigroup generated by , then for all ,
which proves that the operator is compact for all because We have the same formula for , and it suffices to replace by .
If we set then is an operator with finite rank which converges to .
Theorem 3.2. Let be an application from into . One assumes that there exists a sequence of operators of with the following properties: (1)for all and all , is finite rank independent of , (2) is continuous from into for all , (3)for , converges to in for all ,
then the operator is compact from into for all .
Proof. Regarding property (3) of Theorem 3.2, is well defined on , and we have
This proves that converges to in using property (3) of Theorem 3.2.
To show that is compact, it suffices to show that for all , is compact.
Let be a bounded set of , is bounded in , using Ascoli result, it will be relatively compact if is equicontinuous and if for all in , is relatively compact in .
But being bounded and embedded in a subspace of finite dimension of is relatively compact in . Then, let us show the equicontinuity on .
Let and such that For , one has We have which tend to 0 when 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) is of rank for all , (2) which tends to 0 if , (3) if we take an increasing sequence , then one gets Thus, we have which tends to 0 if .
Thus, is compact for all . We have where is bounded in , then belongs to a relatively compact set of .
Therefore, from the sequence , we can extract a subsequence which converges uniformly to for each .
Moreover, we can assume that converges weakly to in .
But and bounded in imply
Thus, the sequence is bounded in . We can assume that converges weakly to in .
Remark that for all , we have
Conclusion 2. One has
Thus, and .
We check that
We have also
For all , is solution of
We take , and therefore ,
The second term in the left side and the right side of the egality converges due to the weak convergence in . The third term in the left-hand side also passes to the limit, due to the convergence in