We establish a convergence theorem for a class of nonlinear reaction–diffusion equations when the diffusion term is the subdifferential of a convex functional in a class of functionals of the calculus of variations equipped with the Mosco-convergence. The reaction term, which is not globally Lipschitz with respect to the state variable, gives rise to bounded solutions, and cover a wide variety of models. As a consequence we prove a homogenization theorem for this class under a stochastic homogenization framework.
Let Ω be a bounded regular domain in , and T any positive real number. The purpose of this paper is to investigate the stability and the stochastic homogenization of the class of reaction–diffusion problems
defined in , when Φ belongs to the class of integral functionals of the type
The reaction functionals F, called CP-structured reaction functionals, have the following form:
where is a locally Lipschitz function, , and .
In the standard presentations, the reaction functional F is assumed to be globally Lipschitz-continuous with respect to the state variable u. This hypothesis provides the existence of a solution through a fixed point procedure. Nevertheless, in many situations, F is only locally Lipschitz-continuous and the expected solutions must be bounded, or at least positive in models involved in population dynamics or theoretical ecology. In this paper, we assume that f is locally Lipschitz-continuous and satisfies a condition (CP) ((CP) for Comparison Principle, see Definition 3.1). Under this condition, admits a unique bounded solution with the initial condition , with a right derivative at every (Theorems 3.1, 3.2). Throughout the paper, the solutions are to be considered in the strong sense, i.e., are absolutely continuous in and satisfy the equation of for a.e. . Problems cover a wide variety of applications in the fields of thermochemistry, combustion, biochemical systems, as well as those of population dynamics and evolution of ecosystems as illustrated in Examples A.1, A.2, A.3, A.4 of Appendix A.
The main first result of the paper, Theorem 4.1, states the stability of the class when the class of functionals Φ is equipped with the Γ-convergence associated both with strong and weak topology of , namely, the Mosco-convergence, and the class of functionals F with some “weak” convergence. Such stability has been first obtained in the context of weak solutions and when the source term does not depend on the solution. Consult [14] and [8, Theorem 17.4.6] for this specific case where the key argument consists in using the convergence of semigroups of contraction. Stochastic homogenization of reaction–diffusion problems is addressed in Section 5 where we set up the basic concepts concerning ergodic dynamical systems. The main theorem, Theorem 5.1, based on Theorem 4.1, states the limit homogenized problem of when . As an example, in Appendix B, the stochastic homogenization of the reaction–diffusion problem describing a food-limited population model is treated in two differents situations. The reaction functional is that of the Fisher model with Allee effect (see Example A.1, a)). In the first situation the small spatial heterogeneities of size ε are distributed following a random patch model, i.e., a random checkerboard-like environment. In the second situation, the discrete dynamical system describes spatial heterogeneities distributed following a Poisson point process. For homogenization of convection-diffusion equations, and parabolic problems in perforated domains or in periodic or random environments, we refer the reader to [1–3,6,16] and references therein. For homogenization of a Fokker–Planck equation with space-time periodic potential, we refer to [27].
A similar analysis will be performed for time delays reaction–diffusion equations and coupled reaction–diffusion systems in forthcoming works. For these problems, the reaction functionals are of the form for all u and v in , where ,1
We denote by the Hadamard (or Schur) product of two elements ξ and in .
under the condition that for fixed , is a CP-structured reaction function.
Existence and uniqueness for reaction–diffusion Cauchy problems in Hilbert spaces
In this section, X denotes a Hilbert space equipped with a scalar product denoted by and the associated norm . In all along the paper we use the same notation to denote the norms of the Euclidean spaces , , and by the standard scalar product of two elements ξ, in .
Let be a convex proper lower semicontinuous (lsc in short) functional, minorized, i.e., satisfying , that we assume to be Gâteaux differentiable so that its subdifferential is single valued. We make this choice in order to simplify the notation but, in this section, we could use the subdifferential of Φ in place of its Gâteaux derivative, denoted by , without additional difficulties. We denote by and the domain of Φ and respectively.
On the other hand, let be a Borel measurable map fulfilling the two following conditions:
there exists such that for all and all ;
the map belongs to .
Given and in , the map F is referred to as the reaction part, and as the diffusion part of the following Cauchy problem:
where denotes the distributional time derivative of u. We say that u is a solution of if is absolutely continuous in time and satisfies . In all the paper, the space is endowed with the sup-norm. The results stated in the theorem below are somewhat well known. The proof is based on [8, Theorems 17.2.5, 17.2.6], or on [13, Theorem 3.7]), together with a fixed point procedure.
(Local existence).
Assume that F satisfies conditions
(C
1
)
,
(C
2
)
. Then, there existsmall enough and a unique solutionofwhich satisfies
for a.e.,
u is almost everywhere differentiable inandfor a.e..
Assume furthermore thatdefined bybelongs to, then u satisfies:
for all,
Denote by a small enough real number so that admits a unique solution in , whose existence is asserted in Theorem 2.1. Under the initial condition we are not assured that the derivative of the solution belongs to . Nevertheless (see [8, Theorem 17.2.5] or [13, Theorem 3.6]). Hence, for , belongs to . Set
Since , we have . We define the maximal time in by and denote by u the maximal solution of in . We have the following alternative:
(Global existence or blow-up in finite time).
Assume that F satisfies
(C
1
)
,
(C
2
)
, then we have the blow-up alternative
(existence of a global solution);
. In this case(blow-up in finite time).
Moreover, for all T,, the restriction of u tosatisfies assertions
(L
1
)
and
(L
2
)
, and furthermore satisfies
(L
3
)
whendefined bybelongs tofor all T,.
We assume that and show that . We argue by contradiction. Assume that u does not fulfill , then there exist and a sequence in E such that and .
Step 1. We show that exists in X.
Let . For a.e. we have
By integration over we obtain
On the other hand, from we infer that
so that for a.e.
Hence we have
From (1), (2), and since , we infer that there exists a constant which does not depend on such that
from which we deduce that
From (3), we deduce that is uniformly continuous. Indeed, let in and choose n large enough (depending on ) so that s and t belong to . We have
so that u is more precisely -Holder continuous. According to the continuous extension principle in the complete normed space X, u possesses a unique continuous extension in , i.e., .
Step 2. (Contradiction) Consider the Cauchy problem
Note that . Indeed, for a.e. t in and (choose with outside the negligible set in which ). Then applying Theorem 2.1, there exists small enough such that admits a solution . Set
Then is a solution of . This leads to a contradiction with the maximality of . □
Proposition 2.1 below provides a condition on which ensures that satisfies (G1). More precisely
Assume thatfor all. Thenadmits a global solution.
According to Theorem 2.2, it suffices to prove that there is no blow-up in finite time. Assume that and let . Taking as a test function, for a.e. we have
Hence, using the fact that (recall that for a.e. ), we infer that
By integrating over for , we deduce
(note that from (C2), belongs to , and belongs to since and ). By using Grönwall’s lemma and the continuity in of
we obatin for all :
Then, if , we have
This makes impossible. Thus . □
From Proposition 2.1, when , (G1) is automatically satisfied. Therefore we obtain the following global existence for non diffusive problems
(Global existence for non diffusive Cauchy problems).
Assume that F satisfies
(C
1
)
,
(C
2
)
. Then, there exists a unique global solutionof the non diffusive Cauchy problemMoreover, for allthe restriction of u tosatisfies assertions
(L
1
)
and
(L
2
)
, and furthermore satisfies
(L
3
)
whendefined bybelongs to.
Existence and uniqueness of bounded solution for a class of reaction–diffusion problems
From now on Ω is a domain of of class and denotes the Lebesgue measure on . We denote by its boundary and by the -dimensional Hausdorff measure. To shorten the notation, we sometimes write X to denote the Hilbert space equipped with its standard scalar product and its associated norm, denoted by and respectively.
The class of diffusion terms associated with convex functionals of the calculus of variations
In all the paper, we focus on the specific case of a standard convex functional Φ of the calculus of variations, i.e., a functional defined by
where2
In the integrals on , we still denote by u the trace of u.
, with -a.e. in and on with for some .
The density is a Borel measurable function which satisfies the following conditions:
there exist and such that for a.e. and every
for a.e. , is a Gâteaux differentiable and convex function (we denote by its Gâteaux derivative), and
By using the subdifferential inequality together with the growth conditions (D1), it is easy to show that there exist nonnegative constants and such that, for all ,
From the second estimate, we infer that if , then the function belongs to .
We assume in (D2) without loss of generality. Indeed, for any Borel measurable function satisfying (D1) with Gâteaux differentiable and convex, define the function by
Then is convex, Gâteaux differentiable with . Moreover satisfies the upper growth condition of (D1) and the lower growth condition up to an additive constant, with two other positive constants and .
Consider the space . It is well known that when Ω is an open domain of class , with outer unit normal n, the normal trace
defined by , has a continuous extension from onto , still denoted by . Moreover, the following Green’s formula holds: for every whose trace denoted by belongs to , we have
In all the paper, for any and any , we (improperly) write, instead of the last term , and, as for regular functions, we denote by and φ the normal trace and the trace of σ and φ respectively. We start by expliciting the subdifferential of the functional Φ (actually its Gâteaux derivative), whose domain contains mixed Dirichlet–Neumann boundary conditions. For a detailed proof we refer the reader to [8, Theorem 17.2.10] where Φ is a more basic integral functional.
The subdifferential of the functional Φ is the operator() defined bywheremust be taken in the trace sense.
The strategy consists in establishing that A is a maximal monotone operator included in the subdifferential , which, in turn, is a maximal monotone operator (for this last point see Theorem 17.4.1 in [8]). Let . Note that . Then, according to the Green formula, to the convexity of , and to the boundary condition expressed in , we infer that
which proves the monotonicity of A.
To establish that , due to the definition of , it is enough to prove that for any and any , . This inequality is easily obtained by integrating over Ω the convexity inequality
and by using Green’s formula, after adding the surface energy .
We claim that the operator A is maximal. By Minty’s theorem (see Theorem 17.2.1 in [8] and references therein) it remains to prove that where denotes the range of the operator . Equivalently, for any f in , we have to establish the existence of a solution of the coercive homogeneous mixed Dirichlet–Neumann problem:
which is a classical result. □
Assume that . Let Γ be a subset of with and define in in the following way:
Then, the integral may be considered as a penalization which forces the function u to belong to . By convention the functional Φ becomes
The subdifferential of Φ contains now the homogeneous Dirichlet–Neumann boundary conditions as stated in the following lemma which can be proved by an easy adaptation of the proof of Lemma 3.1:
The subdifferential of the functional Φ is the operator() defined by
In some particular cases it is possible to characterize the adherence of . Take for example where the measurable matrix valued function satisfies the two borns for all and all . Assume that . In the case of Lemma 3.2, it is easy to show that
Hence , i.e. is dense in .
The class of CP-structured reaction functionals
The reaction–diffusion problems modeling a wide variety of applications, and amenable to analytical manipulation in homogenization (periodic or stochastic), involve a special class of functionals that we define below.
Let denote by the set of all the maps from Ω into . A map is called a CP-structured reaction functional, if there exists a Borel measurable function such that for all and all , , and fulfills the following structure conditions:
with
is a locally Lipschitz continuous function;
for all , r belongs to ;
for all , q belongs to .
Furthermore f must satisfy the following condition:
there exist a pair of functions with and a pair in with , such that each of the two following ordinary differential equations
admits at least one solution denoted by for and by for satisfying for a.e.
The map F is referred to as a CP-structured reaction functional associated with, and f as a CP-structured reaction function associated with. The map F is referred to as a regular CP-structured reaction functional and f as a regular CP-structured reaction function if furthermore, for all , and .
We refer to Appendix A for examples of CP-structured reaction functionals and the way of finding .
Since and are nonincreasing and nondecreasing respectively, for any we have .
We introduce the spaces and because of the specific form of sequences of CP-structured reaction functionals in the framework of homogenization where the scaling appears. Nevertheless, in Section 3, we can replace these two spaces by and respectively. Note that when X is a reflexive space, is exactly the space of absolutely continuous functions from into X (see [13, Corollary A4]).
The reason why we introduce condition (CP) may be summarized as follows: even if CP-structured reaction functionals do not satisfy the Lipschitz condition (C1) invoked in Theorems 2.1, 2.2, according to the comparison principle (Proposition 3.1 and Proposition 3.2 below), we can prove that reaction–diffusion problems associated with a CP-structured reaction functional admits a unique solution which satisfies whenever the initial condition satisfies (see Section 3.4).
The comparison principle
Let us set , and consider two functionals defined by
where are two measurable functions, being Lipschitz continuous uniformly with respect to , i.e., fulfills the condition . Moreover, we are given two functions and in and two functions and in . The following comparison result will be used for proving existence of bounded solutions of reaction–diffusion problems associated with special reaction functionals (see Section 3.2). For similar notion and applications of sub and supersolution related to elliptic boundary valued problems we refer the reader to [10,11] and for parabolic problems, to [24].
Let,andbe a subsolution and a supersolution of the reaction–diffusion problems with respect to the dataand, i.e.,
Then the following comparison principle holds:
Set . We are going to prove that for a.e. . Indeed, for a.e. we have
Take as a test function. By integrating over Ω, and using Green’s formula we obtain
Noticing that , and on , we infer that
from which we deduce
where we have used the relations
in the distributional sense. Noticing that the three last integrands of the first member are nonnegative, and , we obtain
From (6) and the Lipschitz continuity of the function , we deduce that
Integrating this inequality over for , we obtain
Note that since , is continuous. Then, according to Gronwall’s lemma we finally obtain that for all
from which we deduce, since , that in for all , i.e., for all . □
Let us consider the case
with , and set . Then an easy adaptation of the previous proof leads to the following comparison principle
Let,andbe a subsolution and a supersolution of the reaction–diffusion problems with respect to the dataand, i.e.,
Then the following comparison principle holds:
Existence and uniqueness of a bounded solution
Combining Theorems 2.1, 2.2 with the comparison principle we can establish the existence of a bounded solution of the Cauchy problem associated with CP-structured reaction functionals.
Let F be a CP-structured reaction functional, with,and,given by
(CP)
, and let Φ be a standard functional of the calculus of variations (
4
) of Section
3.1
. Assume thaton. Then for any, the Cauchy problemadmits a unique solutionsatisfying assertions
(L
1
)
,
(L
2
)
and the following bounds in:. If furthermore F is a regular CP-structured reaction functional, then u satisfies
(L
3
)
.
The proof of is obtained from a standard calculation (see [5]).
Step 1. We prove existence of a solution u of for small enough, which satisfies (L1), (L2) and the bounds .
By definition of CP-structured reaction functionals, is defined for all , all , and for a.e. by , where for all
and where is locally Lipschitz continuous. Fix arbitrary . The restriction of g to the interval is Lipschitz continuous with some Lipschitz constant .3
To simplify the notation, we do not indicate the dependence on .
Consequently is Lipschitz continuous with respect to ζ, uniformly with respect to in , with
where . According to the Mac Shane extension lemma (see [17, Lemma 3.2]), g can be extended into a Lipschitz continuous function in . Hence the extension of f defined by is Lipschitz continuous with respect to ζ in , uniformly with respect to , with the same Lipschitz constant L. Consequently, the functional defined by satisfies (C1) and (C2) with .
Therefore, according to Theorem 2.1, for small enough, that we can choose such that , the problem
admits a unique solution in which satisfies (L1) and (L2). In particular satisfies the boundary condition
for a.e. .
By using Proposition 3.1, we are going to prove that for all , . From condition (CP), the function , which does not depend on x, is a subsolution of the reaction–diffusion problem in the sense of Proposition 3.1. Indeed since , , and , we have for all
On the other hand is a solution of , thus a supersolution of . From the comparison principle, Proposition 3.1, since , and , we infer that for a.e. . Actually, inequality holds for all (invoke the continuity of ). Reasoning similarly with which is a supersolution of , we obtain that for all . To sum up we have for all .
We claim that is actually solution of in . Indeed, from above, for all we have which in its turn is included in . Therefore so that is solution of . From now on we write u for .
Step 2. We prove that there exists a global solution of . We use the notation of Theorem 2.2, and still denote by the maximal solution of .
By applying Theorem 2.2 it suffices to establish that there is no blow-up in finite time. Assume that . From Step 1 we infer that for all , and all we have . Hence
which makes impossible.
Step 3. We finally establish that if F is a regular CP-structured reaction functional, then , defined by , belongs to for all . According to Theorem 2.1, we will infer that u satisfies (L3).
For all in , and from the fact that q, r and u are absolutely continuous, we have
where4
We still wrote the Lipchitz constant of the restriction of g to .
the function , given by
belongs to . From (7), we easily deduce that G is absolutely continuous, then belongs to . This completes the proof. □
By an easy adaptation of the previous proof, applying this time Proposition 3.2, we obtain the following result.
Let F be a CP-structured reaction functional, with,and,given by
(CP)
, and let Φ be the functional of the calculus of variations (
5
). Assume that. Then for any, the Cauchy problemadmits a unique solutionsatisfying assertions
(L
1
)
,
(L
2
)
and the following bounds in:. If furthermore F is a regular CP-structured reaction functional, then u satisfies
(L
3
)
.
The set of functions satisfying is non empty. For the functional (4) of Theorem 3.1, any constant in is suitable since . The proof of the last equality is a straightforward consequence of the Brønsted–Rockafellar Lemma (see [8, Lemma 17.4.1]). Another direct proof is to apply the convergence of the resolvent as for all , and to note that belongs to (see [8, Remark 17.2.2]). For the functional (5) of Theorem 3.2, is suitable.
In the proof of Theorem 3.1, we have established that if u is the solution of , then, for all , the function belongs to since for all .
In Theorem 3.1, the mixed Dirichlet–Neumann boundary condition fulfilled by the solution u at , is expressed in condition (L3) by the fact that for all , and is given by:
Therefore, when F is a regular CP-structured reaction functional, may be written as
As regards Theorem 3.2, the same remark holds, i.e., problem may be written as
Estimate of the -norm of the right derivative
From above we know that when the CP-structured reaction functional is regular, the solution of admits a right derivative at each . The next estimate below is crucial in the proof of the compactness step (Step 2) of the convergence theorem, Theorem 4.1.
Under hypotheses of Theorems
3.1
,
3.2
, when F is a regular CP-structured reaction functional, for allwe havewhereanddenotes the Lipschitz constant of the restriction of g to.
Step 1. We establish the following lemma.
Let X be a Hilbert space,,andbe a convex proper lower semicontinuous functional. Let u satisfyThen the right derivative of u satisfies for allthe following estimate
For intended to tend to 0, set and let v be the solution of
Clearly (recall that u which solves (10) is the restriction to of a unique global solution of (10) in , and that for all ). From (10), (11), and the monotonicity of , we infer that for a.e.
hence
Integrating over where , we obtain for all
Thus, according to the Grönvall type lemma, [13, Lemma A.5], it follows that for all and all
that is
Dividing by h and letting , we infer that for all and all
(for a justification in order to obtain the last integral, we refer the reader to [13, Proposition A2]). By integration over , we obtain for all
which gives the result for all . This ends the proof of Lemma 3.3. □
Last step. The thesis of Proposition 3.3 follows by combining Lemma 3.3 and the expression of the total variation of G given by (7) and (8) where . □
General convergence theorem for a class of nonlinear reaction–diffusion problems
Consider a sequence of functionals of the calculus of variations defined by
where , , -a.e. in , on with for some , and is a measurable function satisfying the following conditions:
there exist and such that for a.e. , all , and all
for a.e. and every , the function is convex and differentiable with a.e. in ,
is uniformly strongly convex, i.e., for some it holds for all ,
In the following, we fix and we consider a sequence of regular CP-structured reaction functionals, each of them being associated with , i.e., for all , a.e. and all , where
We assume that for all the function is locally Lipschitz continuous, uniformly with respect to n, i.e., for every interval there exists such that for every
This condition is fulfilled for where the scalar functions are convex and satisfy for all , for some and . This is the case of Example A.1 b) where is substitute for .
We assume that the absolute continuity of the functions and holds uniformly with respect to n, i.e.,
Finally, we assume that
and, for all ,
where and are given by condition (CP) fulfilled by each . Recall that these two functions are solution of suitable o.d.e. with initial condition and respectively. When considering the case
and , for all , then (16) has to be replaced by
In order to establish a convergence result for reaction–diffusion problems with diffusion part and reaction part , we take advantage of standard results involving Γ-convergence of the functionals to Φ, and particularly in homogenization framework (see [8, Section 12.4]). More precisely, it is convenient to establish the convergence of the sequence of problems under the hypothesis of the Mosco-convergence of the sequence , introduced in [22,23], i.e., the Γ-convergence of the functionals when is equipped both with its strong and its weak topology. For the definition and variational properties of this notion we refer the reader to [8, Section 17.4.2], and for the connection with Moreau–Yosida approximations we refer to [7,18]. Note that even if is Gâteaux differentiable according to (D2,n), we are not ensured that its Mosco-limit Φ is Gâteaux differentiable. We will denote by the Mosco-convergence of the sequence to Φ. A first important lemma (which is proved in Appendix C) concerns the Mosco-convergence of functionals defined in .
Letbe a reflexive Banach space whose norm together with its dual norm is strictly convex, and such that weak convergence of sequences and convergence of their norms imply strong convergence. Let, ψ be a sequence of convex uniformly proper lower semicontinuous functions from X intosuch thatand consider,defined byThen.
Recall that the sequence , ψ is said to be uniformly proper if ψ is proper and if there exists a bounded sequence in X such that .
Assume thatsatisfies
(D
1,n
)
,
(D
2,n
)
, and
(D
3,n
)
, and that the sequence of CP-structured reaction functionals, of the form (
12
), satisfies (
13
), (
14
), (
15
), (
16
) or (
17
) when, for all,andLetbe the unique solution of the Cauchy problemAssume that
and;
;
there existssuch thatstrongly in;
there exists g such thatpointwise converge to g;
, and there existssuch thatin;
for all,, and there exists q such thatin.
Thenuniformly converges into the unique solution of the problemThe reaction functionalis defined, for all, alland for a.e., byMoreover,weakly inand.
If furthermore,in, andin, thenin.
We only establish the proof for of the form (4). The proof for functionals of the form (5) is slightly shorter, with some easy adaptations. Note that in the statement of Theorem 4.1, we assume that . But , thus . Therefore, according to Theorem 3.1, has a unique solution which satisfies (L2) and (L3) of Theorem 2.1, and the bounds . Finally, note that since , we have the additional regularity: belongs to (see [8, Theorem 17.2.5] or [13, Theorem 3.6]).
Step 1. We establish
(recall that and belong to from (15)).
Inequality (18) follows directly from . Let us establish (19). In what follows the letter C denotes a constant which can vary from line to line. From we deduce that for a.e. ,
We have used the fact that belongs to as stated in Remark 3.4. By integrating this equality over , we obtain
But belongs to and is absolutely continuous (see [13, Theorem 3.6]). Consequently for a.e. , (see [8, Proposition 17.2.5]). Recall that there exists such that . Therefore from (20) and (18) we deduce
By using the structure of the CP-structured reaction functional , we have
where , and is the Lipschitz constant of in the interval . On the other hand, we have clearly
where, from hypothesis (H4) and (H5), C is a nonnegative constant which does not depend on n. Hence
so that, according to hypothesis (H6)
Combining (18) and (25), (23) yields
From (22) and (H2), we infer that
where C is a nonnegative constant which does not depend on n, from which we deduce (19).
Step 2. We prove that there exist , and a subsequence of not relabeled, satisfying in .
We apply the Ascoli–Arzela compactness theorem. From (18), is bounded in . Moreover, for , , we have
which, according to (19), proves the equicontinuity of the sequence . It remains to establish for each , the relative compactness in X of the set . For there is nothing to prove because of hypothesis (H3) on the initial condition. For we are going to use that is compactly embedded in . For that purpose, we are going to see that the boundedness of requires the sharp estimate of Proposition 3.3.
According to Theorem 3.1, satisfies (L3), then possesses a right derivative at each (at , this is due to the fact that belongs to so that the right derivative of at is nothing but the right derivative of the restriction of to ). Moreover,
Taking as a test function, we infer that for all
hence, from the Green formula and the fact that for all ,
Choose , , where γ is the positive constant of the uniform strong convexity condition (D3,n), and is the constant of continuity of the trace operator. From (26), (D3,n), and (18) we infer that for all ,
Hence
To conclude, it suffices to prove that
Indeed, from (27), (18), and the compactness embedding we will conclude the compactness of the set for each . For proving (28), we establish successively
Proof of (
29
). This estimate follows straightforwardly from (18), (23), (24), and hypothesis (H6).
Proof of (
30
). By applying Proposition 3.3, we deduce that there exists such that
From (9), the uniform Lipchitz condition (13) together with (H4), (H5), and (H6), we infer that . Hence (30) follows from (19).
Step 3. We assert that weakly in for a non relabeled subsequence, and that . The first claim is a straightforward consequence of (19) and Step 2. The second one follows easily from inequality and in .
Step 4. We prove that u is the unique solution of . From Step 2, there exists and a (non relabeled) subsequence such that in . To simplify the notation, we still write and we use the subsequence obtained in Step 3, that we do not relabel. According to the Fenchel extremality condition (see [8, Proposition 9.5.1]) is equivalent to
for a.e. (together with the initial condition that we do not write), which is also equivalent to
Note that equivalence above is due to the Legendre–Fenchel inequality which asserts that inequality for a.e. , is always true (see [8, Remark 9.5.1]). Therefore, is equivalent to
or, equivalently, to
From hypothesis (H3) we have
Combining with in , we infer that
We postpone the proof of the following convergence to the last step (see Lemma 4.2)
where and . Passing to the limit in (31), from (32), (33), (34), Step 3, and Lemma 4.1, we obtain5
From hypotheses (H1), (H2) and (H3), the sequence , Φ, is clearly uniformly proper.
or equivalently
But from the Legendre–Fenchel inequality we have , so that (35) yields that for a.e. , which is, according to [8, Proposition 9.5.1], equivalent to
We have already proved that in Step 3. It remains to establish that . From (H1) and (H2), we infer that
which prove the thesis. For the proof of uniqueness of
it is enough to reproduce the proof of uniqueness of Theorem 2.1, with a Lipschitz constant for F given by . Since every subsequence of the subsequence of obtained above converges to the same limit u in , the sequence converges to u in . Idem for the sequence which converges to weakly in .
Step 5. We show that if , strongly in and strongly in , then strongly in . From Step 3 and Step 4 we have weakly in , hence it suffices to establish that to prove the claim. By repeating the proof of Lemma 4.2 below under the hypotheses of strong convergence of and to r and q respectively, it is easily seen that strongly converges to G in . Therefore, passing to the limit on (21), and since , we deduce that
The conclusion follows from the lower semicontinuity of the convex function in .
Last step. We establish the convergence (34) invoked in Step 4.
The functionalweakly converges into G defined bywhere.
Recall that where
Hence, since in , it remains to prove that in where . According to (13) in the interval , we have6
To simplify the notation we write for the function .
Hence
On the other hand, from (13), and hypothesis (H4), we clearly deduce that where C is a nonnegative constant depending only on and . Consequently, applying the Lebesgue dominated convergence theorem and (H4), we infer that
Passing to the limit in (36) we deduce that strongly in . The conclusion of Lemma 4.2 follows from the fact that weakly in . □
In some cases the diffusion term of the limit problem is a single valued subdifferential as seen in Section 5 in the framework of homogenization (see condition () on the Fenchel conjugate of ). The assumption of Gâteaux-differentiability on the density is made to simplify the notation in the proofs. Therefore, strictly speaking, Theorem 4.1 is not a result of stability since the limit problem is a differential inclusion. This is due to the fact that the graph limit of a sequence of single valued maximal monotone operator may be multivalued. Nevertheless, Theorem 4.1 can be generalized without difficulty to differential inclusions : just modify (D2,n) and (D3,n) as follows
for a.e. and every , the function is convex with ;
there exists such that for all , all all and for a.e. :
Under thees assumption, one obtain a stability result.
In some cases, we can specify the domain of the limit functional Φ as in the proposition below.
Let denote bythe domain of the Mosco-limit Φ of the sequence. Then we have
iffor all, then,
if,for thetopology, andweakly in, then.
Let us establish the first assertion. Let , then from , there exists strongly in such that . From the uniform lower growth condition of , and hypotheses (H1) and (H2), we infer that for any ,
Hence choosing ν such that , we obtain for some constant ,
and, finally, from (H3),
Therefore, there exists a subsequence, that we do not relabel, and satisfying weakly in and strongly in . Hence .
On the other hand, for , according to and the growth condition, one has
from which we infer the second assertion. □
For each let us write as follows:
where is defined by . The following result gives sufficient conditions for the Mosco-convergence of when we assume that Γ-converges to with respect to the topology.
Assume that
there existandsuch that the sequencesatisfies
(D
1,n
)
withandfor all;
Γ-converges towhenis equipped with the strong convergence of;
strongly in;
strongly in.
Thenwhereis given by
The proof fall into two steps.
Step 1. Let weakly in , we establish that .
We assume that and we reason with various subsequences that we do not relabel. Moreover C denotes various positive constants. From (), the uniform lower bound of , and the continuity of the trace operator, we have, for ,
Hence
Therefore, choosing , we deduce that
Consequently, there exist a subsequence and such that weakly in and strongly in . Thus so that and strongly in . According to (), we infer that
On the other hand
According to the weak continuity of the trace operator from into and to the lower semicontinuity of the map , we infer that
Finally, since strongly in , and weakly in , we have
The proof of the claim is obtained by collecting (37), (38), and (39).
Step 2. Assume that . We prove that there exists a sequence strongly converging to v in such that .
Since , we infer that , and, according to hypothesis (), there exists a sequence in strongly converging to v in , such that
By using the well known De Giorgi slicing method (see [8, proof of Corollary 11.2.1], it is precisely at this point that we use the uniform growth condition), we can modify into a function in satisfying on and
(see proof of [8, Corollary 11.2.1]). Then clearly , which proof the claim. □
Proposition 4.2 leads straight to the following corollary of Theorem 4.1 which is applied in Theorem 5.1 below.
Under hypotheses of Theorem
4.1
where
(H
1
)
is replaced by (), the same conclusions hold.
We can, in some sense, justify our convention which consists to see the functional
as a particular case of
with and
For this purpose we apply suitably Theorem 4.1. Set and . We have
On the other hand, set and , , . The conditions on become . We claim that Mosco-converges to .
Consider a sequence satisfying strongly in and . In what follows, we reason with various subsequences that we do not relabel. From
we infer that
On the other hand, from
and the lower bound condition of W, we deduce that the sequence is bounded in (recall that in ). Therefore weakly in , and, according to the continuity of the trace operator from into , weakly in . From (40) we infer that in Γ, hence and . Since for all ,
we deduce that .
Take now (otherwise we have nothing to prove), and set . Since , we have , which proves the claim.
Since all other conditions of Theorem 4.1 are fulfilled, we deduce that problem with mixed Dirichlet–Neumann boundary conditions
converges in the sense of Theorem 4.1, to problem with homogeneous Dirichlet–Neumann boundary conditions
Application to stochastic homogenization
The behavior of heterogeneous media in physics or mechanics has been thoroughly analyzed from a mathematical perspective through the framework of homogenization. In this context, diffusion problems with periodic heterogeneities are now well understood, and diffusion in random media has been fairly well analyzed in [20,21,25,26], or [19], [8, Sections 17.4.4, 17.4.5] and references therein, where the diffusion operator is the subdifferential of a random energy.
By contrast, homogenization of reaction–diffusion problems modeling for example biological invasion in the context of food limited population dynamics, does not seem to be addressed. The interplay between environment heterogeneities in the individual evolution of propagation species, plays an essential role. Indeed, empirical observations suggest that growth rates, or various thresholds which appear in the models, are mostly influenced by the environment, and vary in each small habitats (forests, marshes, hedges, etc.). Most of the time, these heterogeneities appear very small compared with the dimension of the domain, and statistically, are homogeneously distributed. Therefore both diffusion and reaction parts in the problems modeling the propagation, present random coefficients and a small parameter ε which accounts for the dimension of heterogeneities. To identify the effective coefficients (effective growth rate, various effective thresholds etc.), the purpose of this section is to describe the equivalent homogenized problem when ε goes to zero. The procedure consists in applying Theorem 4.1 in Section 4.
Probabilistic setting
For any topological space , we denote by its Borel σ-field, and we return to the basic concepts of [8, Section 12.4.3] (see also references therein) concerning ergodic dynamic systems. Let be a probability space. Let be a group of P-preserving transformations on Σ, i.e., for all , the map is measurable and satisfies , where we use the standard notation to denote the image measure (or push forward) of P by . We denote by the σ-algebra of invariant sets of by the group and, for every h in the space of P-integrable functions, by the conditional expectation of h with respect to , i.e., the unique -measurable function in satisfying for every
If is made up of sets with probability 0 or 1, the discrete dynamical system is said to be ergodic. Under this condition, we have where is the mathematical expectation of h.
A sufficient condition to ensure ergodicity is the so called mixing condition which expresses an asymptotic independence: for all sets E and F of
Ergodicity is indeed obtained from (41) by taking in . In what follows we will also need the following technical standard results.
Invariance and-measurability. A function is -measurable if and only if it is invariant under the group , i.e., for all . For implication
the claim is indeed the straightforward consequence of
The other implication is immediate.
The conditional Lebesgue dominated convergence theorem. Let be a sequence in such that , P-a.s. in Σ, and assume that there exists such that for all . Let be a sub σ-algebra of , then , P-a.s. in Σ. The proof follows a similar method as in the proof of the standard Lebesgue dominated convergence theorem, using the conditional Fatou Lemma instead of the standard Fatou Lemma.
In the next two sections, is a given discrete dynamical system.
The random diffusion part
Given , , and , we denote by the class of functions , , satisfying conditions (D1,n), (D2,n), and (D3,n). The class is endowed with the σ-algebra which is the trace of the product σ-algebra of , i.e., the smallest σ-algebra on such that all the evaluation maps
are measurable.
We consider a random convex integrand , i.e., a -measurable function such that for every , the function , belongs to the class . Since is -measurable for all , the map , , is -measurable, and we denote by its law, that is .
We assume that W satisfies the following covariance property with respect to the dynamical system : for all
For all g in and all , let us set . This defines a group which acts on , and clearly, is -measurable for all . Then it is easy to show that the covariance property implies that the law of is invariant under the group , that is for all . We say that the random function W is periodic in law.
We write ε to denote a sequence of positive numbers with , which is denoted by . Then, the following random functional defined by
models a random energy concerning various steady-states situations, where the small parameter ε accounts the size of the randomly distributed heterogeneities in the context of a statistically homogeneous media. The measurability of for all may be obtained by standard arguments (see for instance [8, Section 12.4.3 and Proposition 12.4.1]).
Under the hypotheses above on with respect to the discrete dynamical system , it is now standard, using the subadditive ergodic theorem ([19] or [8, Theorem 12.4.3]), that for P-a.s. ω in Σ the sequence Γ-converges to the integral functional , where is given by
when is equipped with its strong convergence. Let Y denote the unit cell , then, for every , the density is given, for P-a.s. , by
If is ergodic, then is deterministic and given for P-a.s. by
For a proof we refer the reader to [8, Proposition 12.4.3, Theorem 12.4.7] and references therein.
Given , and , , on with , for some , we consider the random functionals and defined by
and
According to Lemma 3.1, for P-a.s. , the subdifferential of (actually its Gâteaux derivative) is the operator defined for every by
and, for all ,
Similarly the subdifferential of is the operator defined for every by
and, for all ,
When W is ergodic, then is deterministic and
Note that we can not guarantee a priori that is Gâteaux-differentiable, hence is possibly multivalued. Nevertheless, to shorten the notation, we write indifferently to denote the subdiffrential or any of its elements. We emphasize the fact that is the P-a.s. graph limit of the operator , and that under the following additional condition on the Fenchel conjugate of , the density is Gâteaux differentiable for P a.e. , so that is univalent and reduced to a pointwise limit:
there exists such that for P a.e. , for a.e. , for all and all .
For a proof, we refer the reader to [8, Proposition 17.4.6].
The random reaction part
We consider a regular CP-structured random functional, i.e., a functional
defined by where
is a -measurable function such that for P-a.s. , is a regular CP-structured reaction function associated with . Furthermore, we make the following additional hypotheses on r and q: we assume that , , that for all bounded Borel sets B of , the real valued functions
belong to , and that r and q, satisfy the covariance property with respect to the dynamical system , i.e., that for all , all , a.e. and P-a.s. ,
We set , and define the functional by . Note that in the expression of the condition (CP), the functions , , , , and the numbers , may depend on ω (we sometimes omit it to shorten the notation), and that is a CP-structured reaction functional whose condition (CP) is exactly that of F, i.e., with the functions , , , , and . Since and do not depend on ε, condition (15) is automatically satisfied. We assume that condition (16) or condition (17) is satisfied, i.e., or . Let us show that (14) holds for P-a.s. . This condition is a straightforward consequence of the following more accurate result.
There existsinwithsuch that for all, we have
We only prove (49) and (50), the proof of (47) and (48) is similar. Consider the set function from the class of bounded Borel subsets of into the space of P-integrable real valued functions, defined by
From (45), the process is well defined. Then, for every with , from additivity of the integral we have
Moreover, from (46) we deduce that
Furthermore, fulfills the following domination property: for all Borel set A included in Y,
with . Therefore, is an additive process indexed by , covariant with respect to (see [8, Definition 12.4.1] and references therein). According to the additive ergodic theorem (see [8, Theorem 12.4.1]), there exists with such that for all7
Strictly speaking the almost sure convergence holds when Ω is a convex set. Using approximation of Ω by finite union of convex subset, it is easy to show that the convergence holds for regular Ω of class (see [15, Remark 3.3]).
,
Hence, a change of scale gives
We obtain (50) by combining (49) with
This completes the proof. □
General homogenization theorem for a class of nonlinear reaction–diffusion equations
Given a sequence of -measurable functions , by combining Theorem 4.1 of the previous section together with the variational convergence of the sequence of random energies specified above, we intend to analyze the asymptotic behavior in of the solution of the random reaction–diffusion problem when :
For each, let us denote bythe unique solution inof the (random) reaction–diffusion problem. Assume that forP-a.s.,strongly converges toin, and that. Then, forP-a.s.,uniformly converges into the unique solution of the reaction–diffusion problemwhereis given bywithMoreover, forP-a.s.,weakly inand.
When the dynamical systemis ergodic, the initial condition is deterministic, i.e.,forP-a.s., together with,,, and, thenis deterministic and is given bywhereis given bywithMoreover, forP-a.s.,weakly inand.
If in addition W satisfies (), thenorare univalent equal toor, and the differential inclusions are equalities.
The proof is a straightforward consequence of Theorem 4.1 and consists in checking (H1), (H5) and (H6). In the whole proof, we reason with the set of full probability where is the P-negligible set given by Lemma 5.1.
Proof of
(H
1
)
: . According to [8, Theorem 12.4.7] in the scalar version, we deduce that for P-a.s. ω in , the sequence of functional , defined in Proposition 4.2, Γ-converges to the random integral functional when is endowed with the strong convergence. We can conclude by using Proposition 4.2.
Proof of
(H
5
)
: We have to establish that for P-a.s. , and in .
The first claim is obvious. To show that in we need the following lemma.
There existswith, such that for everyand every,weakly in.
Fix . From (42) we can apply [15, Theorem 4.2]), straightforward consequence of the additive ergodic theorem (see [8, Theorem 12.4.1]): there exists with such that for every
weakly in . Set . We are going to show that for all , the weak convergence , holds for all . Let , , and be a sequence in converging to t with . We have
with, from the weak convergence above, . Let us set . Since , we infer that
Letting , then in (51), from (52) and (47) of Lemma 5.1, we deduce that
which ends the proof of Lemma 5.2 provided that we justify the convergence
which is a straightforward consequence of the continuity of and the conditional Lebesgue dominated convergence theorem. □
Proof of
(H
5
)
continued. Fix . Let . According to Lemma 5.2, for all we have
and the conclusion follows from the Lebesgue dominated convergence theorem. Indeed, the domination property is obtained as follows: we have
where
and, from (48),
Proof of
(H
6
)
. First, we have to prove that for P-a.s. , and for all
For this, by reproducing the proof of Lemma 5.1, and using this time the additive ergodic theorem for the process , which is well defined according to (43), we easily obtain that there exists with such that for all ,
Then for all and all we have
and the claim follows from (53) and (50). The rest of the proof concerning the weak convergence is exactly that of condition (H5). □
Footnotes
Examples of CP-structured reaction functionals
Examples of stochastic homogenization of a diffusive Fisher food limited population model with Allee effect
As an application of Theorem 5.1, we treat the stochastic homogenization of the reaction–diffusion problem describing the food limited population model whose reaction function corresponds to that of the Fisher model with Allee effect (see Examples A.1 of Appendix A). We assume that the growth rate r, along with the critical threshold a below which the per-capita growth rate turns negative, are influenced by the heterogeneities of the spatial environment and change in each small habitats. However we assume that the carrying capacity K is constant. In a first example, we assume that the distribution of the heterogeneities is following a regular random patch model, i.e., in the probabilistic setting, the dynamical system is that of a random checkerboard-like environment. In a second example, the heterogeneities are distributed following a Poisson point process. In the two examples, in order to simplify the model, we assume that r and a do not depend on the time variable t. Otherwise, it would be sufficient to make the appropriate assumptions concerning the absolute continuity on r and a with respect to the time variable, without changing the constructions below. It is interesting to note that the homogenized critical density is now a function of the growth rate. To shorten the notation we assume that the Fenchel conjugate of satisfies ().
Proof of Lemma 4.1
Since , ψ are uniformly proper, according to [8, Lemma 17.4.5], there exists such that and so that the integrals and Ψ are well defined for all .
Furthermore, for sequences of convex proper and lower semicontinuous functions from a reflexive Banach X spaces into , where X is as in the assumptions, there is equivalence between the Mosco-convergence and convergence of the sequences of the Moreau–Yosida approximations (see [7, Theorem 3.26] or [18]). We are going to apply this result to the spaces X and which fulfill these conditions and to the functionals , Ψ, , ψ which are convex proper and lower semicontinuous.
Step 1. Denote by , , , and the Moreau–Yosida approximation of index of , ψ, , and Ψ respectively (for the definition and properties of Moreau–Yosida approximation see [8, Proposition 17.2.1]). For every , we have
This result is an elementary case of interchange of infimum and integral (see for instance [4] and references therein).
Step 2. We claim that if then . We have (see [7, Theorem 3.24])
Let . Assume that . Then from above, for a.e. and for all , we have .
Let . Since , there exists a sequence in X such that and . Then there exists which depends only on and such that for all
where belongs to . Then, according to the Lebesgue dominated convergence theorem, we deduce that for all
that is, from Step 1, , which ends the proof since u is arbitrary chosen in .
References
1.
G.Allaire, A.Mikelic and A.Piatnitski, Homogenization approach to the dispersion theory for reactive transport through porous media, SIAM J. Math. Anal.42 (2010), 125–144. doi:10.1137/090754935.
2.
G.Allaire, I.Pankratova and A.Piatnitski, Homogenization of a nonstationary convection-diffusion equation in a thin rod and in a layer, SeMA Journal58(1) (2012), 53–95. doi:10.1007/BF03322605.
3.
G.Allaire, I.Pankratova and A.Piatnitski, Homogenization and concentration for a diffusion equation with large convection in a bounded domain, J. Funct. Anal.262 (2012), 300–330. doi:10.1016/j.jfa.2011.09.014.
4.
O.Anza Hafsa and J.-P.Mandallena, Interchange of infimum and integral, Calc. Var. Partial Differential Equations18(4) (2003), 433–449. doi:10.1007/s00526-003-0211-3.
5.
O.Anza Hafsa, J.-P.Mandallena and G.Michaille, Groupe de recherche Mathématiques en Cévennes. Convergence of a class of nonlinear reaction–diffusion equations and stochastic homogenization. http://mipa.unimes.fr/preprints/MIPA-Preprint03-2016.pdf.
6.
S.N.Armstrong and P.E.Souganidis, Concentration phenomena for neutronic multigroup diffusion in random environments, Ann. Inst. H. Poincaré Anal. Non Linéaire30 (2013), 419–439. doi:10.1016/j.anihpc.2012.09.002.
7.
H.Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.
8.
H.Attouch, G.Buttazzo and G.Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, 2nd edn, MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2014.
9.
R.B.Banks, Growth and Diffusion Phenomena: Mathematical Frameworks and Applications, Texts in Applied Mathematics, Vol. 14, Springer-Verlag, Berlin, 1994.
10.
H.Berestycki and P.-L.Lions, Some applications of the method of super and subsolutions, in: Bifurcation and Nonlinear Eigenvalue Problems, Proc., Session, Univ. Paris XIII, Villetaneuse, 1978, Lecture Notes in Math., Vol. 782, Springer, Berlin, 1980, pp. 16–41. doi:10.1007/BFb0090426.
11.
H.Berestycki and P.-L.Lions, Une méthode locale pour l’existence de solutions positives de problèmes semi-linéaires elliptiques dans , J. Analyse Math.38 (1980), 144–187. doi:10.1007/BF03033880.
12.
N.Bouleau, Processus Stochastiques et Applications. Collection Méthodes, Hermann, 2000.
13.
H.Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, Vol. 5, North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1973, Notas de Matemática, 50.
14.
H.Brézis and A.Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, Journal of Functional Analysis9 (1972), 63–74. doi:10.1016/0022-1236(72)90014-6.
15.
E.Chabi and G.Michaille, Ergodic theory and application to nonconvex homogenization, Set-Valued Anal.2(1–2) (1994), 117–134. doi:10.1007/BF01027096.
16.
D.Cioranescu and A.Piatnitski, Homogenization of porus medium with randomly pulsating microstructure, SeMA Journal58(1) (2012), 53–95. doi:10.1007/BF03322605.
17.
B.Dacorogna, Direct Methods in the Calculus of Variations, Appl. Math. Sci., Vol. 78, Springer-Verlag, Berlin, 1989.
18.
G.Dal Maso, An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations and Their Applications, Vol. 8, Birkhäuser Boston Inc., Boston, MA, 1993.
19.
G.Dal Maso and L.Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math.363 (1986), 27–43.
20.
M.Duerinckx and A.Gloria, Stochastic homogenization of nonconvex unbounded integral functionals with convex growth, Archive for Rational Mechanics and Analysis221(3) (2016), 1511–1584. doi:10.1007/s00205-016-0992-0.
U.Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math.3 (1969), 510–585. doi:10.1016/0001-8708(69)90009-7.
23.
U.Mosco, On the continuity of the Young–Fenchel transform, J. Math. Anal. Appl.35 (1971), 518–535. doi:10.1016/0022-247X(71)90200-9.
24.
C.V.Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
25.
G.C.Papanicolaou, Diffusion in random media, in: Surveys in Applied Mathematics, J.B.Keller, D.W.McLaughlin and G.C.Papanicolaou, eds, Springer, Boston, MA, 1995. doi:10.1007/978-1-4899-0436-2.
26.
G.C.Papanicolaou and S.R.S.Varadhan, Boundary value problems with rapidly oscillating random coefficients, in: Proceedings of Conference on Random Fields, Vol. 27, Esztergom, Hungary, 1979, pp. 835–873.
27.
B.Perthame and P.E.Souganidis, A homogenization approach to flashing ratchets, Nonlinear Differ. Equ. Appl.18 (2011), 45–58. doi:10.1007/s00030-010-0083-0.
28.
S.I.Pohožaev, The Dirichlet problem for the equation , Soviet Math. Dokl.1 (1960), 1143–1146.
29.
S.Ruan, Delay differential equations in single species dynamics, in: Delay Differential Equations and Applications, NATO Sci. Ser. II Math. Phys. Chem., Vol. 205, Springer, Dordrecht, 2006, pp. 477–517. doi:10.1007/1-4020-3647-7_11.
30.
A.Tsoularis, Analysis of logistic growth models, in: Research Letters in the Information and Mathematical Sciences, Vol. 2, 2001, pp. 23–46, Massey University.
