of a class of nonlinear reaction-diffusion equations and stochastic homogenization.Abstract. 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.that the derivative du dt of the solution belongs to L 2 (0, T * , X). Nevertheless √ t du dt ∈ L 2 (0, T * , X) (see [8, Theorem 17.2.5] or [13, Theorem 3.6]). Hence, for 0 < δ < T * , du dt belongs to L 2 (δ, T * , X). Set E := {T > δ : ∃u ∈ C ([0, T ], X) solution of (P)} .Since T * ∈ E, we have E = ∅. We define the maximal time in R + by T Max := sup E and denote by u the maximal solution of (P) in C ([0, T Max ), X). We have the following alternative:Theorem 2.2 (Global existence or blow-up in finite time). Assume that F satisfies (C 1 ), (C 2 ), then we have the blow-up alternative (G1) T Max = +∞ (existence of a global solution); (G2) T Max < +∞. In this case lim T →+T Max u C([0,T ],X) = +∞ (blow-up in finite time).Moreover, for all T , 0 < T < T Max , the restriction of u to [0, T ] satisfies assertions (L 1 ) and (L 2 ), and furthermore satisfies (Proof. We assume that T Max < +∞ and show that lim T →+T Max u C([0,T ],X) = +∞. We argue by contradiction. Assume that u does not fulfill lim T →+T Max u C([0,T ],X) = +∞, then there exist G > 0 and a sequence (T n ) n∈N in E such that T n → T Max and u C([0,Tn],X) ≤ G.Step 1. We show that lim t→T Max u (t) exists in X.Let n ∈ N. For a.e. t ∈ (0, T n ) we have
A necessary condition called H 1,p µ-quasiconvexity on p-coercive integrands is introduced for the lower semicontinuity with respect to the strong convergence of L p µ pX; R m q of integral functionals defined on Cheeger-Sobolev spaces. Under polynomial growth conditions it turns out that this condition is necessary and sufficient.
Stability under a variational convergence of nonlinear time delays reaction-diffusion equations is discussed. Problems considered cover various models of population dynamics or diseases in heterogeneous environments where delays terms may depend on the space variable. As a consequence a stochastic homogenization theorem is established and applied to vector disease and logistic models. The results illustrate the interplay between the growth rates and the time delays which are mixed in the homogenized model. Contents 1. Introduction Notation 2. The time-delays operator 2.1. Integration with respect to vector measures 2.2. Time delays-operator associated with vector measures 2.3. Examples of time-delays operators 3. Reaction diffusion problems associated with convex functionals of the calculus of variations and DCP-structured reaction functionals 3.1. The class of DCP-structured reaction functionals 3.2. Some examples of DCP-structured reaction functions coming from ecology and biology models 3.3. Existence and uniqueness of bounded nonnegative solution 4. General convergence theorems for a class of delays nonlinear reaction-diffusion problems 4.1. Stability at the limit 4.2. Non stability at the limit for the reaction functional: convergence with mixing effect between growth rates and time delays at the limit 5. Stochastic homogenization of distributed delays reaction diffusion problems 5.1. The random diffusion part 5.2. The random reaction part 5.3. Almost sure convergence to the homogenized reaction-diffusion problem Appendix A. Vector measures Appendix B. Notion of CP-structured reaction functionals Appendix C. Estimate of the right derivative Appendix D. An alternative proof of Theorem 4.1 in the case of a single time delay Appendix E. Appplication of Theorem 5.1 to some examples E.1. Homogenization of vector disease models E.2. Homogenization of delays logistic equations with immigration References
We establish a convergence theorem for a class of two components nonlinear reactiondiffusion systems. Each diffusion term is the subdifferential of a convex functional of the calculus of variations whose class is equipped with the Mosco-convergence. The reaction terms are structured in such a way that the systems admit bounded solutions, which are positive in the modeling of ecosystems. As a consequence, under a stochastic homogenization framework, we prove two homogenization theorems for this class. We illustrate the results with the stochastic homogenization of a prey-predator model with saturation effect.
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