Convergence of a class of nonlinear time delays reaction-diffusion equations

Abstract

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

Convergence of nonlinear integrodifferential reaction-diffusion equations via Mosco×Γ-convergence

“…In the spirit of [2,4], we investigate the compactness or the stability in terms of convergence, of integrodifferential reaction-diffusion problems defined in 𝐿 2 (0, 𝑇, 𝑋) by under suitable variational convergences on the classes of functionals Φ and Ψ (see Theorem 5.1). As usually the integral in the first member is taken in the sense of Bochner.…”