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.
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.
