Homogenization of a semilinear heat equation

Cesaroni, Annalisa; Dirr, Nicolas; Novaga, Matteo

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 633-660.

Nous considérons l’homogénéisation d’une équation de la chaleur semi-linéaire avec viscosité tendant vers $0$ et un potentiel positif oscillant dépendant de $u/\epsilon$. Suivant le rapport entre la fréquence des oscillations dans le potentiel et le facteur tendant vers $0$ dans la viscosité, nous obtenons différents régimes de l’évolution limite et nous discutons la convergence uniforme locale des solutions du problème effectif. L’aspect intéressant du modèle est que, dans un régime à forte diffusion, l’opérateur effectif est discontinu comme fonction du gradient. Nous obtenons une caractérisation complète de la solution limite en dimension $n=1$, alors qu’en dimension $n>1$ nous analysons les propriétés principales des solution du problème effectif sélectionné à la limite, et nous montrons l’unicité pour certaines classes de données initiales.

We consider the homogenization of a semilinear heat equation with vanishing viscosity and with oscillating positive potential depending on $u/\epsilon$. According to the rate between the frequency of oscillations in the potential and the vanishing factor in the viscosity, we obtain different regimes in the limit evolution and we discuss the locally uniform convergence of the solutions to the effective problem. The interesting feature of the model is that in the strong diffusion regime the effective operator is discontinuous in the gradient entry. We get a complete characterization of the limit solution in dimension $n=1$, whereas in dimension $n>1$ we discuss the main properties of the solutions to the effective problem selected at the limit and we prove uniqueness for some classes of initial data.

Classification : 35B27, 74Q10, 35D40
Keywords: Homogenization, parabolic equations, vanishing viscosity
Mot clés : Homogénéisation, équations paraboliques, viscosité tendant vers $0$

[1] Alibaud, N.; Briani, A.; Monneau, R. Diffusion as a singular homogenization of the Frenkel-Kontorova model, J. Differential Equations, Volume 251 (2011) no. 4-5, pp. 785-815 | DOI | MR | Zbl

[2] Bardi, M.; Capuzzo-Dolcetta, I. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997 | Zbl

[3] Barles, G. Some homogenization results for non-coercive Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, Volume 30 (2007) no. 4, pp. 449-466 | DOI | MR | Zbl

[4] Caffarelli, L. A.; Monneau, R. Counter-example in three dimension and homogenization of geometric motions in two dimension, Arch. Rational Mech. Anal., Volume 212 (2014) no. 2, pp. 503-574 | DOI | MR | Zbl

[5] Cardaliaguet, P.; Lions, P.-L.; Souganidis, P. E. A discussion about the homogenization of moving interfaces, J. Math. Pures Appl. (9), Volume 91 (2009) no. 4, pp. 339-363 | DOI | MR | Zbl

[6] Cesaroni, A.; Dirr, N.; Novaga, M. Asymptotic speed of propagation for a viscous semilinear parabolic equation, Gradient flows: from theory to application (ESAIM Proceedings and Surveys), Volume 54, Société de Mathématiques Appliquées et Industrielles & EDP Sciences, Paris, 2016, pp. 45-53 | MR | Zbl

[7] Cesaroni, A.; Novaga, M. Long-time behavior of the mean curvature flow with periodic forcing, Comm. Partial Differential Equations, Volume 38 (2013) no. 5, pp. 780-801 | DOI | MR | Zbl

[8] Cesaroni, A.; Novaga, M.; Valdinoci, E. Curve shortening flow in heterogeneous media, Interfaces Free Bound., Volume 13 (2011) no. 4, pp. 485-505 | DOI | MR | Zbl

[9] Chen, X.; Namah, G. Periodic travelling wave solutions of a parabolic equation: a monotonicity result, J. Math. Anal. Appl., Volume 275 (2002) no. 2, pp. 804-820 | DOI | MR | Zbl

[10] Dirr, N.; Karali, G.; Yip, N. K. Pulsating wave for mean curvature flow in inhomogeneous medium, European J. Appl. Math., Volume 19 (2008) no. 6, pp. 661-699 | DOI | MR | Zbl

[11] Dirr, N.; Yip, N. K. Pinning and de-pinning phenomena in front propagation in heterogeneous media, Interfaces Free Bound., Volume 8 (2006) no. 1, pp. 79-109 | DOI | MR | Zbl

[12] Giga, Y. Surface evolution equations. A level set approach, Monographs in Mathematics, 99, Birkhäuser Verlag, Basel, 2006 | Zbl

[13] Ibrahim, H.; Monneau, R. On the rate of convergence in periodic homogenization of scalar first-order ordinary differential equations, SIAM J. Math. Anal., Volume 42 (2010) no. 5, pp. 2155-2176 | DOI | MR | Zbl

[14] Imbert, C.; Monneau, R. Homogenization of first-order equations with $(u/\epsilon )$-periodic Hamiltonians. I. Local equations, Arch. Rational Mech. Anal., Volume 187 (2008) no. 1, pp. 49-89 | DOI | MR | Zbl

[15] Jerrard, R. L. Singular limits of scalar Ginzburg-Landau equations with multiple-well potentials, Adv. Differential Equations, Volume 2 (1997) no. 1, pp. 1-38 | MR | Zbl

[16] Lions, P.-L.; Souganidis, P. E. Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 22 (2005) no. 5, pp. 667-677 | DOI | Numdam | MR | Zbl

[17] Lou, B. Periodic travelling wave solutions of a curvature flow equation in the plane, Tôhoku Math. J. (2), Volume 59 (2007) no. 3, pp. 365-377 | DOI | MR | Zbl

[18] Lou, B. Periodic traveling waves of a mean curvature flow in heterogeneous media, Discrete Contin. Dynam. Systems, Volume 25 (2009) no. 1, pp. 231-249 | DOI | MR | Zbl

[19] Namah, G.; Roquejoffre, J.-M. Convergence to periodic fronts in a class of semilinear parabolic equations, NoDEA Nonlinear Differential Equations Appl., Volume 4 (1997) no. 4, pp. 521-536 | DOI | MR

[20] Piccinini, L. C. Homogeneization problems for ordinary differential equations, Rend. Circ. Mat. Palermo (2), Volume 27 (1978) no. 1, pp. 95-112 | DOI | MR | Zbl

[21] Protter, M. H.; Weinberger, H. F. Maximum principles in differential equations, Springer-Verlag, New York, 1984 | DOI | Zbl