A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions
Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 1, p. 17-39
@article{AIHPC_2007__24_1_17_0,
     author = {El Soufi, Ahmad and Jazar, M. and Monneau, R\'egis},
     title = {A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {24},
     number = {1},
     year = {2007},
     pages = {17-39},
     doi = {10.1016/j.anihpc.2005.09.005},
     zbl = {1112.35108},
     mrnumber = {2286557},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2007__24_1_17_0}
}
El Soufi, A.; Jazar, M.; Monneau, R. A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 1, pp. 17-39. doi : 10.1016/j.anihpc.2005.09.005. http://www.numdam.org/item/AIHPC_2007__24_1_17_0/

[1] Alberti G., Variational models for phase transitions, an approach via Γ-convergence, in: Buttazzo G., (Eds.), Calculus of Variations and Partial Differential Equations. Topics on Geometrical Evolution Problems and Degree Theory. Based on a Summer School, Pisa, Italy, September 1996, Springer, Berlin, 2000, pp. 95-114. | Zbl 0957.35017

[2] Amann H., Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations 72 (1988) 201-269. | MR 932367 | Zbl 0658.34011

[3] Avinyo A., Mora X., Geometric inequalities of Cheeger type for the first positive eigenvalue of the n-dimensional free membrane problem, Ital. J. Pure Appl. Math. 2 (1997) 133-140. | MR 1770591 | Zbl 0948.35089

[4] Ball J., Remarks on blow-up and non-existence theorems for nonlinear evolution equations, Quart. J. Math. Oxford 28 (1977) 473-486. | MR 473484 | Zbl 0377.35037

[5] Budd C., Dold B., Stuart A., Blow-up in a partial differential equation with conserved first integral, SIAM J. Appl. Math. 53 (3) (1993) 718-742. | MR 1218381 | Zbl 0784.35009

[6] Budd C., Dold B., Stuart A., Blow-up in a system of partial differential equations with conserved first integral II. Problems with convection, SIAM J. Appl. Math. 54 (3) (1994) 610-640. | MR 1274219 | Zbl 0928.35081

[7] Buser P., On Cheeger’s inequality λ 1 h 2 /4, in: Geometry of the Laplace Operator, Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979, Proc. Sympos. Pure Math., vol. XXXVI, Amer. Math. Soc., 1980, pp. 29-77. | MR 573428 | Zbl 0432.58024

[8] Cazenave T., Haraux A., Introduction aux problèmes d'évolutions semi-linéaires, Ellipses, Paris, 1990. | MR 1299976 | Zbl 0786.35070

[9] Dal Maso G., An Introduction to Γ-Convergence, Progr. Nonlinear Differential Equations Appl., vol. 8, Birkhäuser, Basel, 1993. | Zbl 0816.49001

[10] Davies E.B., Heat Kernels and Spectral Theory, Cambridge Tracts in Math, vol. 92, Cambridge University Press, Cambridge, 1990. | MR 1103113 | Zbl 0699.35006

[11] De Giorgi E., New problems in Gamma-convergence and G-convergence, in: Free Boundary Problems, Proc. Semin. Pavia, 1979, vol. II, 1980, pp. 183-194. | Zbl 0465.35002

[12] Elliott C.M., The Cahn-Hilliard model for the kinetics of phase separation, in: Rodrigues J. (Ed.), Mathematical Models for Phase Change Problems, Birkhäuser-Verlag, Basel, 1989. | Zbl 0692.73003

[13] Fujita H., On the blowing-up of solutions of the Cauchy problem for u t =Δu+u 1+α , J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966) 109-124. | MR 214914 | Zbl 0163.34002

[14] Furter J., Grinfeld M., Local vs. non-local interactions in population dynamics, J. Math. Biol. 27 (1989) 65-80. | MR 984226 | Zbl 0714.92012

[15] Giga Y., Kohn R.V., Characterizing blow-up using similarity variables, Indiana Univ. Math. J. 36 (1987) 1-40. | MR 876989 | Zbl 0601.35052

[16] Gilkey P., Branson T., The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (2) (1990) 245-272. | MR 1032631 | Zbl 0721.58052

[17] Giusti E., Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, Boston, 1984. | MR 775682 | Zbl 0545.49018

[18] Herrero M.A., Velázquez J.J.L., Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (2) (1993) 131-189. | Numdam | MR 1220032 | Zbl 0813.35007

[19] Hu B., Yin H.M., Semi linear parabolic equations with prescribed energy, Rend. Circ. Math. Palermo 44 (1995) 479-505. | MR 1388759 | Zbl 0856.35063

[20] Leray J., Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. J. 63 (1934) 193-248. | JFM 60.0726.05 | MR 1555394

[21] Levine H.A., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu t =-Au+Fu, Arch. Rational Mech. Anal. 51 (1973) 371-386. | MR 348216 | Zbl 0278.35052

[22] Lieberman G.M., Second Order Parabolic Partial Differential Equations, World Scientific, 1996. | MR 1465184 | Zbl 0884.35001

[23] Lunardi A., Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser-Verlag, 1995. | MR 1329547 | Zbl 0816.35001

[24] Merle F., Zaag H., A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann. 316 (2000) 103-137. | MR 1735081 | Zbl 0939.35086

[25] Meyer D., Minoration de la première valeur propre non nulle du problème de Neumann sur les variétés riemanniennes à bord, Ann. Inst. Fourier (Grenoble) 36 (2) (1986) 113-125. | Numdam | MR 850747 | Zbl 0582.58036

[26] Modica L., The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (2) (1987) 123-142. | MR 866718 | Zbl 0616.76004

[27] Modica L., Mortola S., Un esempio di Γ-convergenza, Boll. Un. Mat. Ital. B (5) 14 (1977) 285-299. | Zbl 0356.49008

[28] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol. 44, Springer-Verlag, 1983. | MR 710486 | Zbl 0516.47023

[29] Rubinstein J., Sternberg P., Non-local reaction-diffusion equations and nucleation, IMA J. Appl. Math. 48 (3) (1992) 249-264. | Zbl 0763.35051

[30] Souplet Ph., Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 29 (6) (1998) 1301-1334. | Zbl 0909.35073

[31] Souplet Ph., Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations 153 (1999) 374-406. | MR 1683627 | Zbl 0923.35077

[32] Souplet Ph., Recent results and open problems on parabolic equations with gradient nonlinearities, E.J.D.E. 10 (2001) 1-19. | MR 1824790 | Zbl 0982.35054

[33] Stewart H.B., Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1) (1980) 299-310. | MR 561838 | Zbl 0451.35033

[34] Szegö G., Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954) 343-356. | MR 61749 | Zbl 0055.08802

[35] Wang J., Global heat kernel estimates, Pacific J. Math. 178 (2) (1997) 377-398. | MR 1447421 | Zbl 0882.35018

[36] Wang M., Wang Y., Properties of positive solutions for non-local reaction-diffusion problems, Math. Methods Appl. Sci. 19 (1996) 1141-1156. | Zbl 0990.35066

[37] Weinberger H.F., An isoperimetric inequality for the N-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956) 633-636. | MR 79286 | Zbl 0071.09902

[38] H. Zaag, Sur la description des formations de singularités pour l'équation de la chaleur non linéaire, PhD Thesis, Univ. of Cergy-Pontoise, France, 1998.

[39] Zaag H., On the regularity of the blow-up set for semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002) 505-542. | Numdam | MR 1922468 | Zbl 1012.35039