Blow-up behavior in nonlocal vs local heat equations
Souplet, Philippe
Banach Center Publications, Tome 51 (2000), p. 221-226 / Harvested from The Polish Digital Mathematics Library

We present some recent results on the blow-up behavior of solutions of heat equations with nonlocal nonlinearities. These results concern blow-up sets, rates and profiles. We then compare them with the corresponding results in the local case, and we show that the two types of problems exhibit "dual" blow-up behaviors.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:209060
@article{bwmeta1.element.bwnjournal-article-bcpv52z1p221bwm,
     author = {Souplet, Philippe},
     title = {Blow-up behavior in nonlocal vs local heat equations},
     journal = {Banach Center Publications},
     volume = {51},
     year = {2000},
     pages = {221-226},
     zbl = {0954.35091},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv52z1p221bwm}
}
Souplet, Philippe. Blow-up behavior in nonlocal vs local heat equations. Banach Center Publications, Tome 51 (2000) pp. 221-226. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv52z1p221bwm/

[000] [BBL] J. Bebernes, A. Bressan and A. Lacey, Total blow-up versus single-point blow-up, J. Differ. Equations 73 (1988), 30-44. | Zbl 0674.35051

[001] [BK] J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity 7 (1994), 539-575. | Zbl 0857.35018

[002] [CM] X. Y. Chen and H. A. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differ. Equations 78 (1989), 160-190. | Zbl 0692.35013

[003] [D] K. Deng, Nonlocal nonlinearity versus global blow-up, Math. Applicata (1995), 124-129.

[004] [FM] A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447. | Zbl 0576.35068

[005] [GK1] Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 8 (1985), 297-319. | Zbl 0585.35051

[006] [GK2] Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40. | Zbl 0601.35052

[007] [GK3] Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equation, Comm. Pure Appl. Math. 42 (1989), 845-884. | Zbl 0703.35020

[008] [HV] M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Annali Sc. Norm. Sup. Pisa 19, 3 (1992), 381-450. | Zbl 0798.35081

[009] [MZ1] F. Merle and H. Zaag, Stability of blow-up profile for equation of the type ut=Δu+|u|p-1u, Duke Math. J. 86 (1997), 143-195. | Zbl 0872.35049

[010] [MZ2] F. Merle and H. Zaag, Optimal estimates for blow-up rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), 139-196.

[011] [MW] C. E. Mueller and F. B. Weissler, Single point blow-up for general semilinear heat equation, Indiana Univ. Math. J. 34 (1985), 881-913. | Zbl 0597.35057

[012] [S1] Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 29 (1998), 1301-1334. | Zbl 0909.35073

[013] [S2] Ph. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differ. Equations 153 (1999), 374-406. | Zbl 0923.35077

[014] [S3] Ph. Souplet, Some blow-up results for nonlocal reaction-diffusion equations, in: Actes du 3ème Congrès Européen sur les problèmes elliptiques et paraboliques (Pont-à-Mousson, juin 1997), Pitman Research Notes in Mathematics Series, n° 384, Addison Wesley Longman, 1998.

[015] [V1] J. J. L. Velázquez, Classification of singularities for blowing-up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1993), 441-464. | Zbl 0803.35015

[016] [V2] J. J. L. Velázquez, Estimates on the (n-1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42 (1993), 445-476. | Zbl 0802.35073

[017] [V3] J. J. L. Velázquez, Blow up for semilinear parabolic equations, in: Recent advances in partial differential equations, M. A. Herrero et al. (eds.), Res. Notes Appl. Math. 30, Masson, Paris, 1993, 131-145.

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

[019] [W1] F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differ. Equations 55 (1984), 204-224. | Zbl 0555.35061

[020] [W2] F. B. Weissler, An L blow-up estimate for a nonlinear heat equation, Comm. Pure Appl. Math. 38 (1985), 291-295. | Zbl 0592.35071