Similarity stabilizes blow-up
Journées équations aux dérivées partielles (1999), article no. 12, 7 p.

The blow-up of solutions to a quasilinear heat equation is studied using a similarity transformation that turns the equation into a nonlocal equation whose steady solutions are stable. This allows energy methods to be used, instead of the comparison principles used previously. Among the questions discussed are the time and location of blow-up of perturbations of the steady blow-up profile.

@article{JEDP_1999____A12_0,
     author = {Schochet, Steve},
     title = {Similarity stabilizes blow-up},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {12},
     publisher = {Universit\'e de Nantes},
     year = {1999},
     mrnumber = {1718994},
     zbl = {01810585},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_1999____A12_0/}
}
Schochet, Steve. Similarity stabilizes blow-up. Journées équations aux dérivées partielles (1999), article  no. 12, 7 p. http://www.numdam.org/item/JEDP_1999____A12_0/

[BG] J. Bebernes and V. A. Galaktionov : On classification of blow-up patterns for a quasilinear heat equation, Differential and Integrals Eqs., Vol 9, (1996), p. 655-670. | MR 97e:35077 | Zbl 0851.35057

[CDE] C. Cortázar, M. Del Pino, and M. Elgueta : On the blow-up set for ut = ∆um + um, m > 1, Indiana U. Math. J., Vol 47, (1998), p. 541-561. | MR 99h:35085 | Zbl 0916.35056

[CEF] C. Cortázar, M. Elgueta, and P. Felmer : Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation, Commun. Partial Differential Equations, Vol. 21, (1996), p.507-520. | MR 97d:35053 | Zbl 0854.35033

[LR] D. Levy and P. Rosenau : On a class of thermal blow-up patterns, Physics Letters A, Vol. 236, (1997), p. 483-493. | MR 1489695 | Zbl 0969.35520

[SGKM] A.A Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov : Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin (1995). | MR 96b:35003 | Zbl 00764718

[S] S. Schochet : Similarity stabilizes blow-up in quasilinear parabolic equations with balanced nonlinearity, in preparation.