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On the regularity of the blow-up set for semilinear heat equations

✍ Scribed by Hatem Zaag

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
293 KB
Volume
19
Category
Article
ISSN
0294-1449

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✦ Synopsis

We consider u(x, t) a blow-up solution of u t = u + |u| p-1 u where u : R N × [0, T ) → R, p > 1, (N -2)p < N + 2 and either u(0) 0 or (3N -4)p < 3N + 8. The blow-up set S ⊂ R N of u is the set of all blow-up points. Under a nondegeneracy condition, we show that if S is continuous, then it is a C 1 manifold.  2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. -On considère u(x, t) une solution singulière de u t = u + |u| p-1 u où u : R N × [0, T ) → R, p > 1, (N -2)p < N + 2 et soit u(0) 0, soit (3N -4)p < 3N + 8. On définit l'ensemble singulier S ⊂ R N de u comme étant l'ensemble de tous les points d'explosion. Sous une certaine condition de non dégénérescence, on montre que si S est continu, alors c'est une variété de classe C 1 .  2002 Éditions scientifiques et médicales Elsevier SAS

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## Abstract Theoretical aspects related to the approximation of the semilinear parabolic equation: $u\_t=\Delta u+f(u)$\nopagenumbers\end, with a finite unknown ‘blow‐up’ time __T__~b~ have been studied in a previous work. Specifically, for __ε__ a small positive number, we have considered coupled