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Communications in Mathematical Physics

, Volume 225, Issue 3, pp 523–549 | Cite as

One Dimensional Behavior of Singular N Dimensional Solutions of Semilinear Heat Equations

  • Hatem Zaag

Abstract:

We consider u(x,t) a solution of u t u+|u| p − 1 u that blows up at time T, 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. We are concerned with the behavior of the solution near a non isolated blow-up point, as Tt→ 0. Under a non-degeneracy condition and assuming that the blow-up set is locally continuous and N−1 dimensional, we escape logarithmic scales of the variable Tt and give a sharper expansion of the solution with the much smaller error term (Tt)1, 1/2−η for any η>0. In particular, if in addition p>3, then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow-up set. Finally, we prove that the mere hypothesis that the blow-up set is continuous implies that it is C 1, 1/2−η for any η>0.

Keywords

Error Term Heat Equation Logarithmic Scale Small Error Dimensional Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Hatem Zaag
    • 1
  1. 1.Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USAUS

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