Duke Mathematical Journal

Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation

Hatem Zaag

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Abstract

We consider u(x,t), a solution of ut=Δu+|u|p1u which blows up at some time T>0, where u:RN×[0,T)R, p>1, and (N2)p<N+2. Under a nondegeneracy condition, we show that the mere hypothesis that the blow-up set S is continuous and (N1)-dimensional implies that it is C2. In particular, we compute the N1 principal curvatures and directions of S. Moreover, a much more refined blow-up behavior is derived for the solution in terms of the newly exhibited geometric objects. Refined regularity for S and refined singular behavior of u near S are linked through a new mechanism of algebraic cancellations that we explain in detail

Article information

Source
Duke Math. J., Volume 133, Number 3 (2006), 499-525.

Dates
First available in Project Euclid: 13 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1150201200

Digital Object Identifier
doi:10.1215/S0012-7094-06-13333-1

Mathematical Reviews number (MathSciNet)
MR2228461

Zentralblatt MATH identifier
1096.35062

Subjects
Primary: 35A20: Analytic methods, singularities 35B40: Asymptotic behavior of solutions
Secondary: 35K55: Nonlinear parabolic equations

Citation

Zaag, Hatem. Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation. Duke Math. J. 133 (2006), no. 3, 499--525. doi:10.1215/S0012-7094-06-13333-1. https://projecteuclid.org/euclid.dmj/1150201200


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