## Duke Mathematical Journal

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

Hatem Zaag

#### Abstract

We consider $u(x,t)$, a solution of $u_t =\Delta u +|u|^{p-1}u$ which blows up at some time $T\gt 0$, where $u:{\mathbb R}^N\times [0,T)\to {\mathbb R}$, $p\gt 1$, and $(N-2)p \lt N+2$. Under a nondegeneracy condition, we show that the mere hypothesis that the blow-up set $S$ is continuous and $(N-1)$-dimensional implies that it is $C^2$. In particular, we compute the $N-1$ 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

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

#### 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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