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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Blowup for $u_t = \Delta u + \vert\nabla u\vert^2 u$ from $\mathbb{R}^n$ into $\mathbb{R}^m$


Author: Daisuke Hirata
Journal: Proc. Amer. Math. Soc. 133 (2005), 1823-1827
MSC (2000): Primary 35K45, 35K20; Secondary 58E20
Published electronically: January 14, 2005
MathSciNet review: 2120283
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Abstract: In this note we consider the global regularity of smooth solutions $u=(u^1,\dots , u^m)$ to the vector-valued Cauchy problem

\begin{displaymath}u_t = \Delta u + \vert\nabla u\vert^2 u \quad \text{in } \ma... ...\infty), \quad u(x,0) = u_0(x) \quad \text{in } \mathbb{R}^n. \end{displaymath}

We show that if $n,m \geq 3$, the gradient-blowup phenomenon occurs in finite time for suitably chosen $u_0$ vanishing at infinity. We also present a simple example of the $L^\infty$-blowup solutions for $\vert u_0\vert \equiv 1+\epsilon$for any $\epsilon >0$, if $m \geq 2$.


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Additional Information

Daisuke Hirata
Affiliation: Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba, 278-8510, Japan
Email: dhirata@kurenai.waseda.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07821-4
PII: S 0002-9939(05)07821-4
Keywords: Blowup, parabolic system, Cauchy problem
Received by editor(s): February 25, 2004
Published electronically: January 14, 2005
Additional Notes: The author was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.