Blowup for $u_t = \Delta u + |\nabla u|^2 u$ from $\mathbb {R}^n$ into $\mathbb {R}^m$
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- by Daisuke Hirata PDF
- Proc. Amer. Math. Soc. 133 (2005), 1823-1827 Request permission
Abstract:
In this note we consider the global regularity of smooth solutions $u=(u^1,\dots , u^m)$ to the vector-valued Cauchy problem \[ u_t = \Delta u + |\nabla u|^2 u \quad \text {in } \mathbb {R}^n \times [0,\infty ), \quad u(x,0) = u_0(x) \quad \text {in } \mathbb {R}^n. \] 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 $|u_0| \equiv 1+\epsilon$ for any $\epsilon >0$, if $m \geq 2$.References
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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
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 1823-1827
- MSC (2000): Primary 35K45, 35K20; Secondary 58E20
- DOI: https://doi.org/10.1090/S0002-9939-05-07821-4
- MathSciNet review: 2120283