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November  2014, 34(11): 4617-4645. doi: 10.3934/dcds.2014.34.4617

Blow-up set for a superlinear heat equation and pointedness of the initial data

1. 

Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka 560-8531, Japan

Received  October 2013 Revised  March 2014 Published  May 2014

We study the blow-up problem for a superlinear heat equation \begin{equation} \label{eq:P} \tag{P} \left\{ \begin{array}{ll} \partial_t u = \epsilon \Delta u + f(u),                      x\in\Omega, \,\,\, t>0, \\ u(x,t)=0,                                       x\in\partial\Omega, \,\,\, t>0, \\ u(x,0)=\varphi(x)\ge 0\, (\not\equiv 0),       x\in\Omega, \end{array} \right. \end{equation} where $\partial_t=\partial/\partial t$, $\epsilon>0$ is a sufficiently small constant, $N\ge 1$, $\Omega\subset {\bf R}^N$ is a domain, $\varphi\in C^2(\Omega)\cap C(\overline{\Omega})$ is a nonnegative bounded function, and $f$ is a positive convex function in $(0,\infty)$. In [10], the author of this paper and Ishige characterized the location of the blow-up set for problem (p) with $f(u)=u^p$ ($p>1$) with the aid of the invariance of the equation under some scale transformation for the solution, which played an important role in their argument. However, due to the lack of such scale invariance for problem (p), we can not apply their argument directly to problem (p). In this paper we introduce a new transformation for the solution of problem (p), which is a generalization of the scale transformation introduced in [10], and generalize the argument of [10]. In particular, we show the relationship between the blow-up set for problem (p) and pointedness of the initial function under suitable assumptions on $f$.
Citation: Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617
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show all references

References:
[1]

J. Differential Equations, 78 (1989), 160-190. doi: 10.1016/0022-0396(89)90081-8.  Google Scholar

[2]

J. Differential Equations, 244 (2008), 766-802. doi: 10.1016/j.jde.2007.11.004.  Google Scholar

[3]

J. Math. Anal. Appl., 335 (2007), 418-427. doi: 10.1016/j.jmaa.2007.01.079.  Google Scholar

[4]

SIAM J. Math. Anal., 18 (1987), 711-721. doi: 10.1137/0518054.  Google Scholar

[5]

Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.  Google Scholar

[6]

Differential Integral Equations, 25 (2012), 759-786.  Google Scholar

[7]

J. Differential Equations, 249 (2010), 1056-1077. doi: 10.1016/j.jde.2010.03.028.  Google Scholar

[8]

J. Differential Equations, 250 (2011), 2508-2543. doi: 10.1016/j.jde.2010.12.008.  Google Scholar

[9]

J. Differential Equations, 252 (2012), 1835-1861. doi: 10.1016/j.jde.2011.08.040.  Google Scholar

[10]

Indiana Univ. Math. J., 61 (2012), 627-663. doi: 10.1512/iumj.2012.61.4596.  Google Scholar

[11]

Ann. Inst. H. Poincaré Anal., 31 (2014), 231-247. doi: 10.1016/j.anihpc.2013.03.001.  Google Scholar

[12]

Comm. Pure Appl. Math., 42 (1989), 845-884. doi: 10.1002/cpa.3160420607.  Google Scholar

[13]

Adv. Differential Equations, 7 (2002), 1003-1024.  Google Scholar

[14]

Math. Ann., 327 (2003), 487-511. doi: 10.1007/s00208-003-0463-4.  Google Scholar

[15]

J. Differential Equations, 212 (2005), 114-128. doi: 10.1016/j.jde.2004.10.021.  Google Scholar

[16]

Indiana Univ. Math. J., 50 (2001), 591-610. doi: 10.1512/iumj.2001.50.1905.  Google Scholar

[17]

J. Math. Anal. Appl., 261 (2001), 350-368. doi: 10.1006/jmaa.2001.7530.  Google Scholar

[18]

Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-8442-5.  Google Scholar

[19]

J. Math. Anal. Appl. 343 (2008), 1061-1074. doi: 10.1016/j.jmaa.2008.02.018.  Google Scholar

[20]

Comm. Partial Differential Equations, 17 (1992), 1567-1596. doi: 10.1080/03605309208820896.  Google Scholar

[21]

Indiana Univ. Math. J., 42 (1993), 445-476. doi: 10.1512/iumj.1993.42.42021.  Google Scholar

[22]

J. Differential Equations 55 (1984), 204-224. doi: 10.1016/0022-0396(84)90081-0.  Google Scholar

[23]

J. Math. Soc. Japan, 56 (2004), 993-1005. doi: 10.2969/jmsj/1190905445.  Google Scholar

[24]

J. Math. Soc. Japan, 56 (2004), 1007-1017. doi: 10.2969/jmsj/1190905446.  Google Scholar

[25]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 505-542. doi: 10.1016/S0294-1449(01)00088-9.  Google Scholar

[26]

Comm. Math. Phys., 225 (2002), 523-549. doi: 10.1007/s002200100589.  Google Scholar

[27]

Mathematics mathematics education (Bethlehem, 2000), 337-347, World Sci. Publ., River Edge, NJ, 2002.  Google Scholar

[28]

Duke Math. J., 133 (2006), 499-525. doi: 10.1215/S0012-7094-06-13333-1.  Google Scholar

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