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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 $$\label{eq:P} \tag{P} \left\{ \begin{array}{ll} \partial_t u = \epsilon \Delta u + f(u), x\in\Omega, \,\,\, t>0, \\ u(x,t)=0, x\in\partial\Omega, \,\,\, t>0, \\ u(x,0)=\varphi(x)\ge 0\, (\not\equiv 0), x\in\Omega, \end{array} \right.$$ 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 - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617
##### References:
 [1] X. Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, J. Differential Equations, 78 (1989), 160.  doi: 10.1016/0022-0396(89)90081-8.  Google Scholar [2] T. Cheng and G. F. Zheng, Some blow-up problems for a semilinear parabolic equation with a potential,, J. Differential Equations, 244 (2008), 766.  doi: 10.1016/j.jde.2007.11.004.  Google Scholar [3] C. Cortazar, M. Elgueta and J. D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential,, J. Math. Anal. Appl., 335 (2007), 418.  doi: 10.1016/j.jmaa.2007.01.079.  Google Scholar [4] A. Friedman and A. A. Lacey, The blow-up time for solutions of nonlinear heat equations with small diffusion,, SIAM J. Math. Anal., 18 (1987), 711.  doi: 10.1137/0518054.  Google Scholar [5] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425.  doi: 10.1512/iumj.1985.34.34025.  Google Scholar [6] Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion,, Differential Integral Equations, 25 (2012), 759.   Google Scholar [7] Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation with small diffusion,, J. Differential Equations, 249 (2010), 1056.  doi: 10.1016/j.jde.2010.03.028.  Google Scholar [8] Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$,, J. Differential Equations, 250 (2011), 2508.  doi: 10.1016/j.jde.2010.12.008.  Google Scholar [9] Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$. II,, J. Differential Equations, 252 (2012), 1835.  doi: 10.1016/j.jde.2011.08.040.  Google Scholar [10] Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data,, Indiana Univ. Math. J., 61 (2012), 627.  doi: 10.1512/iumj.2012.61.4596.  Google Scholar [11] Y. Fujishima and K. Ishige, Blow-up set for type I blowing up solutions for a semilinear heat equation,, Ann. Inst. H. Poincaré Anal., 31 (2014), 231.  doi: 10.1016/j.anihpc.2013.03.001.  Google Scholar [12] Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations,, Comm. Pure Appl. Math., 42 (1989), 845.  doi: 10.1002/cpa.3160420607.  Google Scholar [13] K. Ishige, Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion,, Adv. Differential Equations, 7 (2002), 1003.   Google Scholar [14] K. Ishige and N. Mizoguchi, Location of blow-up set for a semilinear parabolic equation with large diffusion,, Math. Ann., 327 (2003), 487.  doi: 10.1007/s00208-003-0463-4.  Google Scholar [15] K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion,, J. Differential Equations, 212 (2005), 114.  doi: 10.1016/j.jde.2004.10.021.  Google Scholar [16] N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic equation,, Indiana Univ. Math. J., 50 (2001), 591.  doi: 10.1512/iumj.2001.50.1905.  Google Scholar [17] N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small diffusion,, J. Math. Anal. Appl., 261 (2001), 350.  doi: 10.1006/jmaa.2001.7530.  Google Scholar [18] P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher, (2007).  doi: 10.1007/978-3-7643-8442-5.  Google Scholar [19] S. Sato, Life span of solutions with large initial data for a superlinear heat equation,, J. Math. Anal. Appl. 343 (2008), 343 (2008), 1061.  doi: 10.1016/j.jmaa.2008.02.018.  Google Scholar [20] J. J. L. Velázquez, Higher-dimensional blow-up for semilinear parabolic equations,, Comm. Partial Differential Equations, 17 (1992), 1567.  doi: 10.1080/03605309208820896.  Google Scholar [21] J. J. L. Velázquez, Estimates on the $(n-1)$-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation,, Indiana Univ. Math. J., 42 (1993), 445.  doi: 10.1512/iumj.1993.42.42021.  Google Scholar [22] F. B. Weissler, Single point blow-up for a semilinear initial value problem,, J. Differential Equations 55 (1984), 55 (1984), 204.  doi: 10.1016/0022-0396(84)90081-0.  Google Scholar [23] H. Yagisita, Blow-up profile of a solution for a nonlinear heat equation with small diffusion,, J. Math. Soc. Japan, 56 (2004), 993.  doi: 10.2969/jmsj/1190905445.  Google Scholar [24] H. Yagisita, Variable instability of a constant blow-up solution in a nonlinear heat equation,, J. Math. Soc. Japan, 56 (2004), 1007.  doi: 10.2969/jmsj/1190905446.  Google Scholar [25] H. Zaag, On the regularity of the blow-up set for semilinear heat equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 505.  doi: 10.1016/S0294-1449(01)00088-9.  Google Scholar [26] H. Zaag, One-dimensional behavior of singular $N$-dimensional solutions of semilinear heat equations,, Comm. Math. Phys., 225 (2002), 523.  doi: 10.1007/s002200100589.  Google Scholar [27] H. Zaag, Regularity of the blow-up set and singular behavior for semilinear heat equations,, Mathematics mathematics education (Bethlehem, (2000), 337.   Google Scholar [28] H. Zaag, Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation,, Duke Math. J., 133 (2006), 499.  doi: 10.1215/S0012-7094-06-13333-1.  Google Scholar

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##### References:
 [1] X. Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, J. Differential Equations, 78 (1989), 160.  doi: 10.1016/0022-0396(89)90081-8.  Google Scholar [2] T. Cheng and G. F. Zheng, Some blow-up problems for a semilinear parabolic equation with a potential,, J. Differential Equations, 244 (2008), 766.  doi: 10.1016/j.jde.2007.11.004.  Google Scholar [3] C. Cortazar, M. Elgueta and J. D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential,, J. Math. Anal. Appl., 335 (2007), 418.  doi: 10.1016/j.jmaa.2007.01.079.  Google Scholar [4] A. Friedman and A. A. Lacey, The blow-up time for solutions of nonlinear heat equations with small diffusion,, SIAM J. Math. Anal., 18 (1987), 711.  doi: 10.1137/0518054.  Google Scholar [5] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425.  doi: 10.1512/iumj.1985.34.34025.  Google Scholar [6] Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion,, Differential Integral Equations, 25 (2012), 759.   Google Scholar [7] Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation with small diffusion,, J. Differential Equations, 249 (2010), 1056.  doi: 10.1016/j.jde.2010.03.028.  Google Scholar [8] Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$,, J. Differential Equations, 250 (2011), 2508.  doi: 10.1016/j.jde.2010.12.008.  Google Scholar [9] Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$. II,, J. Differential Equations, 252 (2012), 1835.  doi: 10.1016/j.jde.2011.08.040.  Google Scholar [10] Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data,, Indiana Univ. Math. J., 61 (2012), 627.  doi: 10.1512/iumj.2012.61.4596.  Google Scholar [11] Y. Fujishima and K. Ishige, Blow-up set for type I blowing up solutions for a semilinear heat equation,, Ann. Inst. H. Poincaré Anal., 31 (2014), 231.  doi: 10.1016/j.anihpc.2013.03.001.  Google Scholar [12] Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations,, Comm. Pure Appl. Math., 42 (1989), 845.  doi: 10.1002/cpa.3160420607.  Google Scholar [13] K. Ishige, Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion,, Adv. Differential Equations, 7 (2002), 1003.   Google Scholar [14] K. Ishige and N. Mizoguchi, Location of blow-up set for a semilinear parabolic equation with large diffusion,, Math. Ann., 327 (2003), 487.  doi: 10.1007/s00208-003-0463-4.  Google Scholar [15] K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion,, J. Differential Equations, 212 (2005), 114.  doi: 10.1016/j.jde.2004.10.021.  Google Scholar [16] N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic equation,, Indiana Univ. Math. J., 50 (2001), 591.  doi: 10.1512/iumj.2001.50.1905.  Google Scholar [17] N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small diffusion,, J. Math. Anal. Appl., 261 (2001), 350.  doi: 10.1006/jmaa.2001.7530.  Google Scholar [18] P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher, (2007).  doi: 10.1007/978-3-7643-8442-5.  Google Scholar [19] S. Sato, Life span of solutions with large initial data for a superlinear heat equation,, J. Math. Anal. Appl. 343 (2008), 343 (2008), 1061.  doi: 10.1016/j.jmaa.2008.02.018.  Google Scholar [20] J. J. L. Velázquez, Higher-dimensional blow-up for semilinear parabolic equations,, Comm. Partial Differential Equations, 17 (1992), 1567.  doi: 10.1080/03605309208820896.  Google Scholar [21] J. J. L. Velázquez, Estimates on the $(n-1)$-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation,, Indiana Univ. Math. J., 42 (1993), 445.  doi: 10.1512/iumj.1993.42.42021.  Google Scholar [22] F. B. Weissler, Single point blow-up for a semilinear initial value problem,, J. Differential Equations 55 (1984), 55 (1984), 204.  doi: 10.1016/0022-0396(84)90081-0.  Google Scholar [23] H. Yagisita, Blow-up profile of a solution for a nonlinear heat equation with small diffusion,, J. Math. Soc. Japan, 56 (2004), 993.  doi: 10.2969/jmsj/1190905445.  Google Scholar [24] H. Yagisita, Variable instability of a constant blow-up solution in a nonlinear heat equation,, J. Math. Soc. Japan, 56 (2004), 1007.  doi: 10.2969/jmsj/1190905446.  Google Scholar [25] H. Zaag, On the regularity of the blow-up set for semilinear heat equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 505.  doi: 10.1016/S0294-1449(01)00088-9.  Google Scholar [26] H. Zaag, One-dimensional behavior of singular $N$-dimensional solutions of semilinear heat equations,, Comm. Math. Phys., 225 (2002), 523.  doi: 10.1007/s002200100589.  Google Scholar [27] H. Zaag, Regularity of the blow-up set and singular behavior for semilinear heat equations,, Mathematics mathematics education (Bethlehem, (2000), 337.   Google Scholar [28] H. Zaag, Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation,, Duke Math. J., 133 (2006), 499.  doi: 10.1215/S0012-7094-06-13333-1.  Google Scholar
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