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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. [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. [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. [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. [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. [6] Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion,, Differential Integral Equations, 25 (2012), 759. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [20] J. J. L. Velázquez, Higher-dimensional blow-up for semilinear parabolic equations,, Comm. Partial Differential Equations, 17 (1992), 1567. doi: 10.1080/03605309208820896. [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. [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. [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. [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. [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. [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. [27] H. Zaag, Regularity of the blow-up set and singular behavior for semilinear heat equations,, Mathematics mathematics education (Bethlehem, (2000), 337. [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.

show all references

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. [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. [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. [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. [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. [6] Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion,, Differential Integral Equations, 25 (2012), 759. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [20] J. J. L. Velázquez, Higher-dimensional blow-up for semilinear parabolic equations,, Comm. Partial Differential Equations, 17 (1992), 1567. doi: 10.1080/03605309208820896. [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. [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. [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. [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. [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. [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. [27] H. Zaag, Regularity of the blow-up set and singular behavior for semilinear heat equations,, Mathematics mathematics education (Bethlehem, (2000), 337. [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.
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