On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation

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March 2018, 17(2): 449-475. doi: 10.3934/cpaa.2018025

On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation

Yohei Fujishima

Department of Mathematical and Systems Engineering, Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561, Japan

Received November 2016 Revised May 2017 Published March 2018

Fund Project: The first author is partially supported by by the Grant-in-Aid for Encouragement of Young Scientists (B)(No. 15K17573) from Japan Society for the Promotion of Science.

This paper concerns the blow-up problem for a semilinear heat equation

$\begin{equation}\label{eq:P}\tag{P}≤\left\{\begin{array}{ll}\partial_t u=Δ u+u^p, &x∈ Ω, \, \, \, t>0, \\ u(x, t)=0, &x∈\partialΩ, \, \, \, t>0, \\ u(x, 0)=u_0(x)≥ 0, &x∈ Ω, \end{array}\right.\end{equation}$

where

$\partial_t=\partial/\partial t$

,

$p>1$

,

$N≥ 1$

,

$Ω\subset {\bf R}^N$

,

$u_0$

is a bounded continuous function in

$\overline{Ω}$

. For the case

$u_0(x)=λ\varphi(x)$

for some function

$\varphi$

and a sufficiently large

$λ>0$

, it is known that the solution blows up only near the maximum points of

$\varphi$

under suitable assumptions. Furthermore, if

$\varphi$

has several maximum points, then the blow-up set for (P) is characterized by

$Δ\varphi$

at its maximum points. However, for initial data

$u_0(x)=λ\varphi(x)$

, it seems difficult to obtain further information on the blow-up set such that effect of higher order derivatives of initial data. In this paper, we consider another type large initial data

$u_0(x)=λ+\varphi(x)$

and study the relationship between the blow-up set for (P) and higher order derivatives of initial data.

Keywords: Semilinear heat equation, blow-up problem, blow-up set, large initial data, comparison principle.

Mathematics Subject Classification: Primary:35K91;Secondary:35B44.

Citation: Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025

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), 160-190. doi: 10.1016/0022-0396(89)90081-8.
[2]	A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.
[3]	Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion, Differential Integral Equations, 25 (2012), 759-786.
[4]	Y. Fujishima, Blow-up set for a superlinear heat equation and pointedness of the initial data, Discrete Continuous Dynamical Systems A, 34 (2014), 4617-4645. doi: 10.3934/dcds.2014.34.4617.
[5]	Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation with small diffusion, J. Differential Equations, 249 (2010), 1056-1077. doi: 10.1016/j.jde.2010.03.028.
[6]	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), 627-663. doi: 10.1512/iumj.2012.61.4596.
[7]	Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845-884. doi: 10.1002/cpa.3160420607.
[8]	K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion, J. Differential Equations, 212 (2005), 114-128. doi: 10.1016/j.jde.2004.10.021.
[9]	N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small diffusion, J. Math. Anal. Appl., 261 (2001), 350-368. doi: 10.1006/jmaa.2001.7530.
[10]	N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic equation, Indiana Univ. Math. J., 50 (2001), 591-610. doi: 10.1512/iumj.2001.50.1905.
[11]	P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher Birkhäuser Verlag, Basel, 2007.
[12]	J. J. L. Velázquez, Higher-dimensional blow-up for semilinear parabolic equations, Comm. Partial Differential Equations, 17 (1992), 1567-1596. doi: 10.1080/03605309208820896.
[13]	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), 445-476. doi: 10.1512/iumj.1993.42.42021.
[14]	F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations, 55 (1984), 204-224. doi: 10.1016/0022-0396(84)90081-0.
[15]	H. Yagisita, Blow-up profile of a solution for a nonlinear heat equation with small diffusion, J. Math. Soc. Japan, 56 (2004), 993-1005. doi: 10.2969/jmsj/1190905445.
[16]	H. Yagisita, Variable instability of a constant blow-up solution in a nonlinear heat equation, J. Math. Soc. Japan, 56 (2004), 1007-1017. doi: 10.2969/jmsj/1190905446.
[17]	H. Zaag, On the regularity of the blow-up set for semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 505-542. doi: 10.1016/S0294-1449(01)00088-9.
[18]	H. Zaag, One-dimensional behavior of singular $N$-dimensional solutions of semilinear heat equations, Comm. Math. Phys., 225 (2002), 523-549. doi: 10.1007/s002200100589.
[19]	H. Zaag, Regularity of the blow-up set and singular behavior for semilinear heat equations, Mathematics & mathematics education (Bethlehem, 2000), 337-347, World Sci. Publ., River Edge, NJ, 2002.
[20]	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), 499-525. 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), 160-190. doi: 10.1016/0022-0396(89)90081-8.
[2]	A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.
[3]	Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion, Differential Integral Equations, 25 (2012), 759-786.
[4]	Y. Fujishima, Blow-up set for a superlinear heat equation and pointedness of the initial data, Discrete Continuous Dynamical Systems A, 34 (2014), 4617-4645. doi: 10.3934/dcds.2014.34.4617.
[5]	Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation with small diffusion, J. Differential Equations, 249 (2010), 1056-1077. doi: 10.1016/j.jde.2010.03.028.
[6]	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), 627-663. doi: 10.1512/iumj.2012.61.4596.
[7]	Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845-884. doi: 10.1002/cpa.3160420607.
[8]	K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion, J. Differential Equations, 212 (2005), 114-128. doi: 10.1016/j.jde.2004.10.021.
[9]	N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small diffusion, J. Math. Anal. Appl., 261 (2001), 350-368. doi: 10.1006/jmaa.2001.7530.
[10]	N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic equation, Indiana Univ. Math. J., 50 (2001), 591-610. doi: 10.1512/iumj.2001.50.1905.
[11]	P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher Birkhäuser Verlag, Basel, 2007.
[12]	J. J. L. Velázquez, Higher-dimensional blow-up for semilinear parabolic equations, Comm. Partial Differential Equations, 17 (1992), 1567-1596. doi: 10.1080/03605309208820896.
[13]	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), 445-476. doi: 10.1512/iumj.1993.42.42021.
[14]	F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations, 55 (1984), 204-224. doi: 10.1016/0022-0396(84)90081-0.
[15]	H. Yagisita, Blow-up profile of a solution for a nonlinear heat equation with small diffusion, J. Math. Soc. Japan, 56 (2004), 993-1005. doi: 10.2969/jmsj/1190905445.
[16]	H. Yagisita, Variable instability of a constant blow-up solution in a nonlinear heat equation, J. Math. Soc. Japan, 56 (2004), 1007-1017. doi: 10.2969/jmsj/1190905446.
[17]	H. Zaag, On the regularity of the blow-up set for semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 505-542. doi: 10.1016/S0294-1449(01)00088-9.
[18]	H. Zaag, One-dimensional behavior of singular $N$-dimensional solutions of semilinear heat equations, Comm. Math. Phys., 225 (2002), 523-549. doi: 10.1007/s002200100589.
[19]	H. Zaag, Regularity of the blow-up set and singular behavior for semilinear heat equations, Mathematics & mathematics education (Bethlehem, 2000), 337-347, World Sci. Publ., River Edge, NJ, 2002.
[20]	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), 499-525. doi: 10.1215/S0012-7094-06-13333-1.

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