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

Yohei Fujishima

doi:10.3934/cpaa.2018025

Previous Article
Nonexistence of nonconstant positive steady states of a diffusive predator-prey model
CPAA Home
This Issue
Next Article
Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model

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 & 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. Google Scholar
[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. Google Scholar
[3]	Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion, Differential Integral Equations, 25 (2012), 759-786. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[3]	Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion, Differential Integral Equations, 25 (2012), 759-786. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	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
[2]	Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
[3]	Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105
[4]	Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585
[5]	Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733
[6]	Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108
[7]	Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
[8]	Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147
[9]	Pierpaolo Esposito, Maristella Petralla. Symmetries and blow-up phenomena for a Dirichlet problem with a large parameter. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1935-1957. doi: 10.3934/cpaa.2012.11.1935
[10]	Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
[11]	Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388
[12]	Marina Chugunova, Chiu-Yen Kao, Sarun Seepun. On the Benilov-Vynnycky blow-up problem. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1443-1460. doi: 10.3934/dcdsb.2015.20.1443
[13]	Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[14]	José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43
[15]	Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1
[16]	Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021060
[17]	Hiroyuki Takamura, Hiroshi Uesaka, Kyouhei Wakasa. Sharp blow-up for semilinear wave equations with non-compactly supported data. Conference Publications, 2011, 2011 (Special) : 1351-1357. doi: 10.3934/proc.2011.2011.1351
[18]	Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315
[19]	Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure & Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435
[20]	Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors