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.
Citation: |
[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.![]() ![]() ![]() |