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August  2011, 4(4): 833-849. doi: 10.3934/dcdss.2011.4.833

Hot spots for the two dimensional heat equation with a rapidly decaying negative potential

Kazuhiro Ishige 1, and  Y. Kabeya 2,

1. 

Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578
2. 

Department of Mathematical Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, 599-8531

Received  August 2009 Revised  January 2010 Published  November 2010

We consider the Cauchy problem of the two dimensional heat equation with a radially symmetric, negative potential $-V$ which behaves like $V(r)=O(r^{-\kappa})$ as $r\to\infty$, for some $\kappa > 2$. We study the rate and the direction for hot spots to tend to the spatial infinity. Furthermore we give a sufficient condition for hot spots to consist of only one point for any sufficiently large $t>0$.
Keywords: heat equation, large time behavior., Hot spots.
Mathematics Subject Classification: Primary: 35B05, 35B40, 35K05; Secondary: 53C3.
Citation: Kazuhiro Ishige, Y. Kabeya. Hot spots for the two dimensional heat equation with a rapidly decaying negative potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 833-849. doi: 10.3934/dcdss.2011.4.833
References:
[1]

I. Chavel and L. Karp, Movement of hot spots in Riemannian manifolds, J. Analyse Math., 55 (1990), 271-286. doi: doi:10.1007/BF02789205.  Google Scholar

[2]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. T. M. A., 11 (1987), 1103-1133. doi: doi:10.1016/0362-546X(87)90001-0.  Google Scholar

[3]

K. Ishige, Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition, J. Differential Equations, 212 (2005), 394-431. doi: doi:10.1016/j.jde.2004.11.002.  Google Scholar

[4]

K. Ishige, Movement of hot spots on the exterior domain of a ball under the Dirichlet boundary condition, Adv. Differential Equations, 12 (2007), 1135-1166.  Google Scholar

[5]

K. Ishige and Y. Kabeya, Large time behaviors of hot spots for the heat equation with a potential, J. Differential Equations, 244 (2008), 2934-2962; Corrigendum in J. Differential Equations, 245 (2008), 2352-2354. doi: doi:10.1016/j.jde.2008.07.023.  Google Scholar

[6]

K. Ishige and Y. Kabeya, Hot spots for the heat equation with a rapidly decaying negative potential, Adv. Differential Equations, 14 (2009), 643-662.  Google Scholar

[7]

K. Ishige and Y. Kabeya, Large time behaviors of hot spots for the heat equation with a potential II: Positive potential case,, in preparation., ().   Google Scholar

[8]

S. Jimbo and S. Sakaguchi, Movement of hot spots over unbounded domains in $R^N$, J. Math. Anal. Appl., 182 (1994), 810-835. doi: doi:10.1006/jmaa.1994.1123.  Google Scholar

[9]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence, 1968.  Google Scholar

show all references

References:
[1]

I. Chavel and L. Karp, Movement of hot spots in Riemannian manifolds, J. Analyse Math., 55 (1990), 271-286. doi: doi:10.1007/BF02789205.  Google Scholar

[2]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. T. M. A., 11 (1987), 1103-1133. doi: doi:10.1016/0362-546X(87)90001-0.  Google Scholar

[3]

K. Ishige, Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition, J. Differential Equations, 212 (2005), 394-431. doi: doi:10.1016/j.jde.2004.11.002.  Google Scholar

[4]

K. Ishige, Movement of hot spots on the exterior domain of a ball under the Dirichlet boundary condition, Adv. Differential Equations, 12 (2007), 1135-1166.  Google Scholar

[5]

K. Ishige and Y. Kabeya, Large time behaviors of hot spots for the heat equation with a potential, J. Differential Equations, 244 (2008), 2934-2962; Corrigendum in J. Differential Equations, 245 (2008), 2352-2354. doi: doi:10.1016/j.jde.2008.07.023.  Google Scholar

[6]

K. Ishige and Y. Kabeya, Hot spots for the heat equation with a rapidly decaying negative potential, Adv. Differential Equations, 14 (2009), 643-662.  Google Scholar

[7]

K. Ishige and Y. Kabeya, Large time behaviors of hot spots for the heat equation with a potential II: Positive potential case,, in preparation., ().   Google Scholar

[8]

S. Jimbo and S. Sakaguchi, Movement of hot spots over unbounded domains in $R^N$, J. Math. Anal. Appl., 182 (1994), 810-835. doi: doi:10.1006/jmaa.1994.1123.  Google Scholar

[9]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence, 1968.  Google Scholar

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