# American Institute of Mathematical Sciences

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

 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 $t>0$.
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.  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.  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.  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.   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.  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.   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.  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., (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.  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.  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.  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.   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.  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.   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.  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., (1968).   Google Scholar
 [1] Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176 [2] Xiao-Hui Li, Huo-Jun Ruan. The "hot spots" conjecture on higher dimensional Sierpinski gaskets. Communications on Pure & Applied Analysis, 2016, 15 (1) : 287-297. doi: 10.3934/cpaa.2016.15.287 [3] Jesus Ildefonso Díaz, Jacqueline Fleckinger-Pellé. Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 193-200. doi: 10.3934/dcds.2004.10.193 [4] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [5] Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601 [6] Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581 [7] Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41 [8] Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93 [9] Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221 [10] Geonho Lee, Sangdong Kim, Young-Sam Kwon. Large time behavior for the full compressible magnetohydrodynamic flows. Communications on Pure & Applied Analysis, 2012, 11 (3) : 959-971. doi: 10.3934/cpaa.2012.11.959 [11] Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143 [12] Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969 [13] Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481 [14] Martin Burger, Marco Di Francesco. Large time behavior of nonlocal aggregation models with nonlinear diffusion. Networks & Heterogeneous Media, 2008, 3 (4) : 749-785. doi: 10.3934/nhm.2008.3.749 [15] Shijin Deng. Large time behavior for the IBVP of the 3-D Nishida's model. Networks & Heterogeneous Media, 2010, 5 (1) : 133-142. doi: 10.3934/nhm.2010.5.133 [16] Ahmed Bonfoh, Cyril D. Enyi. Large time behavior of a conserved phase-field system. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1077-1105. doi: 10.3934/cpaa.2016.15.1077 [17] Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013 [18] Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-15. doi: 10.3934/dcds.2019229 [19] Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091 [20] Fredi Tröltzsch, Daniel Wachsmuth. On the switching behavior of sparse optimal controls for the one-dimensional heat equation. Mathematical Control & Related Fields, 2018, 8 (1) : 135-153. doi: 10.3934/mcrf.2018006

2018 Impact Factor: 0.545