Time-dependent singularities in the heat equation

Previous Article
On the variational $p$-capacity problem in the plane
CPAA Home
This Issue
Next Article
Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model

May 2015, 14(3): 969-979. doi: 10.3934/cpaa.2015.14.969

Time-dependent singularities in the heat equation

1.	Department of Mathematics, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152-8551, Japan
2.	Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551

Received September 2014 Revised February 2015 Published March 2015

Abstract
Related Papers

We consider solutions of the heat equation with time-dependent singularities. It is shown that a singularity is removable if it is weaker than the order of the fundamental solution of the Laplace equation. Some examples of non-removable singularities are also given, which show the optimality of the condition for removability.

Keywords: H\"older continuity., asymptotic behavior, removable singularity, Heat equation, time-dependent singularity.

Mathematics Subject Classification: Primary: 35K05; Secondary: 35A20, 35B4.

Citation: 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

References:

[1]	H. Brézis and L. Véron, Removable singularities for some nonlinear elliptic equations,, \emph{Arch. Rational Mech. Anal.}, 75 (): 1. doi: 10.1007/BF00284616. Google Scholar
[2]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, 34 (1981), 525. doi: 10.1002/cpa.3160340406. Google Scholar
[3]	A. Grigor'yan, Heat Kernel and Analysis on Manifolds,, American Mathematical Society, (2009). Google Scholar
[4]	K. Hirata, Removable singularities of semilinear parabolic equations,, \emph{Proc. Amer. Math. Soc.}, 142 (2014), 157. doi: 10.1090/S0002-9939-2013-11739-9. Google Scholar
[5]	S.-Y. Hsu, Removable singularities of semilinear parabolic equations,, \emph{Adv. Differential Equations}, 15 (2010), 137. Google Scholar
[6]	K. M. Hui, Another proof for the removable singularities of the heat equation,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 2397. doi: 10.1090/S0002-9939-10-10352-9. Google Scholar
[7]	T. Kan and J. Takahashi, On the profile of solutions with time-dependent singularities for the heat equation,, \emph{Kodai Math. J.}, 37 (2014), 568. doi: 10.2996/kmj/1414674609. Google Scholar
[8]	P.-L. Lions, Isolated singularities in semilinear problems,, \emph{J. Differential Equations}, 38 (1980), 441. doi: 10.1016/0022-0396(80)90018-2. Google Scholar
[9]	S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation,, \emph{J. Differential Equations}, 246 (2009), 724. doi: 10.1016/j.jde.2008.09.004. Google Scholar
[10]	L. Véron, Singular solutions of some nonlinear elliptic equations,, \emph{Nonlinear Anal.}, 5 (1981), 225. doi: 10.1016/0362-546X(81)90028-6. Google Scholar
[11]	L. Véron, Singularities of Solutions of Second Order Quasilinear Equations,, Longman, (1996). Google Scholar

show all references

References:

[1]	H. Brézis and L. Véron, Removable singularities for some nonlinear elliptic equations,, \emph{Arch. Rational Mech. Anal.}, 75 (): 1. doi: 10.1007/BF00284616. Google Scholar
[2]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, 34 (1981), 525. doi: 10.1002/cpa.3160340406. Google Scholar
[3]	A. Grigor'yan, Heat Kernel and Analysis on Manifolds,, American Mathematical Society, (2009). Google Scholar
[4]	K. Hirata, Removable singularities of semilinear parabolic equations,, \emph{Proc. Amer. Math. Soc.}, 142 (2014), 157. doi: 10.1090/S0002-9939-2013-11739-9. Google Scholar
[5]	S.-Y. Hsu, Removable singularities of semilinear parabolic equations,, \emph{Adv. Differential Equations}, 15 (2010), 137. Google Scholar
[6]	K. M. Hui, Another proof for the removable singularities of the heat equation,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 2397. doi: 10.1090/S0002-9939-10-10352-9. Google Scholar
[7]	T. Kan and J. Takahashi, On the profile of solutions with time-dependent singularities for the heat equation,, \emph{Kodai Math. J.}, 37 (2014), 568. doi: 10.2996/kmj/1414674609. Google Scholar
[8]	P.-L. Lions, Isolated singularities in semilinear problems,, \emph{J. Differential Equations}, 38 (1980), 441. doi: 10.1016/0022-0396(80)90018-2. Google Scholar
[9]	S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation,, \emph{J. Differential Equations}, 246 (2009), 724. doi: 10.1016/j.jde.2008.09.004. Google Scholar
[10]	L. Véron, Singular solutions of some nonlinear elliptic equations,, \emph{Nonlinear Anal.}, 5 (1981), 225. doi: 10.1016/0362-546X(81)90028-6. Google Scholar
[11]	L. Véron, Singularities of Solutions of Second Order Quasilinear Equations,, Longman, (1996). Google Scholar

[1]	Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178
[2]	Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141
[3]	Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1557-1574. doi: 10.3934/dcds.2012.32.1557
[4]	Jerry Bona, H. Kalisch. Singularity formation in the generalized Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 27-45. doi: 10.3934/dcds.2004.11.27
[5]	Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041
[6]	Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307
[7]	Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
[8]	Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Robustness of time-dependent attractors in H¹-norm for nonlocal problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1011-1036. doi: 10.3934/dcdsb.2018140
[9]	Veronica Felli, Elsa M. Marchini, Susanna Terracini. On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 91-119. doi: 10.3934/dcds.2008.21.91
[10]	Gang Tian. Finite-time singularity of Kähler-Ricci flow. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1137-1150. doi: 10.3934/dcds.2010.28.1137
[11]	Juan C. Jara, Felipe Rivero. Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4127-4137. doi: 10.3934/dcds.2014.34.4127
[12]	I. Baldomá, Tere M. Seara. The inner equation for generic analytic unfoldings of the Hopf-zero singularity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 323-347. doi: 10.3934/dcdsb.2008.10.323
[13]	Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[14]	Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811
[15]	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
[16]	Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4477-4493. doi: 10.3934/dcds.2016.36.4477
[17]	Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639
[18]	Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[19]	Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
[20]	Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences