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

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.
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]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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]

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

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]