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

Time-dependent singularities in the heat equation

Jin Takahashi 1, and  Eiji Yanagida 2,

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.
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

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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

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