Another proof for the removable singularities of the heat equation

Author:
Kin Ming Hui

Journal:
Proc. Amer. Math. Soc. **138** (2010), 2397-2402

MSC (2010):
Primary 35B65; Secondary 35K05

Published electronically:
February 18, 2010

MathSciNet review:
2607869

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give two different simple proofs for the removable singularities of the heat equation in , where is a bounded domain with . We also give a necessary and sufficient condition for removable singularities of the heat equation in for the case .

**1.**Björn E. J. Dahlberg and Carlos E. Kenig,*Nonnegative solutions of generalized porous medium equations*, Rev. Mat. Iberoamericana**2**(1986), no. 3, 267–305. MR**908054****2.**Avner Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836****3.**Meijiao Guan, Stephen Gustafson, and Tai-Peng Tsai,*Global existence and blow-up for harmonic map heat flow*, J. Differential Equations**246**(2009), no. 1, 1–20. MR**2467012**, 10.1016/j.jde.2008.09.011**4.**Nassif Ghoussoub and Yujin Guo,*On the partial differential equations of electrostatic MEMS devices. II. Dynamic case*, NoDEA Nonlinear Differential Equations Appl.**15**(2008), no. 1-2, 115–145. MR**2408347**, 10.1007/s00030-007-6004-1**5.**Nassif Ghoussoub and Yujin Guo,*Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS*, Methods Appl. Anal.**15**(2008), no. 3, 361–376. MR**2500853**, 10.4310/MAA.2008.v15.n3.a8**6.**Yujin Guo,*On the partial differential equations of electrostatic MEMS devices. III. Refined touchdown behavior*, J. Differential Equations**244**(2008), no. 9, 2277–2309. MR**2413842**, 10.1016/j.jde.2008.02.005**7.**Shu-Yu Hsu,*Asymptotic behaviour of solutions of the equation 𝑢_{𝑡}=Δlog𝑢 near the extinction time*, Adv. Differential Equations**8**(2003), no. 2, 161–187. MR**1948043****8.**S.Y. Hsu,*Removable singularities of semilinear parabolic equations*, Advances in Differential Equations 15 (2010), no. 1-2, 137-158.**9.**Kin Ming Hui,*A Fatou theorem for the solution of the heat equation at the corner points of a cylinder*, Trans. Amer. Math. Soc.**333**(1992), no. 2, 607–642. MR**1091707**, 10.1090/S0002-9947-1992-1091707-4**10.**Kin Ming Hui,*Existence of solutions of the equation 𝑢_{𝑡}=Δlog𝑢*, Nonlinear Anal.**37**(1999), no. 7, Ser. A: Theory Methods, 875–914. MR**1695083**, 10.1016/S0362-546X(98)00081-9**11.**Shilong Kuang and Qi S. Zhang,*A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow*, J. Funct. Anal.**255**(2008), no. 4, 1008–1023. MR**2433960**, 10.1016/j.jfa.2008.05.014**12.**O.A. Ladyzenskaya, V.A. Solonnikov, and N.N. Uraltceva,*Linear and quasilinear equations of parabolic type*, Transl. Math. Mono., Vol. 23, Amer. Math. Soc., Providence, R.I., 1968.**13.**J. Sacks and K. Uhlenbeck,*The existence of minimal immersions of 2-spheres*, Ann. of Math. (2)**113**(1981), no. 1, 1–24. MR**604040**, 10.2307/1971131**14.**Shota Sato and Eiji Yanagida,*Solutions with moving singularities for a semilinear parabolic equation*, J. Differential Equations**246**(2009), no. 2, 724–748. MR**2468735**, 10.1016/j.jde.2008.09.004**15.**Joel Smoller,*Shock waves and reaction-diffusion equations*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR**1301779**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
35B65,
35K05

Retrieve articles in all journals with MSC (2010): 35B65, 35K05

Additional Information

**Kin Ming Hui**

Affiliation:
Institute of Mathematics, Academia Sinica, Nankang, Taipei, 11529, Taiwan, Republic of China

DOI:
http://dx.doi.org/10.1090/S0002-9939-10-10352-9

Keywords:
Removable singularities,
heat equation

Received by editor(s):
September 1, 2009

Received by editor(s) in revised form:
September 2, 2009

Published electronically:
February 18, 2010

Communicated by:
Yingfei Yi

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.