Singular solutions of parabolic -Laplacian with absorption

Authors:
Xinfu Chen, Yuanwei Qi and Mingxin Wang

Journal:
Trans. Amer. Math. Soc. **359** (2007), 5653-5668

MSC (2000):
Primary 35K65, 35K15

DOI:
https://doi.org/10.1090/S0002-9947-07-04336-X

Published electronically:
May 8, 2007

MathSciNet review:
2327046

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider, for and , the -Laplacian evolution equation with absorption

- (i)
- When , there does not exist any such singular solution.
- (ii)
- When , there exists, for every , a unique singular solution that satisfies as .

Also, as , where is a singular solution that satisfies as .

Furthermore, any singular solution is either or for some finite positive .

**1.**H. Brezis & A. Friedman,*Nonlinear parabolic equations involving measures as initial conditions*, J. Math. Pures Appl.,**62**(1983), 73-97. MR**700049 (84g:35093)****2.**H. Brezis, L. A. Peletier & D. Terman,*A very singular solution of the heat equation with absorption*,

Arch. Rational Mech. Anal. ,**96**(1985), 185-209.**3.**Xinfu Chen, Yuanwei Qi, & Mingxin Wang,*Self-similar very singular solutions of the parabolic p-Laplacian with absorption,*

J. Differential Equuations,**190**(2003), 1-15. MR**1970953 (2004c:34050)****4.**Xinfu Chen, Yuanwei Qi, & Mingxin Wang,*Classification of singular solutions of porous medium equations with absorption,*

Proc. Roy. Edinburgh, Ser. A,**135**, 2005, 563-584. MR**2153436 (2006d:35125)****5.**J. I. Diaz & J. E. Saa,*Uniqueness of very singular self-similar solution of a quasilinear degenerate parabolic equation with absorption*,

Publ. Math.,**36**(1992), 19-38. MR**1179599 (93g:35079)****6.**E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. MR**1230384 (94h:35130)****7.**M. Escobedo, O. Kavian, & H. Matano,*Large time behavior of a dissipative semilinear heat equation,*Comm. Partial Diff. Eqns.,**20**(1995), 1427-1452. MR**1335757 (96f:35074)****8.**V. A. Galaktionov, S. P. Kurdyumov, & A. A. Samarskii,*On asymptotic ``eigenfunctions'' of the Cauchy problem for a non-linear parabolic equation,*Math. USSR Sbornik,**54**(1986), 421-455.**9.**S. Kamin & L. A. Peletier,*Singular solutions of the heat equation with absorption*,

Proc. Amer. Math. Soc.,**95**(1985), 205-210. MR**801324 (87b:35090)****10.**S. Kamin & L.A. Peletier,*Source-type solutions of degenerate diffusion equations with absorption,*Israel J. Math.,**50**(1985), 219-230. MR**793855 (87a:35112)****11.**S. Kamin, L. A. Peletier & J. L. Vazquez,*Classification of singular solutions of a nonlinear heat equation*,

Duke Math. J.,**58**(1989), 601-615. MR**1016437 (91g:35137)****12.**S. Kamin & J. L. Vazquez,*Fundamental solutions and asymptotic behaviour for the p-Laplacian equation*,

Rev. Mat. Iberoamericana,**4**(1988), 339-352. MR**1028745 (90m:35020)****13.**S. Kamin & J. L. Vazquez,*Singular solutions of some nonlinear parabolic equations*,

J. Analyse Math.,**59**(1992), 51-74. MR**1226951 (94e:35079)****14.**S. Kamin & L. Veron,*Existence and uniqueness of the very singular solution for the porous media equation with absorption*,

J. Analyse Math.,**51**(1988), 245-258. MR**963156 (90f:35097)****15.**M. Kwak,*A porous media equation with absorption. I. Long time behaviour,*

J. Math. Anal. Appl.,**223**(1998), 96-110. MR**1627352 (99e:35096a)****16.**M. Kwak,*A porous media equation with absorption. II. Uniqueness of the very singular solution,*

J. Math. Anal. Appl.,**223**(1998), 111-125. MR**1627348 (99e:35096b)****17.**G. Leoni,*A very singular solution for the porous media equation when ,*

J. Differential Equations,**132**(1996), 353-376. MR**1422124 (97j:35062)****18.**G. Leoni,*On very singular self-similar solutions for the porous media equation with absorption,*

Differential and Integral Equations,**10**(1997), 1123-1140. MR**1608045 (99e:35098)****19.**G. Leoni,*Classification of positive solutions of the problem div in ,**Differ. Uravn.*(Russian),**34**(1998), 1170-1178; translation in*Differential Equations*,**34**(1998), 1172-1180(1999). MR**1693586 (2000b:35065)****20.**L. Oswald,*Isolated, positive singularities for a nonlinear heat equation*,

Houston J. Math.,**14**(1988), 543-572. MR**998457 (90k:35128)****21.**L. A. Peletier & D. Terman,*A very singular solution of the porous media equation with absorption*,

J. Differential Equations,**65**(1986), 396-410. MR**865069 (88b:35090)****22.**L. A. Peletier & J. Wang,*A very singular solution of a quasi-linear degenerate diffusion equation with absorption*,

Trans. Amer. Math. Soc.,**307**(1988), 813-826. MR**940229 (89e:35081)****23.**L. A. Peletier & J. Zhao,*Source-type solutions of the porous media equation with absorption: The fast diffusion case*,

Nonlinear Analysis, TMA,**14**(1990), 107-121. MR**1036202 (91k:35140)****24.**L. A. Peletier & J. Zhao,*Large time behaviour of solutions of the porous media equation with absorption: The fast diffusion case*,

Nonlinear Analysis, TMA,**17**(1991), 991-1009. MR**1135955 (93d:76068)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35K65,
35K15

Retrieve articles in all journals with MSC (2000): 35K65, 35K15

Additional Information

**Xinfu Chen**

Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Email:
xinfu@pitt.edu

**Yuanwei Qi**

Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32816

Email:
yqi@pegasus.cc.ucf.edu

**Mingxin Wang**

Affiliation:
Department of Applied Mathematics, Southeast University, Nanjing 210018, People’s Republic of China

Email:
mxwang@seu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9947-07-04336-X

Keywords:
$p$-Laplacian,
fast diffusion,
absorption,
fundamental solution,
very singular solution.

Received by editor(s):
May 7, 2002

Received by editor(s) in revised form:
May 15, 2006

Published electronically:
May 8, 2007

Additional Notes:
All the authors are grateful to the Hong Kong RGC Grant HKUST 630/95P given to the second author. The first author would like to thank the National Science Foundation for Grant DMS-9971043, USA. The third author thanks the PRC for NSF Grant NSFC-19831060.

Article copyright:
© Copyright 2007
American Mathematical Society

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