Asymptotic behavior of singular solutions for a semilinear parabolic equation

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November 2012, 32(11): 4027-4043. doi: 10.3934/dcds.2012.32.4027

Asymptotic behavior of singular solutions for a semilinear parabolic equation

1.	Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
2.	Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551

Received May 2011 Revised August 2011 Published June 2012

Abstract
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We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. It is known that in some range of parameters, this equation has a family of singular steady states with ordered structure. Our concern in this paper is the existence of time-dependent singular solutions and their asymptotic behavior. In particular, we prove the convergence of solutions to singular steady states. The method of proofs is based on the analysis of a related linear parabolic equation with a singular coefficient and the comparison principle.

Keywords: critical exponent., Semilinear parabolic equation, singular solution, convergence to a steady state.

Mathematics Subject Classification: Primary: 35K58; Secondary: 35B33, 35B3.

Citation: Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027

References:

[1]	L. R. Bragg, The radial heat polynomials and related functions,, Trans. Amer. Math. Soc., 119 (1965), 270. doi: 10.1090/S0002-9947-1965-0181769-4.
[2]	L. R. Bragg, The radial heat equation and Laplace transforms,, SIAM J. Appl. Math., 14 (1966), 986. doi: 10.1137/0114080.
[3]	L. R. Bragg, On the solution structure of radial heat problems with singular data,, SIAM J. Appl. Math., 15 (1967), 1258. doi: 10.1137/0115108.
[4]	L. R. Bragg, The radial heat equation with pole type data,, Bull. Amer. Math. Soc., 73 (1967), 133. doi: 10.1090/S0002-9904-1967-11681-1.
[5]	C.-C. Chen and C.-S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations,, J. Geometric Analysis, 9 (1999), 221.
[6]	D. T. Haimo, Functions with the Huygens property,, Bull. Amer. Math. Soc., 71 (1965), 528. doi: 10.1090/S0002-9904-1965-11318-0.
[7]	D. T. Haimo, Expansions in terms of generalized heat polynomials and their Appell transforms,, J. Math. Mech., 15 (1966), 735.
[8]	O. A. Ladyženskaja, V. A. Solonnikov and N. M. Ural'ceva, "Lineĭnye i Kvazilineĭnye Uravneniya Parabolicheskogo Tipa," (Russian) [Linear and Quasi-linear Equations of Parabolic Type],, Izdat., (1968).
[9]	P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems.Blow-up, Global Existence and Steady States,", Birkh\, (2007).
[10]	S. Sato, A singular solution with smooth initial data for a semilinear parabolic equation,, Nonlinear Anal., 74 (2011), 1383. doi: 10.1016/j.na.2010.10.010.
[11]	S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation,, J. Differential Equations, 246 (2009), 724. doi: 10.1016/j.jde.2008.09.004.
[12]	S. Sato and E. Yanagida, Forward self-similar solution with a movingsingularity for a semilinear parabolic equation,, Discrete Contin. Dyn. Syst., 26 (2010), 313.
[13]	S. Sato and E. Yanagida, Singular backward self-similar solutions of a semilinear parabolic equation,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 897.
[14]	S. Sato and E. Yanagida, Appearance of anomalous singularities in a semilinear parabolic equation,, Commun. Pure Appl. Anal., 11 (2012), 387. doi: 10.3934/cpaa.2012.11.387.
[15]	L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,", Pitman Research Notes in Mathematics Series, 353 (1996).

show all references

References:

[1]	L. R. Bragg, The radial heat polynomials and related functions,, Trans. Amer. Math. Soc., 119 (1965), 270. doi: 10.1090/S0002-9947-1965-0181769-4.
[2]	L. R. Bragg, The radial heat equation and Laplace transforms,, SIAM J. Appl. Math., 14 (1966), 986. doi: 10.1137/0114080.
[3]	L. R. Bragg, On the solution structure of radial heat problems with singular data,, SIAM J. Appl. Math., 15 (1967), 1258. doi: 10.1137/0115108.
[4]	L. R. Bragg, The radial heat equation with pole type data,, Bull. Amer. Math. Soc., 73 (1967), 133. doi: 10.1090/S0002-9904-1967-11681-1.
[5]	C.-C. Chen and C.-S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations,, J. Geometric Analysis, 9 (1999), 221.
[6]	D. T. Haimo, Functions with the Huygens property,, Bull. Amer. Math. Soc., 71 (1965), 528. doi: 10.1090/S0002-9904-1965-11318-0.
[7]	D. T. Haimo, Expansions in terms of generalized heat polynomials and their Appell transforms,, J. Math. Mech., 15 (1966), 735.
[8]	O. A. Ladyženskaja, V. A. Solonnikov and N. M. Ural'ceva, "Lineĭnye i Kvazilineĭnye Uravneniya Parabolicheskogo Tipa," (Russian) [Linear and Quasi-linear Equations of Parabolic Type],, Izdat., (1968).
[9]	P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems.Blow-up, Global Existence and Steady States,", Birkh\, (2007).
[10]	S. Sato, A singular solution with smooth initial data for a semilinear parabolic equation,, Nonlinear Anal., 74 (2011), 1383. doi: 10.1016/j.na.2010.10.010.
[11]	S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation,, J. Differential Equations, 246 (2009), 724. doi: 10.1016/j.jde.2008.09.004.
[12]	S. Sato and E. Yanagida, Forward self-similar solution with a movingsingularity for a semilinear parabolic equation,, Discrete Contin. Dyn. Syst., 26 (2010), 313.
[13]	S. Sato and E. Yanagida, Singular backward self-similar solutions of a semilinear parabolic equation,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 897.
[14]	S. Sato and E. Yanagida, Appearance of anomalous singularities in a semilinear parabolic equation,, Commun. Pure Appl. Anal., 11 (2012), 387. doi: 10.3934/cpaa.2012.11.387.
[15]	L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,", Pitman Research Notes in Mathematics Series, 353 (1996).

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