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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

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.
Citation: Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 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), 270-290. 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), 986-993. 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), 1258-1271. doi: 10.1137/0115108. [4] L. R. Bragg, The radial heat equation with pole type data, Bull. Amer. Math. Soc., 73 (1967), 133-135. 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), 221-246. [6] D. T. Haimo, Functions with the Huygens property, Bull. Amer. Math. Soc., 71 (1965), 528-532. 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), 735-758. [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. "Nauka," Moscow, 1968, 736 pp. [9] P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems.Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007. [10] S. Sato, A singular solution with smooth initial data for a semilinear parabolic equation, Nonlinear Anal., 74 (2011), 1383-1392. 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), 724-748. 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), 313-331. [13] S. Sato and E. Yanagida, Singular backward self-similar solutions of a semilinear parabolic equation, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 897-906. [14] S. Sato and E. Yanagida, Appearance of anomalous singularities in a semilinear parabolic equation, Commun. Pure Appl. Anal., 11 (2012), 387-405. 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, Longman, Harlow, 1996.

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##### References:
 [1] L. R. Bragg, The radial heat polynomials and related functions, Trans. Amer. Math. Soc., 119 (1965), 270-290. 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), 986-993. 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), 1258-1271. doi: 10.1137/0115108. [4] L. R. Bragg, The radial heat equation with pole type data, Bull. Amer. Math. Soc., 73 (1967), 133-135. 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), 221-246. [6] D. T. Haimo, Functions with the Huygens property, Bull. Amer. Math. Soc., 71 (1965), 528-532. 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), 735-758. [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. "Nauka," Moscow, 1968, 736 pp. [9] P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems.Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007. [10] S. Sato, A singular solution with smooth initial data for a semilinear parabolic equation, Nonlinear Anal., 74 (2011), 1383-1392. 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), 724-748. 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), 313-331. [13] S. Sato and E. Yanagida, Singular backward self-similar solutions of a semilinear parabolic equation, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 897-906. [14] S. Sato and E. Yanagida, Appearance of anomalous singularities in a semilinear parabolic equation, Commun. Pure Appl. Anal., 11 (2012), 387-405. 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, Longman, Harlow, 1996.
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