American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity
  • DCDS Home
  • This Issue
  • Next Article
    Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities
November  2012, 32(11): 4027-4043. doi: 10.3934/dcds.2012.32.4027

Asymptotic behavior of singular solutions for a semilinear parabolic equation

Shota Sato 1, and  Eiji Yanagida 2,

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.
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.  Google Scholar

[2]

L. R. Bragg, The radial heat equation and Laplace transforms,, SIAM J. Appl. Math., 14 (1966), 986.  doi: 10.1137/0114080.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

D. T. Haimo, Functions with the Huygens property,, Bull. Amer. Math. Soc., 71 (1965), 528.  doi: 10.1090/S0002-9904-1965-11318-0.  Google Scholar

[7]

D. T. Haimo, Expansions in terms of generalized heat polynomials and their Appell transforms,, J. Math. Mech., 15 (1966), 735.   Google Scholar

[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).   Google Scholar

[9]

P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems.Blow-up, Global Existence and Steady States,", Birkh\, (2007).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,", Pitman Research Notes in Mathematics Series, 353 (1996).   Google Scholar

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.  Google Scholar

[2]

L. R. Bragg, The radial heat equation and Laplace transforms,, SIAM J. Appl. Math., 14 (1966), 986.  doi: 10.1137/0114080.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

D. T. Haimo, Functions with the Huygens property,, Bull. Amer. Math. Soc., 71 (1965), 528.  doi: 10.1090/S0002-9904-1965-11318-0.  Google Scholar

[7]

D. T. Haimo, Expansions in terms of generalized heat polynomials and their Appell transforms,, J. Math. Mech., 15 (1966), 735.   Google Scholar

[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).   Google Scholar

[9]

P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems.Blow-up, Global Existence and Steady States,", Birkh\, (2007).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,", Pitman Research Notes in Mathematics Series, 353 (1996).   Google Scholar

[1]

Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351
[2]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[3]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307
[4]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
[5]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[6]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39
[7]

Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703
[8]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[9]

Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897
[10]

Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459
[11]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[12]

Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159
[13]

Lingwei Ma, Zhong Bo Fang. A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1697-1706. doi: 10.3934/cpaa.2017081
[14]

Michinori Ishiwata. Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent. Conference Publications, 2005, 2005 (Special) : 443-452. doi: 10.3934/proc.2005.2005.443
[15]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
[16]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
[17]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[18]

Qilin Xie, Jianshe Yu. Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension. Communications on Pure & Applied Analysis, 2019, 18 (1) : 129-158. doi: 10.3934/cpaa.2019008
[19]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[20]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences