American Institute of Mathematical Sciences

Advanced
September  2009, 25(3): 991-1001. doi: 10.3934/dcds.2009.25.991

Asymptotic stability of singular solution to nonlinear heat equation

Dominika Pilarczyk 1,

1. 

Instytut Matematyczny, Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland

Received  December 2008 Revised  June 2009 Published  August 2009

In this paper, we discuss the asymptotic stability of singular steady states of the nonlinear heat equation $u_t=\Delta u+u^p$ in weighted $L^r$ - norms.
Keywords: semilinear parabolic equation, asymptotics of solutions, supercritical nonlinearity..
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B33, 35B4.
Citation: Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991
[1]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[2]

Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021060
[3]

Minkyu Kwak, Kyong Yu. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 1996, 2 (4) : 483-496. doi: 10.3934/dcds.1996.2.483
[4]

Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027
[5]

Alfonso Castro, Benjamin Preskill. Existence of solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 649-658. doi: 10.3934/dcds.2010.28.649
[6]

Lorena Bociu, Petronela Radu. Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping. Conference Publications, 2009, 2009 (Special) : 60-71. doi: 10.3934/proc.2009.2009.60
[7]

Haixia Li. Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020088
[8]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
[9]

Zhengce Zhang, Yanyan Li. Gradient blowup solutions of a semilinear parabolic equation with exponential source. Communications on Pure & Applied Analysis, 2013, 12 (1) : 269-280. doi: 10.3934/cpaa.2013.12.269
[10]

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
[11]

Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019
[12]

Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703
[13]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613
[14]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
[15]

Satoshi Hashimoto, Mitsuharu Ôtani. Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 323-333. doi: 10.3934/dcds.2007.19.323
[16]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
[17]

José Caicedo, Alfonso Castro, Rodrigo Duque, Arturo Sanjuán. Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1193-1202. doi: 10.3934/dcdss.2014.7.1193
[18]

Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309
[19]

Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50
[20]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences