American Institute of Mathematical Sciences

Advanced
August  2011, 4(4): 897-906. doi: 10.3934/dcdss.2011.4.897

Singular backward self-similar solutions of 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, Japan

Received  September 2009 Revised  December 2009 Published  November 2010

We consider a parabolic partial differential equation with power nonlinearity. Our concern is the existence of a singular solution whose singularity becomes anomalous in finite time. First we study the structure of singular radial solutions for an equation derived by backward self-similar variables. Using this, we obtain a singular backward self-similar solution whose singularity becomes stronger or weaker than that of a singular steady state.
Keywords: critical exponent., Semilinear parabolic equation, backward self-similar solution, singular solution.
Mathematics Subject Classification: Primary: 35K55; Secondary: 35B3.
Citation: 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
References:
[1]

C. J. Budd and Y.-W. Qi, The existence of bounded solutions of a semilinear elliptic equation, J. Differential Equations, 82 (1989), 207-218. doi: doi:10.1016/0022-0396(89)90131-9.  Google Scholar

[2]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.  Google Scholar

[3]

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.  Google Scholar

[4]

Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equation, Comm. Pure Appl. Math., 42 (1989), 845-884. doi: doi:10.1002/cpa.3160420607.  Google Scholar

[5]

L. A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-conduction equation with distributed parameters, Differentsial'nye Uravneniya, 24 (1988), 1226-1234; English translation: Differential Equation, 24 (1988), 799-805.  Google Scholar

[6]

L. A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model., 2 (1990), 63-74, (in Russian).  Google Scholar

[7]

N. Mizoguchi, Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation, J. Funct. Anal. 257 (2009), 2911-2937. doi: doi:10.1016/j.jfa.2009.07.009.  Google Scholar

[8]

N. Mizoguchi, On backward self-similar blowup solutions to a supercritical semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 821-831. doi: doi:10.1017/S0308210509000444.  Google Scholar

[9]

Y. Naito and T. Suzuki, Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity, J. Differential Equations, 232 (2007), 176-211. doi: doi:10.1016/j.jde.2006.07.012.  Google Scholar

[10]

S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations, 246 (2009), 724-748. doi: doi:10.1016/j.jde.2008.09.004.  Google Scholar

[11]

S. Sato and E. Yanagida, Forward self-similar solution with a moving singularity for a semilinear parabolic equation, Disc. Cont. Dyn. Systems, 26 (2010), 313-331.  Google Scholar

[12]

S. Sato and E. Yanagida, Backward self-similar solution with a moving singularity for a semilinear parabolic equation,, preprint., ().   Google Scholar

[13]

T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. J., 57 (2008), 3365-3396. doi: doi:10.1512/iumj.2008.57.3269.  Google Scholar

[14]

W. C. Troy, The existence of bounded solutions of a semilinear heat equation, SIAM J. Math. Anal., 18 (1987), 332-336. doi: doi:10.1137/0518026.  Google Scholar

[15]

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

show all references

References:
[1]

C. J. Budd and Y.-W. Qi, The existence of bounded solutions of a semilinear elliptic equation, J. Differential Equations, 82 (1989), 207-218. doi: doi:10.1016/0022-0396(89)90131-9.  Google Scholar

[2]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.  Google Scholar

[3]

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.  Google Scholar

[4]

Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equation, Comm. Pure Appl. Math., 42 (1989), 845-884. doi: doi:10.1002/cpa.3160420607.  Google Scholar

[5]

L. A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-conduction equation with distributed parameters, Differentsial'nye Uravneniya, 24 (1988), 1226-1234; English translation: Differential Equation, 24 (1988), 799-805.  Google Scholar

[6]

L. A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model., 2 (1990), 63-74, (in Russian).  Google Scholar

[7]

N. Mizoguchi, Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation, J. Funct. Anal. 257 (2009), 2911-2937. doi: doi:10.1016/j.jfa.2009.07.009.  Google Scholar

[8]

N. Mizoguchi, On backward self-similar blowup solutions to a supercritical semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 821-831. doi: doi:10.1017/S0308210509000444.  Google Scholar

[9]

Y. Naito and T. Suzuki, Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity, J. Differential Equations, 232 (2007), 176-211. doi: doi:10.1016/j.jde.2006.07.012.  Google Scholar

[10]

S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations, 246 (2009), 724-748. doi: doi:10.1016/j.jde.2008.09.004.  Google Scholar

[11]

S. Sato and E. Yanagida, Forward self-similar solution with a moving singularity for a semilinear parabolic equation, Disc. Cont. Dyn. Systems, 26 (2010), 313-331.  Google Scholar

[12]

S. Sato and E. Yanagida, Backward self-similar solution with a moving singularity for a semilinear parabolic equation,, preprint., ().   Google Scholar

[13]

T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. J., 57 (2008), 3365-3396. doi: doi:10.1512/iumj.2008.57.3269.  Google Scholar

[14]

W. C. Troy, The existence of bounded solutions of a semilinear heat equation, SIAM J. Math. Anal., 18 (1987), 332-336. doi: doi:10.1137/0518026.  Google Scholar

[15]

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

[1]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[2]

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
[3]

Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857
[4]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801
[5]

Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101
[6]

Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3331-3355. doi: 10.3934/cpaa.2021108
[7]

Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471
[8]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
[9]

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
[10]

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
[11]

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
[12]

Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323
[13]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[14]

Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401
[15]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036
[16]

D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685
[17]

G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131
[18]

F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91
[19]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[20]

L. Olsen. Rates of convergence towards the boundary of a self-similar set. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 799-811. doi: 10.3934/dcds.2007.19.799

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences