American Institute of Mathematical Sciences

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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

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