2012, 11(1): 387-405. doi: 10.3934/cpaa.2012.11.387

Appearance of anomalous singularities in a semilinear parabolic equation

1. 

Mathematical Institute, Tohoku University, Sendai 980-8578

2. 

Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551

Received  March 2010 Revised  November 2010 Published  September 2011

The Cauchy problem for a parabolic partial differential equation with a power nonlinearity is studied. It is known that in some parameter range, there exists a time-local solution whose singularity has the same asymptotics as that of a singular steady state. In this paper, a sufficient condition for initial data is given for the existence of a solution with a moving singularity that becomes anomalous in finite time.
Citation: Shota Sato, Eiji Yanagida. Appearance of anomalous singularities in a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2012, 11 (1) : 387-405. doi: 10.3934/cpaa.2012.11.387
References:
[1]

M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété riemanniennes,, in, 194 (1971).

[2]

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

[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.

[4]

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

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer, (1998).

[6]

L. A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-conduction equation with distributed parameters,, Differentsial'nye Uravneniya, 24 (1988), 1226.

[7]

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

[8]

N. Mizoguchi, Blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity,, J. Differential Equations, 205 (2004), 298. doi: 10.1016/j.jde.2004.03.001.

[9]

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

[10]

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. doi: 10.1016/j.jde.2006.07.012.

[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.

[12]

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. doi: 10.3934/dcds.2010.26.313.

[13]

S. Sato and E. Yanagida, Singular backward self-similar solution of a semilinear parabolic equation,, Discrete Continuous Dynam. Systems -S, 4 (2011), 897. doi: 10.3934/dcdss.2011.4.897.

[14]

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

[15]

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

[16]

L. Véron, Singularities of solutions of second order quasilinear equations,, in, 353 (1996).

show all references

References:
[1]

M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété riemanniennes,, in, 194 (1971).

[2]

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

[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.

[4]

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

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer, (1998).

[6]

L. A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-conduction equation with distributed parameters,, Differentsial'nye Uravneniya, 24 (1988), 1226.

[7]

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

[8]

N. Mizoguchi, Blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity,, J. Differential Equations, 205 (2004), 298. doi: 10.1016/j.jde.2004.03.001.

[9]

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

[10]

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. doi: 10.1016/j.jde.2006.07.012.

[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.

[12]

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. doi: 10.3934/dcds.2010.26.313.

[13]

S. Sato and E. Yanagida, Singular backward self-similar solution of a semilinear parabolic equation,, Discrete Continuous Dynam. Systems -S, 4 (2011), 897. doi: 10.3934/dcdss.2011.4.897.

[14]

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

[15]

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

[16]

L. Véron, Singularities of solutions of second order quasilinear equations,, in, 353 (1996).

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