American Institute of Mathematical Sciences

January  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:
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References:
 [1] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété riemanniennes, in "Lecture Notes in Math.," 194, Springer-Verlag, 1971.  Google Scholar [2] 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: 10.1016/0022-0396(89)90131-9.  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: 10.1002/cpa.3160420607.  Google Scholar [5] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, Berlin, 1998.  Google Scholar [6] 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 [7] L. A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model., 2 (1990), 63-74, (in Russian).  Google Scholar [8] N. Mizoguchi, Blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity, J. Differential Equations, 205 (2004), 298-328. doi: 10.1016/j.jde.2004.03.001.  Google Scholar [9] 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: 10.1017/S0308210509000444.  Google Scholar [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), 176-211. doi: 10.1016/j.jde.2006.07.012.  Google Scholar [11] S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations, 246 (2009), 724-748. doi: 10.1016/j.jde.2008.09.004.  Google Scholar [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), 313-331. doi: 10.3934/dcds.2010.26.313.  Google Scholar [13] S. Sato and E. Yanagida, Singular backward self-similar solution of a semilinear parabolic equation, Discrete Continuous Dynam. Systems -S, 4 (2011), 897-906. doi: 10.3934/dcdss.2011.4.897.  Google Scholar [14] T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. J., 57 (2008), 3365-3396. doi: 10.1512/iumj.2008.57.3269.  Google Scholar [15] W. C. Troy, The existence of bounded solutions of a semilinear heat equation, SIAM J. Math. Anal., 18 (1987), 332-336. doi: 10.1137/0518026.  Google Scholar [16] L. Véron, Singularities of solutions of second order quasilinear equations, in "Pitman Research Notes in Mathematics Series," 353, Longman, Harlow, 1996.  Google Scholar
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