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Classification of singular solutions of a nonlinear heat equation. (English) Zbl 0701.35083

This is the detailed version of the authors’ note in C. R. Acad. Sci. Paris, Ser. I 305, 595-598 (1987; Zbl 0638.35042).

MSC:

35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35A08 Fundamental solutions to PDEs

Citations:

Zbl 0638.35042
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Full Text: DOI

References:

[1] D. Aronson, M. G. Crandall, and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem , Nonlinear Anal. 6 (1982), no. 10, 1001-1022. · Zbl 0518.35050 · doi:10.1016/0362-546X(82)90072-4
[2] H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions , J. Math. Pures Appl. (9) 62 (1983), no. 1, 73-97. · Zbl 0527.35043
[3] M. Bertsch, R. Kersner, and L. A. Peletier, Positivity versus localization in degenerate diffusion equations , Nonlinear Anal. 9 (1985), no. 9, 987-1008. · Zbl 0596.35073 · doi:10.1016/0362-546X(85)90081-1
[4] M. Bertsch, T. Nanbu, and L. A. Peletier, Decay of solutions of a degenerate nonlinear diffusion equation , Nonlinear Anal. 6 (1982), no. 6, 539-554. · Zbl 0487.35051 · doi:10.1016/0362-546X(82)90047-5
[5] H. Brezis and L. Oswald, Singular solutions for some semilinear elliptic equations , Arch. Rational Mech. Anal. 99 (1987), no. 3, 249-259. · Zbl 0635.35035 · doi:10.1007/BF00284508
[6] H. Brezis, L. A. Peletier, and D. Terman, A very singular solution of the heat equation with absorption , Arch. Rational Mech. Anal. 95 (1986), no. 3, 185-209. · Zbl 0627.35046 · doi:10.1007/BF00251357
[7] H. Brezis and L. Veron, Removable singularities for some nonlinear elliptic equations , Arch. Rational Mech. Anal. 75 (1980/81), no. 1, 1-6. · Zbl 0459.35032 · doi:10.1007/BF00284616
[8] S. Kamin and L. A. Peletier, Source-type solutions of degenerate diffusion equations with absorption , Israel J. Math. 50 (1985), no. 3, 219-230. · Zbl 0581.35035 · doi:10.1007/BF02761403
[9] S. Kamin and L. A. Peletier, Singular solutions of the heat equation with absorption , Proc. Amer. Math. Soc. 95 (1985), no. 2, 205-210. · Zbl 0607.35046 · doi:10.2307/2044513
[10] S. Kamin, L. A. Peletier, and J. L. Vazquez, Classification de solutions singulières à l’origine pour une équation de la chaleur non linéaire , C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 13, 595-598. · Zbl 0638.35042
[11] S. Kamin and L Veron, Existence and uniqueness of the very singular solution of the porous media equation with absorption , J. Analyse Math. 51 (1988), 245-258. · Zbl 0673.35044 · doi:10.1007/BF02791125
[12] L. Oswald, Isolated, positive singularities for a non-linear heat equation , Houston J. Math.; · Zbl 0696.35023
[13] L. A. Peletier and D. Terman, A very singular solution of the porous media equation with absorption , J. Differential Equations 65 (1986), no. 3, 396-410. · Zbl 0617.35115 · doi:10.1016/0022-0396(86)90026-4
[14] L. Veron, Singular solutions of some nonlinear elliptic equations , Nonlinear Anal. 5 (1981), no. 3, 225-242. · Zbl 0457.35031 · doi:10.1016/0362-546X(81)90028-6
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