Abstract.
The Cauchy problem for semilinear heat equations with singular initial data is studied, where N≥2, λ>0 is a parameter, and a≥0, a≠0. We show that when p>(N+2)/N and (N−2)p<N+2, there exists a positive constant such that the problem has two positive self-similar solutions and with if and no positive self-similar solutions if . Furthermore, for each fixed and in L ∞(R N) as λ→0, where w 0 is a non-unique solution to the problem with zero initial data, which is constructed by Haraux and Weissler.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Brezis, H.: Analyse Fonctionnelle. Masson, Paris, 1983
Cazenave, T., Dickstein, F., Escobedo, M., Weissler, F.B.: Self-similar solutions of a nonlinear heat equation. J. Math. Sci. Univ. Tokyo 8, 501–540 (2001)
Cazenave, T., Weissler, F.B.: Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations. Math. Z. 228, 83–120 (1998)
Crandall, M.G., Rabinowitz, P.H.: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Rational Mech. Anal. 58, 207–218 (1975)
DiBenedetto, E.: An Introduction to Partial Differential Equations. Birkhäuser, Boston, 1995
Dohmen, C., Hirose, M.: Structure of positive radial solutions to the Haraux-Weissler equation. Nonlinear Anal. TMA 33, 51–69 (1998)
Escobedo, M., Kavian, O.: Variational problems related to self-similar solutions for the heat equation. Nonlinear Anal. TMA 11, 1103–1133 (1987)
Escobedo, M., Kavian, O., Matano, H.: Large time behavior of solutions of a dissipative semilinear heat equation. Comm. Partial Diff. Eqs. 27, 1427–1452 (1995)
Fujita, H.: On the blowing up of solutions of the Cauchy problem for u t = Δu + u 1 + α. J. Fac. Sci. Univ. Tokyo, Sect. I 13, 109–124 (1966)
Galaktionov, V.A., Vazquez, J.L.: Continuation of blowup solutions of nonlinear heat equations in several space dimensions. Comm. Pure Appl. Math. 50, 1–67 (1997)
Giga, Y.: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Diff. Eqs. 62, 186–212 (1986)
Giga, Y., Miyakawa, T.: Navier-Stokes flow in R 3 with measures as initial vorticity and Morrey spaces. Comm. Partial Diff. Eqs. 14, 577–618 (1989)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of the Second Order. Springer-Verlag, Berlin, 1983
Gmira, A., Veron, L.: Large time behaviour of the solutions of a semilinear parabolic equation in R N. J. Diff. Eqs. 53, 258–276 (1984)
Haraux, A., Weissler, F.B.: Non-uniqueness for a semilinear initial value problem. Indiana Univ. Math. J. 31, 167–189 (1982)
Hartman, P., Wintner, A.: On a comparison theorem for self-adjoint partial differential equations of elliptic type. Proc. Amer. Math. Soc. 6, 862–865 (1955)
Herraiz, L.: Asymptotic behaviour of solutions of some semilinear parabolic problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 49–105 (1999)
Kamin, S., Peletier, L.A.: Large time behaviour of solutions of the heat equation with absorption. Ann. Scuola Norm. Sup. Pisa 12, 393–408 (1985)
Kavian, O.: Remarks on the large time behavior of a nonlinear diffusion equation. Annal. Insitut Henri Poincaré-Analyse nonlinéaire 4, 423–452 (1987)
Kawanago, T.: Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 1–15 (1996)
Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Partial Diff. Eqs. 19, 959–1014 (1994)
Kwak, M.: A semilinear heat equation with singular initial data. Proc. Royal Soc. Edinburgh Sect. A 128, 745–758 (1998)
Lee, T.-Y., Ni, W.-M.: Global existence, large time behavior and life span of solutions of semilinear parabolic Cauchy problem. Trans. Am. Math. Soc. 333, 365–371 (1992)
Naito, Y., Suzuki, T.: Radial symmetry of self-similar solutions for semilinear heat equations. J. Diff. Eqs. 163, 407–428 (2000)
Peletier, L.A., Terman, D., Weissler, F.B.: On the equation . Arch. Rational Mech. Anal. 94, 83–99 (1986)
Protter, M., Weinberger, H.: Maximal Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, NJ, 1967
Snoussi, S., Tayachi, S., Weissler, F.B.: Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient term. Proc. Roy. Soc. Edinburgh Sect. A 129, 1291–1307 (1999)
Snoussi, S., Tayachi, S., Weissler, F.B.: Asymptotically self-similar global solutions of a general semilinear heat equation. Math. Ann. 321, 131–155 (2001)
Souplet, P., Weissler, F.B.: Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state. Ann. Inst. H. Poincare Anal. Non Lineaire 20, 213–235 (2003)
Swanson, C.A.: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York, 1968
Weissler, F.B.: Semilinear evolution equations in Banach spaces. J. Funct. Anal. 32, 277–296 (1979)
Weissler, F.B.: Local existence and nonexistence for semilinear parabolic equations in L p. Indiana Univ. Math. J. 29, 79–102 (1980)
Weissler, F.B.: Existence and non-existence of global solutions for a semilinear heat equation. Israel J. Math. 38, 29–40 (1981)
Weissler, F.B.: Asymptotic analysis of an ordinary differential equation and nonuniqueness for a semilinear partial differential equation. Arch. Rational Mech. Anal. 91, 231–245 (1985)
Weissler, F.B.: Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations. Arch. Rational Mech. Anal. 91, 247–266 (1985)
Weissler, F.B.: L p-energy and blow-up for a semilinear heat equation. Proc. Sympos. Pure Math. 45 Part 2, Amer. Math. Soc., Providence, 1986, pp. 545–551
Yanagida, E.: Uniqueness of rapidly decaying solutions to the Haraux-Weissler equation. J. Diff. Eqs. 127, 561–570 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 35K55, 35J60
Rights and permissions
About this article
Cite this article
Naito, Y. Non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data. Math. Ann. 329, 161–196 (2004). https://doi.org/10.1007/s00208-004-0515-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-004-0515-4