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On bounded and unbounded global solutions of a supercritical semilinear heat equation

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Abstract

We consider the Cauchy problem where u 0 ∈C 0 (ℝN), the space of all continuous functions on ℝN that decay to zero at infinity, and p is supercritical in the sense that N≥11 and \({{p\ge ((N-2)^2-4N+8\sqrt{{N-1}})/{{(N-2)(N-10)}}}}\). We first examine the domain of attraction of steady states (and also of general solutions) in a class of admissible functions. In particular, we give a sharp condition on the initial function u 0 so that the solution of the above problem converges to a given steady state. Then we consider the asymptotic behavior of global solutions bounded above and below by classical steady states (such solutions have compact trajectories in C 0 (ℝN), under the supremum norm). Our main result reveals an interesting possibility: the solution may approach a continuum of steady states, not settling down to any particular one of them. Finally, we prove the existence of global unbounded solutions, a phenomenon that does not occur for Sobolev-subcritical exponents.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Peter Polá č ik

  2. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Eiji Yanagida

Authors
  1. Peter Polá č ik
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  2. Eiji Yanagida
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Correspondence to Eiji Yanagida.

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Mathematics Subject Classification (2000) Primary 35K15, Secondary 35B35, 35B40, 35B33

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Polá č ik, P., Yanagida, E. On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann. 327, 745–771 (2003). https://doi.org/10.1007/s00208-003-0469-y

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  • DOI: https://doi.org/10.1007/s00208-003-0469-y

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