Mathematische Annalen

, Volume 327, Issue 4, pp 745–771

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

Article

Abstract

We consider the Cauchy problem where u0C0(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 u0 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 C0(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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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Mathematical InstituteTohoku UniversitySendaiJapan

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