Nonexistence of radial time-periodic solutions of reaction-diffusion equations with generic nonlinearities

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Abstract

We consider reaction-diffusion equations ut=Δu+f(u) on the entire space RN, N≥4. Assuming that the function f is sufficiently smooth (C2 is sufficient) and has only nondegenerate zeros, we prove that the equation has no bounded solutions u(x,t) which are radial in x, and periodic and nonconstant in t. We also prove some weaker nonexistence results for N=3. In dimensions N=1,2, the nonexistence of time-periodic solutions (radial or not) is known by results of Gallay and Slijepčević.

Original languageEnglish (US)
Pages (from-to)307-326
Number of pages20
JournalJournal of Differential Equations
Volume363
DOIs
StatePublished - Aug 5 2023

Bibliographical note

Publisher Copyright:
© 2023 Elsevier Inc.

Keywords

  • Periodic solutions
  • Reaction-diffusion equations on the entire space

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