Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III: unstable limit at infinity

Antoine Pauthier, Peter Poláčik

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1 Scopus citations

Abstract

This is a continuation, and conclusion, of our study of bounded solutions u of the semilinear parabolic equation ut= uxx+ f(u) on the real line whose initial data u= u(· , 0) have finite limits θ± as x→ ± ∞. We assume that f is a locally Lipschitz function on R satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of u(x, t) as t→ ∞. In the first two parts of this series we mainly considered the cases where either θ-≠ θ+; or θ±= θ and f(θ) ≠ 0 ; or else θ±= θ, f(θ) = 0 , and θ is a stable equilibrium of the equation ξ˙ = f(ξ). In all these cases we proved that the corresponding solution u is quasiconvergent—if bounded—which is to say that all limit profiles of u(· , t) as t→ ∞ are steady states. The limit profiles, or accumulation points, are taken in Lloc∞(R). In the present paper, we take on the case that θ±= θ, f(θ) = 0 , and θ is an unstable equilibrium of the equation ξ˙ = f(ξ). Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on u(· , t) is that it is nonoscillatory (has only finitely many critical points) at some t≥ 0. Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal.

Original languageEnglish (US)
Article number48
JournalPartial Differential Equations and Applications
Volume3
Issue number4
DOIs
StatePublished - Aug 2022

Bibliographical note

Funding Information:
Supported in part by the NSF Grant DMS-1856491.

Publisher Copyright:
© 2022, The Author(s).

Keywords

  • Chains
  • Entire solutions
  • Parabolic equations on R
  • Quasiconvergence
  • Spatial trajectories
  • Zero number

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