Convergence to a steady state for asymptotically autonomous semilinear heat equations on R{double-struck}N

Juraj Földes, P. Polacik

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

We consider parabolic equations of the form. ut=δu+f(u)+h(x,t),(x,t)∈R{double-struck}N×(0,∞), where f is a C1 function with f(0)=0, f<(0)<0, and h is a suitable function on RN×[0,∈) which decays to zero as t→∈ (hence the equation is asymptotically autonomous). We show that, as t→∈, each bounded localized solution u≥0 approaches a set of steady states of the limit autonomous equation ut=δu+f(u). Moreover, if the decay of h is exponential, then u converges to a single steady state. We also prove a convergence result for abstract asymptotically autonomous parabolic equations.

Original languageEnglish (US)
Pages (from-to)1903-1922
Number of pages20
JournalJournal of Differential Equations
Volume251
Issue number7
DOIs
StatePublished - Oct 1 2011

Bibliographical note

Funding Information:
E-mail address: polacik@math.umn.edu (P. Polácˇik). 1 Supported in part by NSF Grant DMS-0900947.

Keywords

  • Asymptotically autonomous
  • Convergence
  • Parabolic equation
  • Quasiconvergence

Fingerprint Dive into the research topics of 'Convergence to a steady state for asymptotically autonomous semilinear heat equations on R{double-struck}<sup>N</sup>'. Together they form a unique fingerprint.

Cite this