Asymptotic spatial behavior for the heat equation on noncompact regions

Robin J. Knops*, Ramon Quintanilla

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
31 Downloads (Pure)


We consider the isotropic initial boundary value problem for the heat equation on open regions with noncompact boundary and construct differential inequalities for a generalized heat flow measure defined over a spherical cross section. Under suitable assumptions, integration of the differential inequality leads to spatial growth and decay rate estimates for mean-square cross-sectional measures of the time-weighted temperature spatial gradient. The estimates are then used to obtain similar results for the time-weighted temperature. In particular, when the base heat flow measure is positive, the time-weighted temperature becomes pointwise unbounded at large spatial distance.

Original languageEnglish
Pages (from-to)1049-1076
Number of pages28
JournalStudies in Applied Mathematics
Issue number4
Early online date16 Jan 2024
Publication statusPublished - May 2024


  • heat equation
  • noncompact region
  • spatial growth and decay rate estimates

ASJC Scopus subject areas

  • Applied Mathematics


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