Asymptotic spatial behavior for the heat equation on noncompact regions

Robin J. Knops; Ramon Quintanilla

doi:10.1111/sapm.12669

Asymptotic spatial behavior for the heat equation on noncompact regions

Robin J. Knops^*, Ramon Quintanilla

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

35 Downloads (Pure)

Abstract

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 language	English
Pages (from-to)	1049-1076
Number of pages	28
Journal	Studies in Applied Mathematics
Volume	152
Issue number	4
Early online date	16 Jan 2024
DOIs	https://doi.org/10.1111/sapm.12669
Publication status	Published - May 2024

Keywords

heat equation
noncompact region
spatial growth and decay rate estimates

ASJC Scopus subject areas

Applied Mathematics

Access to Document

10.1111/sapm.12669Licence: CC BY

Download pdf
© 2024 The Authors.
Final published version, 287 KBLicence: CC BY

Cite this

@article{a4ee09c8bdf14ecea3bbeb3b63f0f0b5,

title = "Asymptotic spatial behavior for the heat equation on noncompact regions",

abstract = "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.",

keywords = "heat equation, noncompact region, spatial growth and decay rate estimates",

author = "Knops, {Robin J.} and Ramon Quintanilla",

note = "Funding Information: The authors are grateful to John Ball and Lyonell Boulton for constructive discussions on several aspects. The work of R. Quintanilla has been funded by the research project PID2019‐105118GB‐100, funded by the Spanish Ministry of Science, Innovation and Universities and FEDER “A way to make Europe.” Publisher Copyright: {\textcopyright} 2024 The Authors. Studies in Applied Mathematics published by Wiley Periodicals LLC.",

year = "2024",

month = may,

doi = "10.1111/sapm.12669",

language = "English",

volume = "152",

pages = "1049--1076",

journal = "Studies in Applied Mathematics",

issn = "0022-2526",

publisher = "Wiley-Blackwell",

number = "4",

}

TY - JOUR

T1 - Asymptotic spatial behavior for the heat equation on noncompact regions

AU - Knops, Robin J.

AU - Quintanilla, Ramon

N1 - Funding Information: The authors are grateful to John Ball and Lyonell Boulton for constructive discussions on several aspects. The work of R. Quintanilla has been funded by the research project PID2019‐105118GB‐100, funded by the Spanish Ministry of Science, Innovation and Universities and FEDER “A way to make Europe.” Publisher Copyright: © 2024 The Authors. Studies in Applied Mathematics published by Wiley Periodicals LLC.

PY - 2024/5

Y1 - 2024/5

N2 - 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.

AB - 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.

KW - heat equation

KW - noncompact region

KW - spatial growth and decay rate estimates

UR - http://www.scopus.com/inward/record.url?scp=85182477772&partnerID=8YFLogxK

U2 - 10.1111/sapm.12669

DO - 10.1111/sapm.12669

M3 - Article

AN - SCOPUS:85182477772

SN - 0022-2526

VL - 152

SP - 1049

EP - 1076

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

IS - 4

ER -