Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum

Montie Avery; Arnd Scheel

doi:10.1137/20M1343476

Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum

Montie Avery, Arnd Scheel

School of Mathematics

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

5 Scopus citations

Abstract

We study nonlinear stability of pulled fronts in scalar parabolic equations on the real line of arbitrary order, under conceptual assumptions on existence and spectral stability of fronts. In this general setting, we establish sharp algebraic decay rates and temporal asymptotics of perturbations to the front. Some of these results are known for the specific example of the Fisher-KPP equation, and our results can thus be viewed as establishing universality of some aspects of this simple model. We also give a precise description of how the spatial localization of perturbations to the front affects the temporal decay rate, across the full range of localizations for which asymptotic stability holds. Technically, our approach is based on a detailed study of the resolvent operator for the linearized problem, through which we obtain sharp linear time decay estimates that allow for a direct nonlinear analysis.

Original language	English (US)
Pages (from-to)	2206-2242
Number of pages	37
Journal	SIAM Journal on Mathematical Analysis
Volume	53
Issue number	2
DOIs	https://doi.org/10.1137/20M1343476
State	Published - Apr 19 2021

Bibliographical note

Funding Information:
∗Received by the editors June 8, 2020; accepted for publication (in revised form) December 14, 2020; published electronically April 19, 2021. https://doi.org/10.1137/20M1343476 Funding: The work of the authors was supported by the National Science Foundation through the Graduate Research Fellowship Program under grant 00074041, as well as through grant DMS-1907391. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. †School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (avery142@ umn.edu, scheel@math.umn.edu).

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Keywords

Nonlinear stability
Pulled fronts
Resolvent expansions
Traveling waves

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10.1137/20M1343476

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title = "Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum",

abstract = "We study nonlinear stability of pulled fronts in scalar parabolic equations on the real line of arbitrary order, under conceptual assumptions on existence and spectral stability of fronts. In this general setting, we establish sharp algebraic decay rates and temporal asymptotics of perturbations to the front. Some of these results are known for the specific example of the Fisher-KPP equation, and our results can thus be viewed as establishing universality of some aspects of this simple model. We also give a precise description of how the spatial localization of perturbations to the front affects the temporal decay rate, across the full range of localizations for which asymptotic stability holds. Technically, our approach is based on a detailed study of the resolvent operator for the linearized problem, through which we obtain sharp linear time decay estimates that allow for a direct nonlinear analysis.",

keywords = "Nonlinear stability, Pulled fronts, Resolvent expansions, Traveling waves",

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note = "Funding Information: ∗Received by the editors June 8, 2020; accepted for publication (in revised form) December 14, 2020; published electronically April 19, 2021. https://doi.org/10.1137/20M1343476 Funding: The work of the authors was supported by the National Science Foundation through the Graduate Research Fellowship Program under grant 00074041, as well as through grant DMS-1907391. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. †School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (avery142@ umn.edu, scheel@math.umn.edu). Publisher Copyright: Copyright {\textcopyright} by SIAM. Unauthorized reproduction of this article is prohibited.",

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N1 - Funding Information: ∗Received by the editors June 8, 2020; accepted for publication (in revised form) December 14, 2020; published electronically April 19, 2021. https://doi.org/10.1137/20M1343476 Funding: The work of the authors was supported by the National Science Foundation through the Graduate Research Fellowship Program under grant 00074041, as well as through grant DMS-1907391. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. †School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (avery142@ umn.edu, scheel@math.umn.edu). Publisher Copyright: Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

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Y1 - 2021/4/19

N2 - We study nonlinear stability of pulled fronts in scalar parabolic equations on the real line of arbitrary order, under conceptual assumptions on existence and spectral stability of fronts. In this general setting, we establish sharp algebraic decay rates and temporal asymptotics of perturbations to the front. Some of these results are known for the specific example of the Fisher-KPP equation, and our results can thus be viewed as establishing universality of some aspects of this simple model. We also give a precise description of how the spatial localization of perturbations to the front affects the temporal decay rate, across the full range of localizations for which asymptotic stability holds. Technically, our approach is based on a detailed study of the resolvent operator for the linearized problem, through which we obtain sharp linear time decay estimates that allow for a direct nonlinear analysis.

AB - We study nonlinear stability of pulled fronts in scalar parabolic equations on the real line of arbitrary order, under conceptual assumptions on existence and spectral stability of fronts. In this general setting, we establish sharp algebraic decay rates and temporal asymptotics of perturbations to the front. Some of these results are known for the specific example of the Fisher-KPP equation, and our results can thus be viewed as establishing universality of some aspects of this simple model. We also give a precise description of how the spatial localization of perturbations to the front affects the temporal decay rate, across the full range of localizations for which asymptotic stability holds. Technically, our approach is based on a detailed study of the resolvent operator for the linearized problem, through which we obtain sharp linear time decay estimates that allow for a direct nonlinear analysis.

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KW - Resolvent expansions

KW - Traveling waves

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Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum

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