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A remark on the long-time asymptotics of global-in-time solutions for semilinear parabolic equations in ${\mathbb {R}^N}$

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  10 October 2025

Michinori Ishiwata  and
Ikkei Shimizu Open the ORCID record for Ikkei Shimizu [Opens in a new window]
Michinori Ishiwata
Affiliation:
Department of System Innovation, Graduate School of Engineering Science, The University of Osaka , Osaka, Japan e-mail: ishiwata.michinori.es@osaka-u.ac.jp
Ikkei Shimizu*
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University , Kyoto, Japan
*
e-mail: shimizu.ikkei.8s@kyoto-u.ac.jp
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Abstract

We consider the asymptotics of long-time behavior of a solution u of the semilinear parabolic problem $\partial _tu=\Delta u-u+u|u|^{p-2}$ in ${\mathbb {R}^N}\times (0,\infty )$, $u(0)=u_0\in H^1({\mathbb {R}^N})\cap L^\infty ({\mathbb {R}^N})$. Since the spatial domain on which the problem is posed is noncompact, we cannot expect the relative compactness of the solution orbit, e.g., in $H^1({\mathbb {R}^N})$ in general. In this article, we prove that the compactness of the orbit holds up to the ground state energy level, namely, if $\lim _{t\to \infty }I(u(t))\leq d_\infty $, where I is the energy functional associated with (P) and $d_\infty $ its ground state energy, then the orbit of $u(t)$ is compact in $H^1({\mathbb {R}^N})$. Our result includes the previous results in [4, 5].

Keywords

Semilinear parabolic equationasymptoticsground statenoncompactnessŁojasiewicz–Simon inequality

MSC classification

Primary: 35B40: Asymptotic behavior of solutions 35K58: Semilinear parabolic equations

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 21
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work was supported by JSPS KAKENHI Grant Numbers JP19H05599. The second author is supported by JSPS KAKENHI Grant-in-Aid for JSPS Fellows 23KJ1416.

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