Nonexistence of radial time-periodic solutions of reaction-diffusion equations with generic nonlinearities

Peter Poláčik

doi:10.1016/j.jde.2023.03.018

Nonexistence of radial time-periodic solutions of reaction-diffusion equations with generic nonlinearities

Peter Poláčik

School of Mathematics

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

1 Scopus citations

Abstract

We consider reaction-diffusion equations u_t=Δu+f(u) on the entire space R^N, N≥4. Assuming that the function f is sufficiently smooth (C² is sufficient) and has only nondegenerate zeros, we prove that the equation has no bounded solutions u(x,t) which are radial in x, and periodic and nonconstant in t. We also prove some weaker nonexistence results for N=3. In dimensions N=1,2, the nonexistence of time-periodic solutions (radial or not) is known by results of Gallay and Slijepčević.

Original language	English (US)
Pages (from-to)	307-326
Number of pages	20
Journal	Journal of Differential Equations
Volume	363
DOIs	https://doi.org/10.1016/j.jde.2023.03.018
State	Published - Aug 5 2023

Bibliographical note

Publisher Copyright:
© 2023 Elsevier Inc.

Keywords

Periodic solutions
Reaction-diffusion equations on the entire space

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10.1016/j.jde.2023.03.018

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title = "Nonexistence of radial time-periodic solutions of reaction-diffusion equations with generic nonlinearities",

abstract = "We consider reaction-diffusion equations ut=Δu+f(u) on the entire space RN, N≥4. Assuming that the function f is sufficiently smooth (C2 is sufficient) and has only nondegenerate zeros, we prove that the equation has no bounded solutions u(x,t) which are radial in x, and periodic and nonconstant in t. We also prove some weaker nonexistence results for N=3. In dimensions N=1,2, the nonexistence of time-periodic solutions (radial or not) is known by results of Gallay and Slijep{\v c}evi{\'c}.",

keywords = "Periodic solutions, Reaction-diffusion equations on the entire space",

author = "Peter Pol{\'a}{\v c}ik",

note = "Publisher Copyright: {\textcopyright} 2023 Elsevier Inc.",

year = "2023",

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day = "5",

doi = "10.1016/j.jde.2023.03.018",

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T1 - Nonexistence of radial time-periodic solutions of reaction-diffusion equations with generic nonlinearities

AU - Poláčik, Peter

PY - 2023/8/5

Y1 - 2023/8/5

N2 - We consider reaction-diffusion equations ut=Δu+f(u) on the entire space RN, N≥4. Assuming that the function f is sufficiently smooth (C2 is sufficient) and has only nondegenerate zeros, we prove that the equation has no bounded solutions u(x,t) which are radial in x, and periodic and nonconstant in t. We also prove some weaker nonexistence results for N=3. In dimensions N=1,2, the nonexistence of time-periodic solutions (radial or not) is known by results of Gallay and Slijepčević.

AB - We consider reaction-diffusion equations ut=Δu+f(u) on the entire space RN, N≥4. Assuming that the function f is sufficiently smooth (C2 is sufficient) and has only nondegenerate zeros, we prove that the equation has no bounded solutions u(x,t) which are radial in x, and periodic and nonconstant in t. We also prove some weaker nonexistence results for N=3. In dimensions N=1,2, the nonexistence of time-periodic solutions (radial or not) is known by results of Gallay and Slijepčević.

KW - Periodic solutions

KW - Reaction-diffusion equations on the entire space

UR - https://www.scopus.com/pages/publications/85150413951

UR - https://www.scopus.com/pages/publications/85150413951#tab=citedBy

U2 - 10.1016/j.jde.2023.03.018

DO - 10.1016/j.jde.2023.03.018

M3 - Article

AN - SCOPUS:85150413951

SN - 0022-0396

VL - 363

SP - 307

EP - 326

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

Nonexistence of radial time-periodic solutions of reaction-diffusion equations with generic nonlinearities

Abstract

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Keywords

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