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Asymptotic profiles for positive solutions of diffusive logistic equations

Part of: Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  15 February 2023

Jian-Wen Sun  and
Peng-Fei Fang
Jian-Wen Sun
Affiliation:
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China (jianwensun@lzu.edu.cn)
Peng-Fei Fang
Affiliation:
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China (jianwensun@lzu.edu.cn)
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Abstract

In this paper, we study the asymptotic profiles of positive solutions for diffusive logistic equations. The aim is to study the sharp effect of linear growth and nonlinear function. Both the classical reaction-diffusion equation and nonlocal dispersal equation are investigated. Our main results reveal that the linear and nonlinear parts of reaction term play quite different roles in the study of positive solutions.

Keywords

Reaction-diffusion equationpositive solutionlogistic

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35K57: Reaction-diffusion equations 92D25: Population dynamics (general)

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 1 , February 2024 , pp. 273 - 284
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Amann, H.. Existence of multiple solutions for nonlinear elliptic boundary value problems. Indiana Univ. Math. J. 21 (1971-72), 925–935.CrossRefGoogle Scholar
Andreu-Vaillo, F., Mazón, J. M., Rossi, J. D. and Toledo-Melero, J.. Nonlocal diffusion problems, mathematical surveys and monographs (AMS, Providence, Rhode Island, 2010).CrossRefGoogle Scholar
Berestycki, H.. Le nombre de solutions de certains problèmes semi-linéaires elliptiques. J. Funct. Anal. 40 (1981), 1–29.CrossRefGoogle Scholar
Brézis, H. and Oswald, L.. Remarks on sublinear elliptic equations. Nonlinear Anal. 10 (1986), 55–64.CrossRefGoogle Scholar
Cortazar, C., Elgueta, M., Rossi, J. D. and Wolanski, N.. Boundary fluxes for nonlocal diffusion. J. Differ. Equ. 234 (2007), 360–390.CrossRefGoogle Scholar
Chasseigne, E., Chaves, M. and Rossi, J. D.. Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. 86 (2006), 271–291.CrossRefGoogle Scholar
Daners, D. and López-Gómez, J.. Global dynamics of generalized logistic equations. Adv. Nonl. Studies 18 (2018), 217–236.CrossRefGoogle Scholar
Du, Y.. Spatial patterns for population models in a heterogeneous environment. Taiwanese J. Math. 8 (2004), 155–182.CrossRefGoogle Scholar
Du, Y.. Order structure and topological methods in nonlinear partial differential equations. Maximum principle and applications, vol 1, Fang-Hua Lin (Courant Institute of Math. Sci., New York University) (Series ed.) (Singapore: World Scientific Publishing, 2006).CrossRefGoogle Scholar
Fife, P.. Some nonclassical trends in parabolic and parabolic-like evolutions, In: Trends in Nonlinear Analysis (Springer, Berlin, 2003), pp. 153–191.CrossRefGoogle Scholar
Garcia-Melian, J. and Rossi, J. D.. A logistic equation with refuge and nonlocal diffusion. Commun. Pure Appl. Anal. 8 (2009), 2037–2053.CrossRefGoogle Scholar
Kao, C. Y., Lou, Y. and Shen, W. X.. Random dispersal vs. non-local dispersal. Discrete Contin. Dyn. Syst. 26 (2010), 551–596.CrossRefGoogle Scholar
Henry, D.. Geometric theory of semilinear parabolic equations, Lecture Notes Math., Vol. 840 (Springer-Verlag, 1981).CrossRefGoogle Scholar
Hutson, V., Martinez, S., Mischaikow, K. and Vickers, G. T.. The evolution of dispersal. J. Math. Biol. 47 (2003), 483–517.CrossRefGoogle ScholarPubMed
Li, W. T., López-Gómez, J. and Sun, J. W.. Sharp blow-up profiles of positive solutions for a class of semilinear elliptic problems. Adv. Nonlinear Stud. 21 (2021), 751–765.CrossRefGoogle Scholar
López-Gómez, J.. Metasolutions of Parabolic Equations in Population Dynamics (Boca Raton: CRC Press, 2016).Google Scholar
López-Gómez, J. and Rabinowitz, P.. The effects of spatial heterogeneities on some multiplicity results. Discrete Contin. Dyn. Syst. 127 (2016), 941–952.Google Scholar
Murray, J.. Mathematical Biology, 2nd ed. (New York: Springer-verlag, 1998).Google Scholar
Sun, J. W.. Limiting solutions of nonlocal dispersal problem in inhomogeneous media. J. Dynam. Differ. Equ. 34 (2022), 1489–1504.CrossRefGoogle Scholar
Sun, J. W.. Asymptotic profiles in diffusive logistic equations. Z. Angew. Math. Phys. 72 (2021), 152.CrossRefGoogle Scholar
Sun, J. W.. Effects of dispersal and spatial heterogeneity on nonlocal logistic equations. Nonlinearity 34 (2021), 5434–5455.CrossRefGoogle Scholar
Sun, J. W., Li, W. T. and Wang, Z. C.. A nonlocal dispersal logistic equation with spatial degeneracy. Discrete Contin. Dyn. Syst. 35 (2015), 3217–3238.CrossRefGoogle Scholar
Zhang, G. B., Li, T. T. and Sun, Y. J.. Asymptotic behavior for nonlocal dispersal equations. Nonlinear Anal. 72 (2010), 4466–4474.CrossRefGoogle Scholar