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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The periodic logistic equation with spatial and temporal degeneracies
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by Yihong Du and Rui Peng PDF
Trans. Amer. Math. Soc. 364 (2012), 6039-6070 Request permission

Abstract:

In this article, we study the degenerate periodic logistic equation with homogeneous Neumann boundary conditions: \begin{equation*} \begin {cases} \partial _t u-\Delta u=a u-b(x,t)u^p & \text {in $\Omega \times (0,\infty )$},\\ \partial _\nu u=0 & \text {on $\partial \Omega \times (0,\infty )$},\\ u(x,0) = u_0(x)\geq , \nequiv 0 & \text {in $\Omega $}, \end{cases} \end{equation*} where $\Omega \subset \mathbb {R}^N\ (N\geq 2)$ is a bounded domain with smooth boundary $\partial \Omega$, $a$ and $p>1$ are constants. The function $b\in C^{\theta ,\theta /2}(\overline \Omega \times \mathbb {R})$ $(0<\theta <1)$ is T-periodic in $t$, nonnegative, and vanishes (i.e., has a degeneracy) in some subdomain of $\Omega \times \mathbb {R}$. We examine the effects of various natural spatial and temporal degeneracies of $b(x,t)$ on the long-time dynamical behavior of the positive solutions. Our analysis leads to a new eigenvalue problem for periodic-parabolic operators over a varying cylinder and certain parabolic boundary blow-up problems not known before. The investigation in this paper shows that the temporal degeneracy causes a fundamental change of the dynamical behavior of the equation only when spatial degeneracy also exists; but in sharp contrast, whether or not temporal degeneracy appears in the equation, the spatial degeneracy always induces fundamental changes of the behavior of the equation, though such changes differ significantly according to whether or not there is temporal degeneracy.
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Additional Information
  • Yihong Du
  • Affiliation: Department of Mathematics, School of Science and Technology, University of New England, Armidale, NSW 2351, Australia
  • Email: ydu@turing.une.edu.au
  • Rui Peng
  • Affiliation: Department of Mathematics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, People’s Republic of China
  • MR Author ID: 728442
  • Email: pengrui_seu@163.com
  • Received by editor(s): June 3, 2009
  • Received by editor(s) in revised form: March 21, 2011
  • Published electronically: June 15, 2012
  • Additional Notes: The first author was partially supported by the Australian Research Council
    The second author was partially supported by the National Natural Science Foundation of China and the Program for New Century Excellent Talents in Universities of China
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 6039-6070
  • MSC (2010): Primary 35K20, 35K58, 35J75
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05590-5
  • MathSciNet review: 2946942