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Transactions of the American Mathematical Society

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The periodic logistic equation with spatial and temporal degeneracies

Authors: Yihong Du and Rui Peng
Journal: Trans. Amer. Math. Soc. 364 (2012), 6039-6070
MSC (2010): Primary 35K20, 35K58, 35J75
Published electronically: June 15, 2012
MathSciNet review: 2946942
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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

Rui Peng
Affiliation: Department of Mathematics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, People’s Republic of China
MR Author ID: 728442

Keywords: Degenerate logistic equation, positive periodic solution, asymptotic behavior, boundary blow-up, principal eigenvalue
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.