## The periodic logistic equation with spatial and temporal degeneracies

HTML articles powered by AMS MathViewer

- 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.## References

- Herbert Amann,
*Linear and quasilinear parabolic problems. Vol. I*, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. MR**1345385**, DOI 10.1007/978-3-0348-9221-6 - Yihong Du,
*Order structure and topological methods in nonlinear partial differential equations. Vol. 1*, Series in Partial Differential Equations and Applications, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. Maximum principles and applications. MR**2205529**, DOI 10.1142/9789812774446 - Yihong Du and Qingguang Huang,
*Blow-up solutions for a class of semilinear elliptic and parabolic equations*, SIAM J. Math. Anal.**31**(1999), no. 1, 1–18. MR**1720128**, DOI 10.1137/S0036141099352844 - Yihong Du and Yoshio Yamada,
*On the long-time limit of positive solutions to the degenerate logistic equation*, Discrete Contin. Dyn. Syst.**25**(2009), no. 1, 123–132. MR**2525171**, DOI 10.3934/dcds.2009.25.123 - José M. Fraile, Pablo Koch Medina, Julián López-Gómez, and Sandro Merino,
*Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation*, J. Differential Equations**127**(1996), no. 1, 295–319. MR**1387267**, DOI 10.1006/jdeq.1996.0071 - Avner Friedman,
*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836** - Peter Hess,
*Periodic-parabolic boundary value problems and positivity*, Pitman Research Notes in Mathematics Series, vol. 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. MR**1100011** - O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva,
*Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa*, Izdat. “Nauka”, Moscow, 1967 (Russian). MR**0241821** - Gary M. Lieberman,
*Second order parabolic differential equations*, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR**1465184**, DOI 10.1142/3302 - Moshe Marcus and Laurent Véron,
*The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case*, Arch. Rational Mech. Anal.**144**(1998), no. 3, 201–231. MR**1658392**, DOI 10.1007/s002050050116 - Moshe Marcus and Laurent Véron,
*Existence and uniqueness results for large solutions of general nonlinear elliptic equations*, J. Evol. Equ.**3**(2003), no. 4, 637–652. Dedicated to Philippe Bénilan. MR**2058055**, DOI 10.1007/s00028-003-0122-y

## 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