Taiwanese Journal of Mathematics

Long Time Behavior of Entire Solutions to Bistable Reaction Diffusion Equations

Yang Wang and Xiong Li

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

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Abstract

The first aim in the paper is to prove the local exponential asymptotic stability of some entire solutions to bistable reaction diffusion equations via the super-sub solution method. If the integral of the reaction term $f$ over the interval $[0,1]$ is positive, we not only obtain the similar asymptotic stability result found by Yagisita in 2003, but also simplify the proof. The asymptotic stability result for the case $\int^1_{0} f(u) \, du \lt 0$ is also obtained, which is not considered by Yagisita. After that, the asymptotic behavior of entire solutions as $t \to +\infty$ is investigated, since the other side was completely known. Here, the result is established by use of the asymptotic stability of constant solutions and pairs of diverging traveling front solutions, instead of constructing the super-sub solutions as usual. Finally, for the special bistable case $f(u) = u(1-u)(u-\alpha)$, $\alpha \in (0,1)$, we prove the entire solution continuously depends on $\alpha$.

Article information

Source
Taiwanese J. Math., Advance publication (2020), 22 pages.

Dates
First available in Project Euclid: 11 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1589184077

Digital Object Identifier
doi:10.11650/tjm/200502

Subjects
Primary: 35K57: Reaction-diffusion equations 35B35: Stability 35B41: Attractors

Keywords
entire solutions stability reaction diffusion equations

Citation

Wang, Yang; Li, Xiong. Long Time Behavior of Entire Solutions to Bistable Reaction Diffusion Equations. Taiwanese J. Math., advance publication, 11 May 2020. doi:10.11650/tjm/200502. https://projecteuclid.org/euclid.twjm/1589184077


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