Abstract
In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has attracted a lot of attention and many new types of nonplanar traveling waves have been observed for scalar reaction-diffusion equations with various nonlinearities. In this paper, by using the comparison argument and constructing appropriate super- and subsolutions, we study the existence, uniqueness and stability of threedimensional traveling fronts of pyramidal shape for monotone bistable systems of reaction-diffusion equations in R3. The pyramidal traveling fronts are characterized as either a combination of planar traveling fronts on the lateral surfaces or a combination of two-dimensional V-form waves on the edges of the pyramid. In particular, our results are applicable to some important models in biology, such as Lotka-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of multiple obligate mutualists.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1-40 |
| Number of pages | 40 |
| Journal | Science China Mathematics |
| DOIs | |
| State | Accepted/In press - Aug 3 2016 |
Fingerprint
Keywords
- bistability
- existence
- pyramidal traveling fronts
- reaction-diffusion systems
- stability
- uniqueness
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems. / Wang, Zhi Cheng; Li, Wan Tong; Ruan, Shigui.
In: Science China Mathematics, 03.08.2016, p. 1-40.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems
AU - Wang, Zhi Cheng
AU - Li, Wan Tong
AU - Ruan, Shigui
PY - 2016/8/3
Y1 - 2016/8/3
N2 - In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has attracted a lot of attention and many new types of nonplanar traveling waves have been observed for scalar reaction-diffusion equations with various nonlinearities. In this paper, by using the comparison argument and constructing appropriate super- and subsolutions, we study the existence, uniqueness and stability of threedimensional traveling fronts of pyramidal shape for monotone bistable systems of reaction-diffusion equations in R3. The pyramidal traveling fronts are characterized as either a combination of planar traveling fronts on the lateral surfaces or a combination of two-dimensional V-form waves on the edges of the pyramid. In particular, our results are applicable to some important models in biology, such as Lotka-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of multiple obligate mutualists.
AB - In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has attracted a lot of attention and many new types of nonplanar traveling waves have been observed for scalar reaction-diffusion equations with various nonlinearities. In this paper, by using the comparison argument and constructing appropriate super- and subsolutions, we study the existence, uniqueness and stability of threedimensional traveling fronts of pyramidal shape for monotone bistable systems of reaction-diffusion equations in R3. The pyramidal traveling fronts are characterized as either a combination of planar traveling fronts on the lateral surfaces or a combination of two-dimensional V-form waves on the edges of the pyramid. In particular, our results are applicable to some important models in biology, such as Lotka-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of multiple obligate mutualists.
KW - bistability
KW - existence
KW - pyramidal traveling fronts
KW - reaction-diffusion systems
KW - stability
KW - uniqueness
UR - http://www.scopus.com/inward/record.url?scp=84982860675&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84982860675&partnerID=8YFLogxK
U2 - 10.1007/s11425-016-0015-x
DO - 10.1007/s11425-016-0015-x
M3 - Article
AN - SCOPUS:84982860675
SP - 1
EP - 40
JO - Science China Mathematics
JF - Science China Mathematics
SN - 1674-7283
ER -