Existence,Uniqueness, and Stability of Generalized Traveling Waves in Time Dependent Monostable Equations |
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Authors: | Wenxian Shen |
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Institution: | 1.Department of Mathematics and Statistics,Auburn University,Auburn,USA |
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Abstract: | The current paper is devoted to the study of traveling wave solutions of spatially homogeneous monostable reaction diffusion
equations with ergodic or recurrent time dependence, which includes periodic and almost periodic time dependence as special
cases. Such an equation has two spatially homogeneous and time recurrent solutions with one of them being stable and the other
being unstable. Traveling wave solutions are a type of entire solutions connecting the two spatially homogeneous and time
recurrent solutions. Recently, the author of the current paper proved that a spatially homogeneous time almost periodic monostable
equation has a spreading speed in any given direction. This result can be easily extended to monostable equations with recurrent
time dependence. In this paper, we introduce generalized traveling wave solutions for time recurrent monostable equations
and show the existence of such solutions in any given direction with average propagating speed greater than or equal to the
spreading speed in that direction and non-existence of such solutions of slower average propagating speed. We also show the
uniqueness and stability of generalized traveling wave solutions in any given direction with average propagating speed greater
than the spreading speed in that direction. Moreover, we show that a generalized traveling wave solution in a given direction
with average propagating speed greater than the spreading speed in that direction is unique ergodic in the sense that its
wave profile and wave speed are unique ergodic, and if the time dependence of the monostable equation is almost periodic,
it is almost periodic in the sense that its wave profile and wave speed are almost periodic. |
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