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Published in:
2017 | OriginalPaper | Chapter
10. Traveling Waves in Bistable Nonlinearities
Author : Xiao-Qiang Zhao
Published in: Dynamical Systems in Population Biology
Publisher: Springer International Publishing
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Abstract
We consider the asymptotic behavior, as t → ∞, of the solutions of the problem where f(u, ⋅ ) is ω-periodic for some ω > 0, i.e., \(f(u,\omega +t) = f(u,t),\ \forall (u,t) \in \mathbb{R}^{2}\), and g is an arbitrary bounded function having certain asymptotic behavior as z → ±∞.
$$\displaystyle{\begin{array}{rl} \,&u_{t} - u_{zz} - f(u,t) = 0,\quad z \in \mathbb{R},\,\,t > 0, \\ \,&u(z,0) = g(z),\quad z \in \mathbb{R}, \end{array} }$$