Existence of traveling wave solutions for a nonlocal bistable equation: an abstract approach
Existence of traveling wave solutions for a nonlocal bistable equation: an abstract approach
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DOI：
10.2977/prims/1260476649
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发表时间：
2008-10
影响因子：
1.2
通讯作者：
Hiroki Yagisita
中科院分区：
文献类型：
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作者：
Hiroki Yagisita
我们考虑移动前沿到非局部双稳态方程 ut = µ * u - u + f(u)，其中 µ 是 R 上的 Borel 测度，μ(R) = 1，并且 f 满足 f(0) = f(1) = 0，对于某个常数 α ∈ (0, 1)，f 0 in (α, 1)。我们不假设 µ 对于勒贝格测度绝对连续。我们证明，存在常数 c 和单调函数 φ，且 φ(-∞) = 0 且 φ(+∞) = 1，使得 u(t, x) := φ(x + ct) 是方程的一个解，前提是 f′(α) > 0。为了证明这个结果，我们将开发一种抽象单调动力系统的递归方法并将其应用于方程。
We consider traveling fronts to the nonlocal bistable equation ut = µ * u - u + f(u), where µ is a Borel-measure on R with µ(R) = 1 and f satisfies f(0) = f(1) = 0, f 0 in (α, 1) for some constant α ∈ (0, 1). We do not assume that µ is absolutely continuous with respect to the Lebesgue measure. We show that there are a constant c and a monotone function φ with φ(-∞) = 0 and φ(+∞) = 1 such that u(t, x) : = φ(x + ct) is a solution to the equation, provided f′(α) > 0. In order to prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation.