A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations
A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations
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DOI：
10.1090/s0002-9947-2010-04931-1
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发表时间：
2010-04
影响因子：
1.3
通讯作者：
Xing Liang;Xiaotao Lin;H. Matano
中科院分区：
文献类型：
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作者：
Xing Liang;Xiaotao Lin;H. Matano
我们考虑方程ut = uxx + B（x）u（1 − u）; x2 R;其中B（x）是R上的非负测度，它在x中是周期性的：在B（x）是光滑周期函数的情况下，已知对于任意的cc，存在速度为c的行波。（B）;其中c（B）是取决于B的某个正数：这样的行波通常被称为“脉动行波”或“周期行波”，而c（B）被称为“最小速度”。在本文中，我们首先通过证明对于任何周期为L的非负测度B，最小速度c ′（B）的存在性来扩展这个理论：接下来我们研究在约束R（0，L）B（x）dx = λ L下最大化c ′（B）的问题;其中λ是任意给定的常数。这一问题与80年代后期数学生态学家研究的问题密切相关，但其答案一直没有得到解答。我们回答这个问题，证明最大值是通过周期性排列的狄拉克δ函数L P k2 Z�（x + kL）获得的：
We consider the equation ut = uxx + b(x)u(1 − u); x2 R;where b(x) is a nonnegative measure on R that is periodic in x:In the case where b(x) is a smooth periodic function, it is known that there exists a travelling wave with speed c for any cc � (b); where c � (b) is a certain positive number depending on b:Such a travelling wave is often called a "pulsating travelling wave" or a "periodic travelling wave", and c � (b) is called the "minimal speed". In this paper, we first extend this theory by showing the existence of the minimal speed c � (b) for any nonnegative measure b with period L:Next we study the question of maximizing c � (b) under the constraint R (0,L) b(x)dx = �L; where � is an arbitrarily given constant. This question is closely related to the problem studied by mathematical ecologists in late 1980's but its answer has not been known. We answer this question by proving that the maximum is attained by periodically arrayed Dirac's delta functionsL P k2Z �(x + kL):