Mathematics – Analysis of PDEs
Scientific paper
2010-04-05
Mathematics
Analysis of PDEs
Scientific paper
We consider the equation $u_t=u_{xx}+b(x)u(1-u),$ $x\in\mathbb R,$ where $b(x)$ is a nonnegative measure on $\mathbb R$ 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 $c\geq c^*(b),$ where $c^*(b)$ is a certain positive number depending on $b.$ Such a travelling wave is often called a \lq\lq pulsating travelling wave" or a \lq\lq periodic travelling wave", and $c^*(b)$ is called the \lq\lq 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 $\int_{[0,L)}b(x)dx=\alpha L,$ where $\alpha$ 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 functions $\alpha L\sum_{k\in\mathbb Z}\delta(x+kL).$
Liang Xing
Lin Xiaotao
Matano Hiroshi
No associations
LandOfFree
A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-636883