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A variational problem associated with the minimal speed of traveling waves for spatially periodic KPP type equations. (arXiv:1712.09778v1 [math.AP])
来源于:arXiv
We consider a variational problem associated with the minimal speed of
pulsating traveling waves of the equation $u_t=u_{xx}+b(x)(1-u)u$,
$x\in{\mathbb R},\ t>0$, where the coefficient $b(x)$ is nonnegative and
periodic in $x\in{\mathbb R}$ with a period $L>0$. It is known that there
exists a quantity $c^*(b)>0$ such that a pulsating traveling wave with the
average speed $c>0$ exists if and only if $c\geq c^*(b)$. The quantity $c^*(b)$
is the so-called minimal speed of pulsating traveling waves. In this paper, we
study the problem of maximizing $c^*(b)$ by varying the coefficient $b(x)$
under some constraints. We prove the existence of the maximizer under a certain
assumption of the constraint and derive the Euler--Lagrange equation which the
maximizer satisfies under $L^2$ constraint $\int_0^L b(x)^2dx=\beta$. The limit
problems of the solution of this Euler--Lagrange equation as $L\rightarrow0$
and as $\beta\rightarrow0$ are also considered. Moreover, we also consider the 查看全文>>