2 research outputs found
Determining the optimal coefficient of the spatially periodic Fisher-KPP equation that minimizes the spreading speed
This paper is concerned with the spatially periodic Fisher-KPP equation
, , where and are
periodic functions with period . We assume that has positive mean
and . It is known that there exists a positive number ,
called the minimal wave speed, such that a periodic traveling wave solution
with average speed exists if and only if . In the
one-dimensional case, the minimal speed coincides with the
``spreading speed'', that is, the asymptotic speed of the propagating front of
a solution with compactly supported initial data. In this paper, we study the
minimizing problem for the minimal speed by varying under a
certain constraint, while arbitrarily. We have been able to obtain an
explicit form of the minimizing function . Our result provides the first
calculable example of the minimal speed for spatially periodic Fisher-KPP
equations as far as the author knows
A variational problem associated with the minimal speed of traveling waves for spatially periodic KPP type equations
We consider a variational problem associated with the minimal speed of
pulsating traveling waves of the equation ,
, where the coefficient is nonnegative and
periodic in with a period . It is known that there
exists a quantity such that a pulsating traveling wave with the
average speed exists if and only if . The quantity
is the so-called minimal speed of pulsating traveling waves. In this paper, we
study the problem of maximizing by varying the coefficient
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 constraint . The limit
problems of the solution of this Euler--Lagrange equation as
and as are also considered. Moreover, we also consider the
variational problem in a certain class of step functions under constraint
when or tends to infinity