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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 thecase where $b(x)$ is a smooth periodic function, it is known that there existsa travelling wave with speed $c$ for any $c\geq c^*(b),$ where $c^*(b)$ is acertain positive number depending on $b.$ Such a travelling wave is oftencalled a \lq\lq pulsating travelling wave" or a \lq\lq periodic travellingwave", and $c^*(b)$ is called the \lq\lq minimal speed". In this paper, wefirst 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 ofmaximizing $c^*(b)$ under the constraint $\int_{[0,L)}b(x)dx=\alpha L,$ where$\alpha$ is an arbitrarily given constant. This question is closely related tothe problem studied by mathematical ecologists in late 1980's but its answerhas not been known. We answer this question by proving that the maximum isattained by periodically arrayed Dirac's delta functions $\alphaL\sum_{k\in\mathbb Z}\delta(x+kL).$

A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations