Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations

Assume a single reaction-diffusion equation has zero as an asymptotically stable stationary point. Then we prove that there exist no localized travelling waves with non-zero speed. If $[\liminf_{|x|\to\infty}u(x),\limsup_{|x|\to\infty}u(x)]$ is included in an open interval of zero that does not include other stationary points, then the speed has to be zero or the travelling profile $u$ has to be identically zero.