*(English)*Zbl 0811.35156

Parabolic systems of the form

are considered, where $f:C([-r,0],{\mathbb{R}}^{N})\to {\mathbb{R}}^{N}$ and $D$ is an $N\times N$ real matrix with spectrum in the right half plane. The first part is concerned mainly with fairly standard existence and uniqueness results for equation (1) using the abstract form $\dot{u}=Au+f\left({u}_{t}\right)$ and the variation of constants formula.

In the second part, the existence of travelling waves of (1) is proved under the assumption that the reaction equation $\dot{u}\left(t\right)=f\left({u}_{t}\right)$ has a non-constant periodic solution. The destabilizing effect of the $D$ matrix is shown by considering a variational form of (1) and the corresponding reaction equation. If a characteristic multiplier has absolute value greater than 1 then the travelling wave is unstable.