Instability of homogeneous periodic solutions of parabolic-delay equations
Parabolic systems of the form \[ {\partial u \over \partial t} = D \Delta u(x,t) + f \bigl( u_ t(x, \cdot) \bigr) \tag{1} \] 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 \(\dot u = Au + f(u_ t)\) 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 \(\dot u(t) = f(u_ t)\) 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.
scientific article; zbMATH DE number 589981