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Instability of homogeneous periodic solutions of parabolic-delay equations. (English) Zbl 0811.35156

Parabolic systems of the form

u t=DΔu(x,t)+fu t (x,·)(1)

are considered, where f:C([-r,0], N ) N and D is an N×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 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 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.

MSC:
35R10Partial functional-differential equations
35K57Reaction-diffusion equations