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Zbl 0811.35156
De Oliveira, Luiz A.F.
Instability of homogeneous periodic solutions of parabolic-delay equations.
(English)
[J] J. Differ. Equations 109, No.1, 42-76 (1994). ISSN 0022-0396

Parabolic systems of the form $${\partial u \over \partial t} = D \Delta u(x,t) + f \bigl( u\sb t(x, \cdot) \bigr) \tag 1$$ are considered, where $f : C([-r,0], \bbfR\sp N) \to \bbfR\sp N$ 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 $\dot u = Au + f(u\sb t)$ and the variation of constants formula.\par In the second part, the existence of travelling waves of (1) is proved under the assumption that the reaction equation $\dot u(t) = f(u\sb t)$ 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.
[S.P.Banks (Sheffield)]
MSC 2000:
*35R10 Difference-partial differential equations
35K57 Reaction-diffusion equations

Keywords: existence and uniqueness results; existence of travelling waves; variational form

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