| Abstract: |
| We present a method justifying that the Ginzburg-Landau equation is a valid modulation equation for a wide class of quasilinear dissipative systems. The modulation equation can be derived near the first instability of a Turing or Turing-Hopf bifurcation of spatially homogeneous steady states and correctly predicts the behavior of the solution on the long timescale. The proof of the approximation result relies on maximal regularity and a fixed point argument that can be expressed in a rather abstract way. This allows, for instance, to validate the modulation equation for initially nonlocalized perturbations. We show that the arguments can be applied to the quasilinear Klausmeier-Gray-Scott system as well as a quasilinear version of the B\`enard-Rayleigh convection model. |
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