We investigate the time dependent Ginzburg-Landau (TDGL) equation for a nonconserved order parameter on an infinitely ramified (deterministic) fractallattice employing two alternative methods: the auxiliary field approach and anumerical method of integration of the equations of evolution. In the firstcase the domain size evolves with time as $L(t)\sim t^{1/d_w}$, where $d_w$ isthe anomalous random walk exponent associated with the fractal and differs fromthe normal value 2, which characterizes all Euclidean lattices. Such a powerlaw growth is identical to the one observed in the study of the spherical modelon the same lattice, but fails to describe the asymptotic behavior of thenumerical solutions of the TDGL equation for a scalar order parameter. In fact,the simulations performed on a two dimensional Sierpinski Carpet indicate that,after an initial stage dominated by a curvature reduction mechanism \`a laAllen-Cahn, the system enters in a regime where the domain walls betweencompeting phases are pinned by lattice defects. The lack of translational invariance determines a rough free energylandscape, the existence of many metastable minima and the suppression of themarginally stable modes, which in translationally invariant systems lead topower law growth and self similar patterns. On fractal structures as thetemperature vanishes the evolution is frozen, since only thermally activatedprocesses can sustain the growth of pinned domains.
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