In this paper, we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For irrational rotations, we prove a different behaviour depending on the irrational exponent of the angle of the rotation. For random processes, this problem corresponds to the longest common substring problem. We extend the result of [5] on sequence matching to α-mixing processes with exponential decay.

We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by utt(x, t) − uxx(x, t) + µ 1 (t)ut(x, t) + µ 2 (t)ut(x, t − τ (t)) = 0 in a bounded domain. Under proper conditions on nonlinear weights µ 1 (t), µ 2 (t) and non-constant delay τ (t), we prove global existence and estimative the decay rate for the energy.

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