Exact string solutions in $2+1$-dimensional de Sitter spacetime

Exact and explicit string solutions propagating in $2+1$ dimensional de Sitter spacetime are presented and physically analized. (In this case the string equations reduce to a $sinh$-Gordon model.) Strings generically tend to inflate or either to collapse. The world-sheet time $\tau$ interpolates between the cosmic $\tau \to \pm \infty$ and conformal $\tau \to 0$ times. For $\tau \to 0$, the typical string instability is found, while for $\tau \to \pm \infty$, a new string behavior appears. In that regime, the string expands (or contracts) but not with the same rate as the universe does. The string constraints select periodic solutions of the $sinh$-Gordon equation associated to the lower boundary of the allowed zone, therefore excluding elliptic solutions.