show that these constraints propagate, [*i.e.,*]{} that if they are satisfied by a solution to the Kasner-like evolution equations at some time $t_0>0$, then they are satisfied for all time $t>0$. In the set of Kasner-like solutions, one expects that there is a subset, denoted by $V$, of solutions which have the property of being asymptotic to solutions of the complete Einstein-dilaton-$p$-form equations as $t \rightarrow 0$, i.e as one goes to the singularity. This subset is characterized by inequalities on some of the initial data, which, however, are not always consistent. The existence of a non-empty $V$ requires the dilaton couplings to belong to some range, the “subcritical range". When $V$ is non-empty and open, the solutions in $V$ involve as many arbitrary functions of space as a “general solution" of the full Einstein equations should. On the other hand, if $V$ is empty the construction given in this paper breaks down and the dynamical system is expected to be not Kasner-like but rather oscillatory.

To show that indeed, the solutions in $V$ (when it is non-empty) are asymptotic to true solutions, the third step is to identify decaying quantities such that these decaying quantities along with the leading 