  We study the well-posedness of Cauchy problems on the upper half space
$\mathbb{R}^{n+1}_+$ associated to higher order systems $\partial_t u
=(-1)^{m+1}\mbox{div}_m A\nabla ^m u$ with bounded measurable and uniformly
elliptic coefficients. We address initial data lying in $L^p$ ($1<p<\infty$)
and $BMO$ ($p=\infty$) spaces and work with weak solutions. Our main result is
the identification of a new well-posedeness class, given for $p\in(1,\infty]$
by distributions satisfying $\nabla^m u \in T^{p,2}_m$, where $T^{p,2}_m$ is a
parabolic version of the tent space of Coifman--Meyer--Stein. In the range
$p\in [2,\infty]$, this holds without any further constraints on the operator
and for $p=\infty$ it provides a Carleson measure characterization of $BMO$
with non-autonomous operators. We also prove higher order $L^p$ well-posedness,
previously only known for the case $m = 1$. The uniform $L^p$ boundedness of
propagators of energy solutions plays an important role in the well-podesness
theory and we discover that such bounds hold for $p$ close to $2$. This is a
consequence of local weak solutions being locally H\"older continuous with
values in spatial $L^p_{loc}$ for some $p>2$, what is also new for the case
$m>1$.
