  We study first-order symmetrizable hyperbolic $N\times N$ systems in a
spacetime cylinder whose lateral boundary is totally characteristic. In local
coordinates near the boundary at $x=0$, these systems take the form \[
  \partial_t u + \mathcal A(t,x,y,xD_x,D_y) u = f(t,x,y), \quad
(t,x,y)\in(0,T)\times\mathbb R_+\times\mathbb R^d, \] where $\mathcal
A(t,x,y,xD_x,D_y)$ is a first-order differential operator with coefficients
smooth up to $x=0$ and the derivative with respect to $x$ appears in the
combination $xD_x$. No boundary conditions are required in such a situation and
corresponding initial-boundary value problems are effectively Cauchy problems.
  We introduce a certain scale of Sobolev spaces with asymptotics and show that
the Cauchy problem for the operator $\partial_t + \mathcal A(t,x,y,xD_x,D_y)$
is well-posed in that scale. More specifically, solutions $u$ exhibit formal
asymptotic expansions of the form \[
  u(t,x,y) \sim \sum_{(p,k)} \frac{(-1)^k}{k!} x^{-p} \log^k \!x \, u_{pk}(t,y)
\quad \text{as $x\to+0$} \] where $(p,k)\in\mathbb C\times\mathbb N_0$ and $\Re
p\to-\infty$ as $|p|\to\infty$, provided that the right-hand side $f$ and the
initial data $u|_{t=0}$ admit asymptotic expansions as $x \to +0$ of a similar
form, with the singular exponents $p$ and their multiplicities unchanged. In
fact, the coefficient $u_{pk}$ are, in general, not regular enough to write the
terms appearing in the asymptotic expansions as tensor products. This
circumstance requires an additional analysis of the function spaces. In
addition, we demonstrate that the coefficients $u_{pk}$ solve certain
explicitly known first-order symmetrizable hyperbolic systems in the lateral
boundary.
  Especially, it follows that the Cauchy problem for the operator
$\partial_t+\mathcal A(t,x,y,xD_x,D_y)$ is well-posed in the scale of standard
Sobolev spaces $H^s((0,T)\times\mathbb R_+^{1+d})$.
