We consider the following degenerate parabolic equation
$$
v_{t}=v\Delta v-\gamma|\nabla
v|^{2}\quad\mbox{in $\mathbb{R}^{N} \times(0,\infty)$,}
$$
whose behaviour depends strongly on the parameter $\gamma$. While
the range $\gamma < 0$ is well understood, qualitative and
analytical novelties appear for $\gamma>0$. Thus, the standard
concepts of weak or viscosity solution do not produce uniqueness.
Here we show that for $\gamma>\max\{N/2,1\}$ the initial value
problem is well posed in a precisely defined setting: the
solutions are chosen in a class $\mathcal{W}_s$ of local weak
solutions with constant support; initial data can be any
nonnegative measurable function $v_{0}$ (infinite values also
accepted); uniqueness is only obtained using a special concept of
initial trace, the $p$-trace with $p=-\gamma < 0$, since the
standard concepts of initial trace do not produce uniqueness.
Here are some additional properties: the solutions turn out to be
classical for $t>0$, the support is constant in time, and not all
of them can be obtained by the vanishing viscosity method. We also
show that singular measures are not admissible as initial data,
and study the asymptotic behaviour as $t\to \infty$.