  We study a nonstandard formulation of the Neumann initial value problem
\begin{equation} \begin{array}{rl} u_t(x,t) = \Delta \phi(u(x,t)), & x \in
\Omega \subseteq \mathbb{R}^k, \ t \in \mathbb{R} \label{abstract}\\ u(x,0) =
u_0(x), & x \in \Omega. \end{array} \end{equation} with Neumann boundary
conditions. The function $\phi \in C^1(\mathbb{R})$ is assumed to be decreasing
either in a bounded interval $(u^-,u^+)$, or in an unbounded interval
$(u^-,+\infty)$: under this hypothesis, the aforementioned problem is ill-posed
and only allows for measure-valued solutions. Moreover, such solutions are in
general not unique. By using nonstandard analysis, we derive from very simple
physical principles a continuous-in-time and discrete-in-space model for the
ill-posed pde, and we prove that this model is well-posed. We will also prove
that the solution of the nonstandard formulation is coherent with the
measure-valued solutions and still retains relevant physical properties,
chiefly among them an entropy condition that characterizes physically
admissible solutions to the original problem. We then study the asymptotic
behaviour of the nonstandard solutions. In doing so, we will give a positive
answer to a conjecture by Smarrazzo on the coarsening of the solutions to the
ill-posed problem under the hypothesis that $\phi$ is decreasing in the
interval $(u^-,+\infty)$.
