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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 boundaryconditions. The function $\phi \in C^1(\mathbb{R})$ is assumed to be decreasingeither in a bounded interval $(u^-,u^+)$, or in an unbounded interval$(u^-,+\infty)$: under this hypothesis, the aforementioned problem is ill-posedand only allows for measure-valued solutions. Moreover, such solutions are ingeneral not unique. By using nonstandard analysis, we derive from very simplephysical principles a continuous-in-time and discrete-in-space model for theill-posed pde, and we prove that this model is well-posed. We will also provethat the solution of the nonstandard formulation is coherent with themeasure-valued solutions and still retains relevant physical properties,chiefly among them an entropy condition that characterizes physicallyadmissible solutions to the original problem. We then study the asymptoticbehaviour of the nonstandard solutions. In doing so, we will give a positiveanswer to a conjecture by Smarrazzo on the coarsening of the solutions to theill-posed problem under the hypothesis that $\phi$ is decreasing in theinterval $(u^-,+\infty)$.

A grid function formulation of a class of ill-posed parabolic equations