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

Abstract

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 $(u^-,+\infty)$.