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

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