For a project, I am obtaining the numerical solutions to the following (Cahn-Morral) equations for $y = (y_1, y_2)$:
$$\frac{\partial y_i}{\partial t}=\Delta\left[f_i(y)-\alpha\Delta(2y_i+y_j)\right]; \qquad i\ne j $$
where $\Delta$ is the Laplacian operator, $\alpha$ is some positive constant (of the order unity) and the $f_i$s are functions of the $y$s.In my particular case I have
$$f_i=\frac{3}{10}\ln[y_i/(1-y_1-y_2)]-(2y_i+y_j)$$
where again $i\ne j$. This particular expression for the $f_i$s is obtained from Thermodynamic considerations (they are the partial derivatives of the bulk free energy density for a ternary regular mixture, where $y_3$ was substituted by $1-y_1-y_2$).
My question is if there is any way to know if the solutions are going to be of a differentiability class higher than $C^4$ (which I already know they have to be because the equation is fourth order). I am solving these over a square domain with periodic boundary conditions.