On a phase field model for RNA-Protein dynamics
A phase field model which describes the formation of protein-RNA complexes subject to phase segregation is analyzed. A single protein, two RNA species, and two complexes are considered. Protein and RNA species are governed by coupled reaction-diffusion equations which also depend on the two complexes. The latter ones are driven by two Cahn-Hilliard equations with Flory-Huggins potential and reaction terms depending on the solution variables. The resulting nonlinear coupled system is equipped with no-flux boundary conditions and suitable initial conditions. The former ones entail some mass conservation constraints which are also due to the nature of the reaction terms. The existence of global weak solutions in a bounded (two- or) three-dimensional domain is established. In dimension two, some weighted-in-time regularity properties are shown. In particular, the complexes instantaneously get uniformly separated from the pure phases. Taking advantage of this result, a unique continuation property is proven. Among the many technical difficulties, the most significant one arises from the fact that the two complexes are initially nonexistent, so their initial conditions are zero i.e., they start from a pure phase. Thus we must solve, in particular, a system of two coupled Cahn-Hilliard equations with singular potential and nonlinear sources without the usual assumption on the initial datum, i.e., the initial phase cannot be pure. This novelty requires a new approach to estimate the chemical potential in a suitable $L^p(L^2)$-space with $p\in(1,2)$. This technique can be extended to other models like, for instance, the well-known Cahn-Hilliard-Oono equation.