I have the PDE-ODE system below:
$\frac{\partial c}{\partial t}= D \Delta c - \eta \nabla.(c\nabla v)+g(c,v)$
$\frac{dv}{dt}=-\alpha cv+\xi(c,v)$
with initial conditions and Neumann boundary condition for the first one. How can I tell if this system has a solution or not? Is there any theorem for this?