Parabolic abstract evolution equations in cylindrical domains and uniformly local Sobolev spaces

In this article, we consider parabolic equations of the type $$\partial_t u(x,t)=Δu(x,t) - Bu(x,t) + F(u(x,t))$$ where $u$ is valued in a transverse Hilbert space $Y$ and $B$ is a positive self-adjoint operator on $Y$, allowing a different diffusion mechanism in the transverse direction. We aim at c