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

In this article, we consider parabolic equations of the type $$\partial_t u(x,t)=\Delta u(x,t) - Bu(x,t) + F(u(x,t))$$ 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 considering solutions with infinite energy and we study the Cauchy problem in the uniformly local spaces associated with the norm $$\|u\|_{L^2_{\text{ul}}(\mathbb{R},Y)}= \sup_{a\in\mathbb{R}^d} \|u(x)\|_{L^2(B(a,1),Y)}.$$ For the classical parabolic equation, i.e. if $Y=\mathbb{R}$, it is known that the Cauchy problem is ill-posed in the weak version of the uniformly local spaces but well-posed in a stronger version, where additional uniform continuity is required. In this paper, we show that the linear operator $\partial^2_{xx} - B$ is not necessarily a sectorial operator in any version of the uniformly local Lebesgue space, due to the possible non-density of its domain. Then, we use the theory of parabolic abstract evolution equations to set a well-posed Cauchy problem, even in the weak version of the uniformly local space. In particular, we believe that this paper offers a new perspective on the comparison between both versions of the uniformly local spaces and also provides a new natural example of differential operators with non-dense domain.