In my research I came across the type of differential equation described below. I've looked in introductory ODE and PDE books for a treatment of this type of equations, but I had no success. So any insight as to what class of ODE's these are or any references that deal with them would be much appreciated.
Let $y(t,x)$ be a scalar quantity that depends on a temporal ($t$) and spatial variable ($x \in \Omega$). The dynamics of $y$ are given by $$ \frac{\partial y(t,x)}{\partial t} = a(x) \bar y(t) + b(x) y(t,x), $$ where $\bar y$ is the space average ($\bar y = \int_{\Omega} y(t,x) dx$), a(x) is a parameter, and b(x) is a parameter.
Additional rambling (not part of the question, but I think it's interesting):
What is throwing me off is that the pointwise dynamics depend on the space average variable $\bar y$ (almost like there is a diffusion of spatial information even though there are no differential operators). Specifically,I'd like to show that $|| y(t,x)||_{\mathcal{L^2}} \rightarrow 0$ exponentially fast, but when computing the time derivative of $|| y(t,x)||_{\mathcal{L^2}}^2$ one gets a term of the form $$ \big[\int_\Omega a(x) y(t,x) dx \big]\bar y(t), $$ which is sign indefinite and thus problematic. This almost suggests that the space averaged term can "destabilize" the dynamics.
Any insight is much appreciated.