$\newcommand{\R}{\mathbb R}$I am looking for any possible references (modelling, mathematical and numerical analysis, well-posedness, regularity, anything really!) for heat equation-like problems with nonlocal Dirichlet boundary conditions. The typical problem I might write down is for example $$ \begin{cases} \partial_t u=\Delta u & \mbox{in }(0,T)\times\Omega\\ u|_{\partial\Omega}=\int_\Omega u\, \mathrm d x & \mbox{on }(0,T)\times\partial\Omega\\ u|_{t=0}=u_0 & \mbox{in }\Omega \end{cases} $$ Here $\Omega\subset \R^d$ is as usual your favourite smooth, bounded, open set. This looks like a standard Dirichlet boundary condition $u|_{\partial\Omega}=g$ except that $g=\int_\Omega u$ depends nonlocally on the values of $u$ inside the domain at any given time $t>0$. Has anyone seen anything vaguely related to this kind of models, and if so can anyone provide references? Any help is much appreciated!

I am sorry I cannot really provide more details or a more focussed problem. This popped up in my research as a very secondary question, and as usual I would rather avoid reinventing the wheel if possible. And a quick bibliographical search did not provide much useful information (except for a few exotic papers in dimension $d=1$ only).