On the dynamics of capital accumulation across space

We solve an optimal growth model in continuous space, continuous and bounded
time. The optimizer chooses the optimal trajectories of capital and consumption
across space and time by maximizing an objective function with both space and
time discounting. We extract the corresponding Pontryagin conditions and prove
their sufficiency. We end up with a system of two parabolic differential equations
with the corresponding boundary conditions. We propose a simple numerical set-up
to simulate PDE systems which we employ to study the roles of initial capital and
technology distributions over space in various scenarios.