| Contents |
| We consider the initial-boundary value problem for the partial differential equation
$$
u_t + \Delta^2 u = \det(D^2 u) + \lambda f(x,t),
$$
where the forcing term obeys suitable summability conditions. This problem might present a gradient flow structure depending on the boundary conditions as well as the geometry of the spatial domain where it is posed. We will first summarize our results on the existence and regularity of stationary solutions. Then, for the evolution problem, we will show how to prove local existence of solutions for arbitrary data and global existence of solutions for small data. Depending on the boundary conditions and the concomitant presence of a variational structure in the equation as well as on the size of the data we prove blow-up of the solution in finite time and convergence to a stationary solution in the long time limit. |
|