Asymptotic growth for the parabolic equation of prescribed mean curvature (Q797050)

The authors consider the behavior for large times of the (pseudo) solution of the Dirichlet problem for the equation \(u_ t=Au+h(x),\) \(x\in \Omega\), \(t>0\) where A is the minimal surface operator \(Au=div((1+| \text{grad} u|^ 2)^{-{1\over2}} \text{grad} u)\) and h is a given function. Conjectures are first given: when h is small enough the solution tends to a stationary solution of the corresponding elliptic problem while when h is large enough the solution develops a ''rising elliptic cap'' on a maximal subset of \(\Omega\). Under hypotheses on h yielding a concave u(.,t) on \(\Omega\) and in the radially symmetric setting the above conjectured behavior is proved. Finally, numerical computations show the development of the rising elliptic cap.