Nonexistence for the Laplace equation with a dynamical boundary condition of fractional type

We consider the Laplace equation in $\mathbb R^{d-1}\times\mathbb R^+\times(0,+\infty)$ with a dynamical nonlinear boundary condition of order between 1 and 2. Namely, the boundary condition is a fractional differential inequality involving derivatives of noninteger order as well as a nonlinear source. Nonexistence results and necessary conditions are established for local and global existence. In particular, we show that the critical exponent depends only on the fractional derivative of the least order.