Absence of local and global solutions to an elliptic system with time-fractional dynamical boundary conditions

This paper presents extensions of some nonexistence results for elliptic systems with dynamical boundary conditions involving the time-derivatives of integer orders to the case of noninteger order. In particular, we consider a system of Poisson’s equations with time-fractional derivatives of order less than one in the boundary conditions and specify the thresholds of the nonlinearities which lead to the absence of global solutions. The fractional derivatives here are meant in the Riemann-Liouville sense (or in the Caputo sense). We also present necessary conditions for the existence of local solutions.