Abstract
We first consider a nonlinear time fractional wave equation with a fractional damping posed on the Heisenberg group with Caputo fractional derivatives that interpolate the heat equation and the wave equation with the linear damping. It extends the Caputo-Wisner, the modified Szabo and the fractional Zener models. The non-linearity accounts for a nonlinear medium. We present the Fujita exponent for blow-up which sheds light on the admissible nonlinearity in practice. Then, we establish sufficient conditions ensuring non-existence of local solutions. using the nonlinear capacity method. Furthermore, we extend the analysisto the case of the 2x2 system to show the flexibility of the method used.