On the absence of weak solutions for a sequential time-fractional Klein–Gordon equation with quadratic nonlinearity

The Klein–Gordon equation with quadratic nonlinearity is frequently used to model various problems of physics. On the other hand, it was shown that in many situations, the use of fractional derivatives instead of ordinary derivatives provides more realistic models. In this work, a one-dimensional sequential time-fractional Klein–Gordon equation posed on a bounded interval, subject to certain initial and boundary conditions, is investigated. The fractional derivatives are considered in the Caputo sense and a weight function $h>0$ is allowed in front of the quadratic nonlinearity. Namely, we first establish general sufficient conditions for the nonexistence of solutions. Next, some particular cases of weight functions $h$ are discussed. In our approach, we make use of the test function method. This method requires an appropriate choice of a family of test functions. In this paper, we make a judicious choice of test functions taking into consideration the nonlocal property of the Caputo fractional derivative, the initial and boundary conditions, and the geometry of the domain.