On a one-dimensional time-fractionally damped wave equation with a singular potential

A nonlinear time-fractionally damped wave equation with an inverse-square potential posed on the interval (0, 1) is investigated. The time-fractionally derivative is considered in the Caputo sense. A weight function of the form x − σ ( σ ∈ R ) is allowed in front of the nonlinearity ∣ u ( t , x )∣ p ( p > 1). The problem is studied under certain initial conditions and a homogeneous boundary condition. Namely, we obtain sufficient conditions under which the considered problem admits no weak solution. The cases u (0, · ) ≡ 0 and u (0, · ) ≢ 0 are studied separately.