Nonexistence results for a time-fractional biharmonic diffusion equation

We consider weak solutions of the nonlinear time-fractional biharmonic diffusion equation partial derivative(alpha)(t)u+partial derivative(beta)(t)u+u(xxxx)=h(t,x)|u|(p) in (0,infinity)x(0,1) subject to the initial conditions u(0,x)=u(0)(x), u(t)(0,x)=u(1)(x) and the Navier boundary conditions u(t,1)=u(xx)(t,1)=0, where beta is an element of (0,1), beta is an element of (1,2), partial derivative(alpha)(t) (resp. partial derivative(beta)(t)) is the fractional derivative of order alpha (resp. beta) with respect to the time-variable in the Caputo sense, p >1 and h is a measurable positive weight function. Using nonlinear capacity estimates specifically adapted to the fourth-order differential operator partial derivative(4)/partial derivative x(4), the domain, the initial conditions and the boundary condition, a general nonexistence result is established. Next, some special cases of weight functions h are discussed.