Nonexistence results for a time-fractional biharmonic diffusion equation.

We consider weak solutions of the nonlinear time-fractional biharmonic diffusion equation ∂ t α u + ∂ t β u + u x x x x = h (t , x) | u | p in (0 , ∞) × (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 x x (t , 1) = 0 , where α ∈ (0 , 1) , β ∈ (1 , 2) , ∂ t α (resp. ∂ t β ) is the fractional derivative of order α (resp. β) 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 ∂ 4 ∂ 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. [ABSTRACT FROM AUTHOR]