Nonexistence results for a time-fractional biharmonic diffusion equation

We consider weak solutions of the nonlinear time-fractional biharmonic diffusion equation ${\partial }_t^{\alpha }u+{\partial }_t^{\beta }u+u_{xxxx}=h(t,x){|u|}^p$ $\partial _{t}^{\alpha }u+\partial _{t}^{\beta }u+u_{xxxx}=h(t,x)|u|^{p}$ in $(0,\mathrm{\infty })\times (0,1)$ $(0,\infty )\times (0,1)$ subject to the initial conditions $u(0,x)=u_0(x)$ $u(0,x)=u_{0}(x)$ , $u_t(0,x)=u_1(x)$ $u_{t}(0,x)=u_{1}(x)$ and the Navier boundary conditions $u(t,1)=u_{xx}(t,1)=0$ $u(t,1)=u_{xx}(t,1)=0$ , where $\alpha \in (0,1)$ $\alpha \in (0,1)$ , $\beta \in (1,2)$ $\beta \in (1,2)$ , ${\partial }_t^{\alpha }$ $\partial _{t}^{\alpha}$ (resp. ${\partial }_t^{\beta }$ $\partial _{t}^{\beta}$ ) is the fractional derivative of order α (resp. β) with respect to the time-variable in the Caputo sense, $p>1$ $p>1$ and h is a measurable positive weight function. Using nonlinear capacity estimates specifically adapted to the fourth-order differential operator $\frac{{\partial }^4}{\partial x^4}$ $\frac{\partial ^{4}}{\partial x^{4}}$ , 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.