On the critical behavior for time-fractional reaction diffusion problems

We first investigate the existence and nonexistence of weak solutions to time-fractional reaction diffusion problem \begin{document}$ \frac{\partial^\alpha u}{\partial t^\alpha}-\frac{\partial^2 x^2}+u\geq x^{-a}|u|^p, \, t>0, x\in (0, 1], \quad u(0, x) = u_0(x), 1] $\end{document} under inhomogeneous Dirichlet boundary condition u(t, 1) \delta, where $ u $, 0<\alpha<1 }{\partial t^\alpha} is time-Caputo fractional derivative order \alpha a\geq 0 p>1 \delta>0 $. show that, if a\leq 2 holds for all while a>2 then dividing line with respect or given by critical exponent p^* a-1 The proof result based on nonlinear capacity estimates specifically adapted nonlocal nature problem, modified Helmholtz operator -\frac{\partial^2}{\partial x^2}+I considered condition. part proved construction explicit solutions. next extend our study case systems.