On nonexistence of global solutions for a semilinear equation with Hilfer- Hadamard fractional derivative

For the following semilinear equation with Hilfer- Hadamard fractional derivative begin{equation*} mathcal{D}^{alpha_1,beta}_{a^+} u-Deltamathcal{D}^{alpha_2,beta}_{a^+} u-Delta u =vert uvert^p, qquad t>a>0, qquad xinOmega, end{equation*} where $Omegasubset mathbb{R}^N$ $(Ngeqslant 1)$, $p>1$, $0<alpha _{2}<alpha _{1}<1$ and $0<beta <1$. $mathcal{D}^{alpha_i,beta}_{a^+}$ $(i=1,2)$ is the Hilfer- Hadamard fractional derivative of order $alpha_i$ and of type $beta$, we establish the necessary conditions for the existence of global solutions.