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

Abstract

For the following semilinear equation with Hilfer- Hadamard fractional derivative \begin{equation*} \mathcal{D}^{α_1,β}_{a^+} u-Δ\mathcal{D}^{α_2,β}_{a^+} u-Δu =\vert u\vert^p, \qquad t>a>0, \qquad x\inΩ, \end{equation*} where $\Omega \subset {\mathbb{R}}^N$ $(N⩾1)$, $p>1$, $0<{\alpha }_2<{\alpha }_1<1$ and $0<\beta <1$. ${\mathcal{D}}_{a^{+}}^{{\alpha }_i,\beta }$ $(i=1,2)$ 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.