Blowing-up solutions of a time-space fractional semi-linear equation with a structural damping and a nonlocal in time nonlinearity

Abstract

In this paper, we investigate the semilinear equation with a time-space fractional structural damping and a nonlocal in time nonlinearity \begin{equation*} {\mathbf{D}}_{0|t}^{1+α_1}u + (-Δ)^σu+(-Δ)^δ\mathbf{D}_{0|t}^{α_2} u = I_{0|t}^{1-γ}|u|^{p}, \qquad (t,x)\in (0,\infty) \times \mathbb{R}^N, \end{equation*} where $p>1$, $α_i, γ in (0,1)$, $δ, σ\in (0,1)$, ${\mathbf{D}}_{0|t}^{α_i}$ is the Caputo fractional derivative and $I_{0|t}^{1-γ}$ is the Riemann-Liouville fractional integral of order $1-γ$. We prove the non-existence of global solutions if \begin{equation*} 1 1$, $0<δ_i$, $σ_i<1$ and $γ_2\in (0,1)$. Also, we present the necessary conditions for the existence of local or global solutions.