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 $p>1$, ${\alpha }_i,\gamma in(0,1)$, $\delta ,\sigma \in (0,1)$, ${\mathbf{D}}_{0|t}^{{\alpha }_i}$ is the Caputo fractional derivative and $I_{0|t}^{1-\gamma }$ is the Riemann-Liouville fractional integral of order $1-\gamma$. We prove the non-existence of global solutions if \begin{equation*} 1<p\leqslant \frac{2(2+α_1-γ)}{(\frac{α_1+1}σ N+2γ-2α_1-2)_+ }+1, \end{equation*} for any space dimension $N⩾1.$ Then, we extend the result to the system \begin{align*} &{\mathbf{D}}_{0|t}^{1+α_1}u + (-Δ)^{σ_1} u + (-Δ)^{δ_1}{\mathbf{D}}_{0|t}^{α_2} u = I_{0|t}^{1-γ_{1}}|v|^{p},\qquad (t,x)\in (0,\infty) \times \mathbb{R}^N, \\ &{\mathbf{D}}_{0|t}^{1+β_{1}}v+(-Δ)^{σ_2} v + (-Δ)^{δ_2}{\mathbf{D}}_{0|t}^{β_2}v = I_{0|t}^{1-γ_2}|u|^{q},\qquad (t,x)\in (0,\infty )\times \mathbb{R}^{N}, \end{align*} where $p,q>1$, $0<{\delta }_i$, ${\sigma }_i<1$ and ${\gamma }_2\in (0,1)$. Also, we present the necessary conditions for the existence of local or global solutions.