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+\alpha_1}u + (-\Delta)^\sigma u+(-\Delta )^\delta\mathbf{D}_{0|t}^{\alpha _2} u = I_{0|t}^{1-\gamma }|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+\alpha_1-\gamma)}{(\frac{\alpha_1+1}{\sigma} N+2\gamma-2\alpha_1-2)_+ }+1, \end{equation*} for any space dimension $N\geqslant 1.$ Then, we extend the result to the system \begin{align*} &{\mathbf{D}}_{0|t}^{1+\alpha_1}u + (-\Delta)^{\sigma_1} u + (-\Delta )^{\delta_1}{\mathbf{D}}_{0|t}^{\alpha_2} u = I_{0|t}^{1-\gamma _{1}}|v|^{p},\qquad (t,x)\in (0,\infty) \times \mathbb{R}^N, \\ &{\mathbf{D}}_{0|t}^{1+\beta _{1}}v+(-\Delta)^{\sigma_2} v + (-\Delta )^{\delta_2}{\mathbf{D}}_{0|t}^{\beta_2}v = I_{0|t}^{1-\gamma_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.