Nonexistence of global weak solutions of a system of nonlinear wave equations with nonlinear fractional damping (Q777015)

Summary: We consider the system of nonlinear wave equations with nonlinear time fractional damping \[ \begin{cases} u_{tt} + (-\Delta)^m u +^C D_{0,t}^\alpha (t^\sigma |u|^q) = |v|^p,\, t >0,\, x \in \mathbb{R}^N, \\ v_{tt} + (-\Delta)^m v +^C D_{0,t}^\beta (t^\delta |v|^r) = |v|^s,\, t > 0,\, x \in \mathbb{R}^N, \\ (u(0,x),\, u_t (0,x)) = (u_0 (x), u_1 (x)),\, x \in \mathbb{R}^N, \\ (u(0,x),\, u_t (0,x)) = (u_0(x), u_1 (x)),\, x \in \mathbb{R}^N, \end{cases} \] where \((u,v)=(u(t,x),v (t,x))\), \(m\) and \(N\) are positive natural numbers, \(p\), \(q\), \(r\), \(s>1\), \(\sigma\), \(\delta \geq 0\), \(0<\alpha\), \(\beta<1\), and \(^C D_{0,t}^\kappa\), \(0<\kappa<1\), is the Caputo fractional derivative of order \(\kappa\). Namely, sufficient criteria are derived so that the system admits no global weak solution. To the best of our knowledge, the considered system was not previously studied in the literature.