Non-Existence of Solutions for a Fourth Order Structurally Damped Equation Under Nonlinear Memory Effect

Abstract In this paper, we study a mixed wave-plate equation with rotational inertia, fractional damping and memory non-linearity. This research is a non-existence counterpart to a paper by D’Abbicco and Longen [1], in search of the critical exponent for the global in-time existence of small data solutions to the Cauchy problem for the equation $$\begin{aligned} u_{tt}-\Delta u_{tt}+\Delta ^2 u-\Delta u+(-\Delta )^\theta u_t=\int _0^t(t-s)^{-\gamma }|u(s,x)|^p\,\text {d}s, \\ t\in \mathbb {R}_+^*, x\in \mathbb {R}^n, \end{aligned}$$ with $$\theta \in [0,\frac{1}{2}),\gamma \in (0,1)$$ and $$p>1.$$ We employ a modified test function argument to show that there are arbitrarily small initial data for which there are no global in-time solutions for the problem in the subcritical case $$p