Higher order evolution inequalities with nonlinear convolution terms

We are concerned with the study of existence and nonexistence of weak solutions to $$\begin{cases} &\displaystyle \frac{\partial^k u}{\partial t^k}+(-\Delta)^m u\geq (K\ast |u|^p)|u|^q \quad\mbox{ in } \mathbb R^N \times \mathbb R_+,\\[0.1in] &\displaystyle \frac{\partial^i u}{\partial t^i}(x,0) = u_i(x) \,\, \text{ in } \mathbb R^N,\, 0\leq i\leq k-1,\\ \end{cases}$$ where $N,k,m\geq 1$ are positive integers, $p,q>0$ and $u_i\in L^1_{\rm loc}(\mathbb{R}^N)$ for $0\leq i\leq k-1$. We assume that $K$ is a radial positive and continuous function which decreases in a neighbourhood of infinity. In the above problem, $K\ast |u|^p$ denotes the standard convolution operation between $K(|x|)$ and $|u|^p$. We obtain necessary conditions on $N,m,k,p$ and $q$ such that the above problem has solutions. Our analysis emphasizes the role played by the sign of $\displaystyle \frac{\partial^{k-1} u}{\partial t^{k-1}}$.