For the following wave equations with structural damping and nonlinear memory source terms \[ u_{tt}+(-\Delta)^{\frac{\alpha}{2}}u +(-\Delta)^{\frac{\beta}{2}}u_t =\int_{0}^{t}(t-s)^{\gamma-1} \vert u (s)\vert^{p}\,\text{d}s, \] and \[ u_{tt}+(-\Delta)^{\frac{\alpha}{2}}u +(-\Delta)^{\frac{\beta}{2}}u_t = \int_{0}^{t}(t-s)^{\gamma-1} \vert u_s (s)\vert^{p}\,\text{d}s, \] posed in $(x,t) \in \mathbb{R}^N \times [0,\infty) $, where $u=u(x,t)$ is real-value unknown function, $p>1$, $\alpha,\beta\in (0, 2]$, $\gamma\in (0,1)$, we prove the nonexistence of global solutions. Moreover, we give an upper bound estimate of the life span of solutions.

We first consider the nonlinear time fractional diffusion equation D^{1+α}u+D^β u−∆_{H} u=|u|^p posed on the Heisenberg group H, where 1 < p is a positive real nimber to be specified later; D^δ_{0|t} is the Liouville-Caputo derivative of order δ. For 0 < α < 1,0 < β ≤ 1. This equation interpolates the heat equation and the wave equation with the linear damping D^β_{0|t}u. We present the Fujita exponent for blow-up. Then establish sufficient conditions ensuring non-existence of local solutions. We extend the analysis to the case of the system D^{1+α}u+D^β u−∆_{H} u=|v|^q D^{1+δ}v+D^γ v−∆_{H} v=|u|^p. Our method of proof is based on the nonlinear capacity method.

In this paper, we investigate a fractional p-Laplacian system with different critical Sobolev-Hardy exponents. By variational methods and the Mountain-Pass lemma, positive minimizers of the related best Sobolev constants and the existence of positive solutions of the system are found.