Absence of global solutions to wave equations with structural damping and nonlinear memory

We prove the nonexistence of global solutions for the following wave equations with structural damping and nonlinear memory source term utt+(−Δ)α2u+(−Δ)β2ut=∫0t(t−s)δ−1∣u(s)∣pds{u}_{tt}+{\left(-\Delta )}^{\tfrac{\alpha }{2}}u+{\left(-\Delta )}^{\tfrac{\beta }{2}}{u}_{t}=\underset{0}{\overset{t}{\int }}{\left(t-s)}^{\delta -1}{| u\left(s)| }^{p}{\rm{d}}s and utt+(−Δ)α2u+(−Δ)β2ut=∫0t(t−s)δ−1∣us(s)∣pds,{u}_{tt}+{\left(-\Delta )}^{\tfrac{\alpha }{2}}u+{\left(-\Delta )}^{\tfrac{\beta }{2}}{u}_{t}=\underset{0}{\overset{t}{\int }}{\left(t-s)}^{\delta -1}{| {u}_{s}\left(s)| }^{p}{\rm{d}}s, posed in (x,t)∈RN×[0,∞)\left(x,t)\in {{\mathbb{R}}}^{N}\times \left[0,\infty ), where u=u(x,t)u=u\left(x,t) is the real-valued unknown function, p>1p\gt 1, α,β∈(0,2)\alpha ,\beta \in \left(0,2), δ∈(0,1)\delta \in \left(0,1), by using the test function method under suitable sign assumptions on the initial data. Furthermore, we give an upper bound estimate of the life span of solutions.