Blow-up analysis for a class of plate viscoelastic $p(x)-$Kirchhoff type inverse source problem with variable-exponent nonlinearities

In this work, we study the blow-up analysis for a class of plate viscoelastic $p(x)$-Kirchhoff type inverse source problem of the form:
\begin{align*} u_{tt}+\Delta^{2}u&-\left(a+b\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(x)}u-\int_{0}^{t}g(t-\tau)\Delta^{2}u(\tau)d\tau \\ & +\beta|u_{t}|^{m(x)-2}u_{t}=\alpha|u|^{q(x)-2}u+f(t)\omega(x). \end{align*}
Under suitable conditions on kernel of the memory, initial data and variable exponents, we prove the blow up of solutions in two cases: linear damping term ($m(x)\equiv2$) and nonlinear damping term ($m(x)>2$). Precisely, we show that the solutions with positive initial energy blow up in a finite time when $m(x)\equiv2$ and blow up at infinity if $m(x)>2$.