Global and blow-up results for a quasilinear parabolic equation with variable sources and memory terms

The paper presents a general model of quasi-linear parabolic equations with variable exponents for the source and dissipative term types \(\begin{aligned} \textrm{L}\left( t\right) \left| u_{t}\right| ^{m\left( x\right) -2}u_{t}-\Delta u+\int _{0}^{t}g(t-s)\Delta u(x,s)\textrm{d}s=\left| u\right| ^{p\left( x\right) -2}u. \end{aligned}\) $$\begin{array}{c}\hfill \text{L}\left(t\right){\left|u_t\right|}^{m\left(x\right)-2}{u}_t-\mathrm{\Delta }u+{\int }_0^{t}g(t-s)\mathrm{\Delta }u(x,s)\text{d}s={\left|u\right|}^{p\left(x\right)-2}u.\end{array}$$ When \(p(x)\ge m(x)\ge 2\) $p(x)\ge m(x)\ge 2$ , the matrix \(\textrm{L}(t)\) $\text{L}(t)$ is both positive definite and bounded, while the function g is continuously differentiable and decays over time. The paper shows that the blow-up result occurs at two different finite times and provides an upper bound for the blow-up time. Finally, it establishes that the energy function decays globally for solutions, with both positive and negative initial energy.