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

Abstract

The paper presents a general model of quasi-linear parabolic equations with variable exponents for the source and dissipative term types

$$\begin{aligned} \textrm{L}\left( t\right) \left| u_{t}\right| ^{m\left( x\right) -2}u_{t}-\Delta u+\int _{0}^{t}g(t-s)\Delta u(x,s)\textrm{d}s=\left| u\right| ^{p\left( x\right) -2}u. \end{aligned}$$

When \(p(x)\ge m(x)\ge 2\), the matrix \(\textrm{L}(t)\) 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.