Note on a quasilinear higher-order parabolic equation with variable exponents

In this paper, we investigate weak solutions to an initial-boundary value problem for a quasilinear parabolic equation involving the p(x)-Laplacian and nonlinear absorption, which models phenomena such as epitaxial thin-film growth and lubrication theory. By employing the potential well method and auxiliary function technique, we classify the initial data into those leading to blow-up solutions and those admitting global weak solutions. Furthermore, we study asymptotic estimates for weak solutions, including the decay rate of global solutions, extinction time, and extinction rate, providing a comprehensive analysis of the solution behavior across different regimes.