Asymptotic estimate of solutions to a fourth-order parabolic equation with double logarithm terms¹

Abstract

This paper investigates a fourth-order parabolic equation with double logarithm nonlinearities. Using the Galerkin method, we establish the global existence and uniqueness of weak solutions. By classifying the initial energy, we derive blow-up criteria of solutions and obtain explicit upper bounds for the blow-up time of solutions. Moreover, we present new threshold criteria for the extinction and non-extinction behavior of solutions. Specifically, we give the upper bound and lower bound for the extinction rate of solutions. These results generalize and improve some earlier related results in the literature.