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This paper deals with an initial-boundary problem of the fourth-order parabolic equation involving two logarithm terms. First, we give some results for blow-up or global solutions through classifying the initial energy and the Nehari energy. Second, we show asymptotic estimates about blow-up time and a large time estimate of solutions, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study extinction and blow-up properties of weak solutions for the doubly nonlinear parabolic problem involving p(x, t)-Laplacian operator, subject to homogeneous Dirichlet boundary conditions. We first prove the existence and uniqueness of weak solutions in some anisotropic Orlicz-Sobolev spaces and also establish suitable regularity of weak solutions with respect to the introduced control functional. Second, we use the energy estimate and the comparison principle of the ordinary differential equation to study the extinction or non-extinction of solutions. Third, we combine some new energy functions with the embedding inequalities to show blow-up criteria of solutions. Moreover, the singular rates of weak solutions are determined. It is interesting that blow-up or extinction phenomena depend sensitively on the sign of the precise relationship among the different anisotropic variable exponents, which discovers a key clue to the different effects of diffusion and reaction of the equations on the singularities of weak solutions. [ABSTRACT FROM AUTHOR]