Blow-up analyses in parabolic equations with anisotropic nonstandard damping source

In this paper, we consider the nonlinear parabolic problems with anisotropic nonstandard growth conditions and damping terms. After obtaining well-posedness of solutions, we give blow-up criteria of solutions through constructing different control functions and generalizing eigenfunction method, respectively. We also classify global solutions in all scope of the variable exponents. Moreover, the sharp blow-up rates, blow-up time and blow-up set are determined, which seem to be rarely studied for parabolic problems with anisotropic nonstandard damping sources. It is interesting that the asymptotic estimates of blow-up solutions rely not only on maxima and minima of the anisotropic exponents, but also on the geometry properties of the spatial domain and the scope of variable coefficients.