Abstract
In this paper, we study the parabolic problems with anisotropic nonstandard growth nonlinearities. We first give the existence and uniqueness of weak solutions in variable Sobolev spaces. Second, we use the energy methods to show the existence of blow-up solutions with negative or positive initial energy, respectively. Both the variable exponents and the coefficients make important roles in Fujita blow-up phenomena. Moreover, asymptotic properties of the blow-up solutions are determined.
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