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We consider a nonlinear parabolic equation with an exponential nonlinearitywhich is critical with respect to the growth of the nonlinearity and theregularity of the initial data. After showing the equivalence of the notions ofweak and mild solutions, we derive decay estimates and the asymptotic behaviorfor small global-in-time solutions.

Asymptotic Behavior and Decay Estimates of the Solutions for a Nonlinear Parabolic Equation with Exponential Nonlinearity