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Dynamic behaviour of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary

In this paper we study a reaction-diffusion problem with nonlinear absorption through the boundary. We are interested in the existence of positive global solutions. We find out how the existence of such a solution depends on the relation of the initial datum with the nonnegative solution/solutions to the associated stationary problem. We give a precise description of the set of positive stationary solutions and the long time behavior of positive global solutions to the evolutionary problem. This description depends on some parameters appearing in the problem. The problem we consider is