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Abstract
We associate with an abstract evolution equation (E) in a Banach space a normalized equation (N), defined as the equation fulfilled by u/ϕ(u), where u is a solution of (E), and ϕ is a prescribed functional. After establishing a number of qualitative properties of normalized equations, we specialize to the case in which (E) is a linear parabolic equation of the second order. We prove results concerning the behaviour of the solutions for large times and for small and large values of a specified (diffusion) parameter.
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