Abstract
We develop a general approach for the study of the boundedness and stability of solution to general parabolic semi-linear evolution equations in real-interpolation spaces. Our approach yields a unified method to handle problems in Newtonian viscous incompressible fluid flows. Moreover, our method will be applied to derive new results for the existence and polynomial stability of bounded solutions to the Ornstein–Uhlenbeck semi-linear equations and various diffusion equations with rough coefficients. The method is based on a combination of interpolation functors, duality estimates, smoothing properties of the linearized equation, and fixed point arguments.
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