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Let $$\varphi $$ be an entire function, the superposition operator is defined by $$S_{\varphi }(f)=\varphi \circ f$$ . In this paper, we characterize the entire functions $$\varphi $$ that transform weighted Bloch spaces of analytic functions $${\mathcal {B}}_{\mu }$$ into another space of the same kind $${\mathcal {B}}_{\nu }$$ by superposition. Both $$\mu $$ and $$\nu $$ are normal functions or both belong to a certain class of functions. We also obtain several results about the boundedness of superposition operators acting between Logarithmic-type Bloch spaces, weight Banach spaces among others.

We give a proof for the asymptotic exponential stability in admissible interpolation spaces of equilibrium solutions to quasilinear parabolic evolution equations.