This treatise gives an exposition of the functional analytical approach to quasilinear parabolic evolution equations, developed to a large extent by the author during the 1980s. This approach is based on the theory of nonlinear non-autonomous parabolic evolution equations and on interpolation-extrapolation techniques. It is the only general method that applies to non-coercive quasilinear parabolic systems under nonlinear boundary conditions. This first volume is devoted to a detailed study of non-autonomous linear parabolic evolution equations in general Banach spaces. It contains an exposition of the constant domain case, leading to some improvements of the classical Sobolevski-Tanabe results. It also includes recent results for equations possessing constant interpolation spaces. In addition, the book provides systematic presentations of the theory of maximal regularity in spaces of continuous and Hoelder continuous functions, and in Lebesgue spaces. It includes related recent theorems in the field of harmonic analysis in Banach spaces and on operators possessing bounded imaginary powers.
Lastly, there is a complete presentation of the technique of interpolation-extrapolation spaces and of evolution equations in those spaces, containing new results.