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 last 10 years. This approach is based on the theory of linear nonautonomous parabolic evolution equations and on interpolation-extrapolation techniques. It is the only general method that applies to noncoercive quasilinear parabolic systems under nonlinear boundary conditions.
The present first volume is devoted to a detailed study of nonautonomous linear parabolic evolution equations in general Banach spaces. It contains a careful exposition of the constant domain case, leading to some improvements of the classical Sobolevskii-Tanabe results. It also includes recent results for equations possessing constant interpolation spaces. In addition, systematic presentations of the theory of maximal regularity in spaces of continuous and H”lder continuous functions, and in Lebesgue spaces, are given. 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 many new results.
Part 1 Generators and interpolation: generators of analytic semigroups; interpolation functors. Part 2 Cauchy problems and evolution operators: linear Cauchy problems; parabolic evolution operators; linear Volterra integral equations; existence of evolution operators; stability estimates; invariance and positivity. Part 3 Maximal regularity: general principles; maximal Hoelder regularity; maximal continuous regularity; maximal Sobolev regularity. Part 4 Variable domains: higher regularity; constant interpolation spaces; maximal regularity. Part 5 Scales of Banach spaces: Banach scales; evolution equations in Banach scales.