Maximal regularity and quasilinear parabolic boundary value problems

∂tw +A(v)w = f(v) in J̊ , w(0) = 0 has a unique solution w = w(v) ∈W(J). Clearly, a fixed point of the map v 7→ w(v) is a solution of (1). Fixed point arguments of this type are, of course, omnipresent in the study of evolution equations of type (1). The new feature of our work, which distinguishes it from all previous investigations, is that we assume that W(J) is a maximal regularity space and that A(·) is only defined on W(J). In all the works published so far, it is always assumed that the domain of A(·) is a larger space than W(J) on which the problem is relatively easy to handle. Our approach gives great flexibility in