## Quasilinear parabolic problems via maximal regularity.(English)Zbl 1103.35059

Summary: We use maximal $$L_p$$ regularity to study quasilinear parabolic evolution equations. In contrast to all previous work we only assume that the nonlinearities are defined on the space in which the solution is sought for. It is shown that there exists a unique maximal solution depending continuously on all data, and criteria for global existence are given as well. These general results possess numerous applications, some of which will be discussed in separate publications.

### MSC:

 35K90 Abstract parabolic equations 47H20 Semigroups of nonlinear operators 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

### Keywords:

continuous dependence; maximal $$L_p$$ regularity