© 2000 by London Mathematical Society
© The London Mathematical Society
Smooth Solutions for a Class of Nonlinear Parabolic Evolution Equations
Fachbereich Mathematik, Universität Dortmund D-44221 Dortmund, Germany, poppenberg{at}math.uni-dortmund.de
Received 28 July 1997. Revision received 20 January 1998.
The purpose of the paper is to introduce a new linearization method for local well-posedness of nonlinear evolution equations. This approach is based on an implicit function theorem of Nash–Moser type. The technique is illustrated by an application to a general class of fully nonlinear parabolic partial differential equations of arbitrary order on Rn. The estimates required by the Nash–Moser technique are derived for the higher order Sobolev norms of the solutions of the linearized parabolic equation using semigroup theory and elliptic theory. In particular, a priori estimates, resolvent estimates and commutator estimates are involved. The general method based on a combination of Nash–Moser techniques with semigroup theory is applicable to other problems and has already been used to prove short-time solvability for some nonlinear Schrödinger type equation. This approach might be useful in other situations as well since it compensates for a loss of derivatives in the estimates of the solutions of the linearized equation.