Nichtlineare Evolutionsgleichungen für q-holomorphe Vektoren. (Nonlinear evolution equations for q-holomorphic vectors) (Q1098990)

Using the method of successive approximations in scales of Banach spaces, the author proves the existence of a unique solution to the initial problem \(\partial w/\partial t=A(w,z,t)\partial w/\partial z+F(w,z,t)\), \(w(z,0)=w_ 0(z)\) of the Cauchy-Kowalewskaja type in the space of q- holomorphic vectors. (The latter vectors \(w=(w^ 1,...,w^ n)\) are satisfying a system \(\partial w/\partial z^*=q(z)\partial w/\partial z\) where \(q=(q_{ij}(z))\) is a quasidiagonal matrix with upper triangular blocks on the diagonal and \(\sum_{i}| q_{ij}| \leq const.<1\) ensuring the ellipticity.) The nonlinear systems \(\partial w/\partial t=F(\partial w/\partial z,w,u,t)\) can be transferred (under certain stronger restrictions) to the above quasilinear case.