Kreiss' mixed problems with nonzero initial data

Jeffrey Rauch
1971 Bulletin of the American Mathematical Society  
In [3] , Kreiss has shown that a large class of mixed initial boundary-value problems of hyperbolic type are well-posed in the £2 sense. However only zero initial data were considered. For the same class of problems we show that if square integrable initial data are prescribed then there is a unique solution which is square integrable for each positive time. The differential operators under consideration are of the form where u is a complex fe-vector, and A y and B are kXk matrix-valued
more » ... s. The operator (L) is assumed strictly hyperbolic, that is ]T) Ajh has k distinct real eigenvalues for each ££i£ w \0. The coefficients are assumed to be smooth functions which are constant outside a compact set. In addition, we require that det Ai^O when #i = 0. The following notation is employed: Boundary conditions are prescribed with the aid of a boundary operator M(t, x') which is a smooth iXk matrix-valued function where / = number of negative eigenvalues of Ai. We suppose that M has rank / and is independent of t, x' for 11\ + | x'\ large. The basic problem is to show that for given F G £ 2 ([0, T] X 12), g G £ 2 ([0, r] X dQ), ƒ G £2(0).
doi:10.1090/s0002-9904-1971-12849-5 fatcat:hxxxz4ihwbh7hejoeyovtuxnsi