On the Duhamel principle. (Q1174412)

The paper deals with Cauchy problems \[ LU=F,\quad s\leq t\leq T,\quad x\in\mathbb{R}^ n,\quad U=\Phi,\quad t=s,\quad x\in\mathbb{R}^ n, \] where \(L=L(t,x,\partial_ t,\partial_ x)=\partial_ tI-A(t,x,\partial_ x)\), and the operator \(A\) is an \(N\times N\) matrix. The author proves that the problem is well posed in Sobolev space \(H^ \infty\) for \(0\leq s\leq T\) if and only if the homogeneous problem, \(F=0\), is uniformly well posed for \(0\leq s\leq T\) or, equivalently, that the homogeneous problem is well posed and the Duhamel principle is valid. An example of a scalar equation which is well posed, but not uniformly, is given.