Test for propriety in a layer of a boundary problem with integral condition (Q1174068)

Necessary and sufficient conditions are given for the well-posedness of the problem consisting of the differential equation \(\partial_ tu(x,t)=P(-iD_ x)u(x,t)\) on the slab \(\mathbb{R}^ n\times[0,T]\) with the condition that \[ \int^ T_ 0B(-iD_ x)u(x,t)\exp\{-at\}dt=u_ 0(x) \] for all \(x\) in \(R^ n\). Here, a is a complex constant and \(P\) and \(B\) are polynomials with constant, complex coefficients. The well-posedness is shown in the Sobolev space \(W_ m^ \infty\).