7.5. Mildly Ill-Posed Half-Space Problems

Consider a parabolic problem

in 0 ? x< ?, t ?0, with initial condition

and boundary conditions at x=0,

All matrices are constant (in general, complex), A, B, C have size n n, , have size r n, and has size (n ?r) n. We make the reasonable assumption that and have full rank in order to have n linearly independent boundary conditions. Furthermore, we assume for simplicity that the eigenvalues

of A are distinct. First we want to use the ideas of the previous section to derive a formal solution formula. Second, we will show that the problem is not strongly well-posed in the generalized sense (see Definition 3 of Section 7.3) unless

Here

Hence there are two different ways in which a problem (7.5.1) (7.5.3) can become ill-posed: First, there can be a sequence of eigenvalues s _? with Re s _? ??; see Lemma 7.4.3. Second, no such sequence exists, but det C ₀=0. In the second case the ill-posedness is of a milder nature: One can still estimate the solution if one uses derivatives of the data. In contrast to the weakly ill-posed Cauchy problems of Chapter 2, a perturbation by lower-order terms still does not lead to arbitrarily fast exponential growth.

7.5.1. Formal Solution

Laplace transformation of (7.5.1) (7.5.3) gives us

For each fixed s with sufficiently large real part, we want to solve...