Nonuniqueness and wellposedness of abstract Cauchy problems in a Fréchet space (Q2716968)

Let \(X\) be a Fréchet space and let \(A\) be a closed linear operator in \(X\). The author considers the abstract Cauchy problem NEWLINE\[NEWLINE \dot{u} (t) = A u (t), \quad u (0) = x, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \dot{u} (t) = A u (t) + f (t), \quad u (0) = x NEWLINE\]NEWLINE where he emphasize the case that \(A\) is not the generator of a strongly continuous semigroup. His main theorem reads as follows:NEWLINENEWLINENEWLINETheorem: Let \(A\) be as above. Then there exists a subspace \(Z\) of \(X\) which is a Fréchet space for a stronger topology such that \(Z\) is a quotient of a Fréchet space \(F\) and the part \(A_Z\) of \(A\) in \(Z\) is the quotient of a closed linear operator \(B\) in \(F\) for which the abstract Cauchy problem is well-posed. The subspace \(Z\) is maximal in the sense that it contains all \(x \in X\) for which there is a mild solution of the first equation.NEWLINENEWLINENEWLINEMoreover for all \(x \in Z\) and \(f \in C (\mathbb R_+ , Z)\) there exists a mild solution of the inhomogenious equation. NEWLINENEWLINENEWLINEThe space \(F\) is a subspace of \(C (\mathbb R_+ , X)\), the semigroup is the shift and \(B\) is therefore differentiation with respect to \(t\). \(Z = \{ x \in X : \exists f \in F [ f (0) = x]\}\). For the details we refer to the paper which contains an interesting and striking example concerning the heat equation in spaces of even entire functions.