[For the following notions, notations and the terminology see the paper itself, mainly Section 0 of the unabridged French version!]
Let \(X\) be a general Banach space and \(A \in {\mathcal L}(X)\) a generator of an irregular bounded analytic semigroup \(Q(t)\) and such that there is \(A^{-1} \in {\mathcal B}(X)\), with in general \(\overline{D(A)} = X_ 0 \neq X\). Let also a linear subspace \(H \subset X\) and the “boundary” operator \(B \in {\mathcal B}(X)\) such that \(B: X \to H\), \(X_ 0 \to \{0\}\) and \(H \to H\) identically. Then define the operator \(\widehat{A} \in {\mathcal L}(X)\) by \(D(\widehat{A}) = D(A) \oplus H\), and for \(\widehat{f} \in D(\widehat{A})\) decomposed as \(f+h\), \(f \in D(A)\), \(h \in H\) set \(\widehat{A}\widehat{f} = Af\).
The author considers in the above mentioned general abstract setting, models (i.e. equations allowing one to model) that concern phenomena of diffusion type with shocks (that is sudden changes in boundary conditions). More specifically in his model, the evolution of the phenomenon he studies, is governed between the shock points \(t_ 0\) and \(t_ 1\) by (1) \(\{u' = \widehat{A}u + F(t)\), \(u(t_ 0) = f\), \(Bu(t) = \varphi\}\) with given \(f\in X\), \(F \in L^ 1_{\text{loc}}[(- \infty,+\infty),X]\) and \(\varphi \in H\).
In Section 2 he proves that (1) has a unique optimal regular solution while in Section 3 he proves that there exists a unique periodic orbit which attracts asymptotically all other orbits. In Sections 4 and 5 he treats (1) in a classical setting and with some modification for the hypotheses he provides analogous but interesting results.