Formule de Duhamel abstraite. (Abstract Duhamel formula)
The authors give a rigorous proof of the Duhamel principle, in a general framework that includes, in particular, the elliptic operators with continuous coefficients. Consider an abstract problem of the following type: \((*)\quad \mu '=Au;\quad \mu (0)=u_ 0;\quad Bu=g,\) where A is a linear operator in a Banach space X, B is a linear operator from X into a Banach space Y and \(u_ 0\in X\) is an initial condition. They find some conditions in order to have (formally) \(\mu (t)=(d/dt)\int^{t}_{0}\mu (s,t-s)ds\) where \(\mu(s,\cdot)\) is the solution of (*) corresponding to \(g(s)\in Y\), when s is hold fixed (i.e. with time-independent constraints).
scientific article; zbMATH DE number 3884610