Duhamel principle and uniformly well-posed Cauchy problems
Mimiking the variation of constants approach to inhomogeneous initial value problems for ordinary differential equations Duhamel's principle yields a solution of a general inhomogeneous evolution equation of the form \(\partial_ t u(t)- A(t)u(t)= F(t)\), \(u(0+)= u_ 0\), (\(F\), \(u_ 0\) given) in terms of the homogeneous solution of a pure initial value problem \((F\equiv 0)\). The author analyzes the underlying relation between the homogeneous \((\text{PCH})_ s\) and the inhomogeneous evolution problem \((\text{PC})_ s\) posed in a Fréchet space \(X\). Here the parameter \(s\in [0,T_ 0[\), \(T_ 0> 0\), denotes the initial time. Based on the concept of uniform well-posedness versus well-posedness, the author demonstrates (under suitable assumptions on \(A(\cdot)\) in terms of the family of semi-norms defining the topology of \(X\)) that, roughly speaking, uniform well-posedness of the problems \((\text{PCH})_ s\) for \(s\in [0,T_ 0[\), is equivalent to well-posedness of the problems \((\text{PC})_ s\) for \(s\in[ 0,T_ 0[\), \(T_ 0> 0\). The result applies to Cauchy problems with time-dependent coefficients.
scientific article; zbMATH DE number 567066