Non-autonomous inhomogeneous boundary Cauchy problems

Abstract

In this paper we prove existence and uniqueness of classical solutions for the non-autonomous inhomogeneous Cauchy problem $$displaylines{ frac{d}{dt}u(t)=A(t)u(t)+f(t), quad 0 leq sleq tleq T, cr L(t)u(t)=Phi(t)u(t)+g(t) , quad 0leq sleq tleq T, cr u(s)=x. }$$ The solution to this problem is obtained by a variation of constants formula.

Keywords

• Boundary Cauchy problem
• evolution families
• solution
• well posedness
• variation of constants formula.