Weak solution for an abstract Cauchy problem of parabolic type

Abstract

We define weak solutions for an abstract linear Cauchy problem of parabolic type: (d/dtu)+A(t)u=f, u(0)=u0,with u0in H and f in\(L^2 (0, \infty , V') + H^{ - \tfrac{1}{2}} (0, \infty , H)\),where V ⊂ H ⊂ V′ are Hilbert spaces. We prove the existence of such a solution and isomorphism theorems in spaces of type\(L^2 (0, \infty , V) \cap (H_{00}^{\tfrac{1}{2}} (0, \infty , H) + H^1 (0, \infty , V'))\),and study the approximation of the solution in norms of this type.