Well-posedness of parabolic differential and difference equations

We consider the abstract parabolic differential equation u0.t/ C Au.t/ D f .t/;􀀀1 < t < 1 in a Banach space E with 􀀀A the infinitesimal generator of an analytic, exponentially decreasing semigroup expf􀀀tAg .t 0/: The main purpose of this paper is to establish the well-posedness of this equation in C .R; E /; . ; 2 T0; 1U/; and the well-posedness of the corresponding Rothe difference scheme in C .R ; E /; . ; 2 T0; 1U/: Moreover, we apply our theoretical results to obtain new coercivity inequalities for the solution of parabolic difference equations.