The author consider an abstract parabolic equation
where the initial condition is replaced by the nonlocal condition
. All variables and constants takes values in a Hilbert space
is a linear and possible unbounded operator on this space. Under the assumption that the operator
generates an analytic semigroup
with exponential decay, it is shown that the solutions to the nonlocal parabolic equation satify a coercivity estimate in terms of
with the implication that the problem is well-posed. In addition, first and second order difference schemes are given and so called almost coercive inequalities are established for these (the multiplier in the inequality contains the factor
is the time step).