Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces (Q862242)

The author consider an abstract parabolic equation \(v'(t)+A v(t)=f(t)\) where the initial condition is replaced by the nonlocal condition \(v(0)=v(\lambda)+\mu\). All variables and constants takes values in a Hilbert space \(E\) and \(A\) is a linear and possible unbounded operator on this space. Under the assumption that the operator \(-A\) generates an analytic semigroup \(\{\exp(-At)\}_{t\geq0}\) with exponential decay, it is shown that the solutions to the nonlocal parabolic equation satify a coercivity estimate in terms of \(f\) and \(\mu\) 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 \(\min\{1/\tau,| \ln \| A\| _{E\to E}| \}\), where \(\tau\) is the time step).