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Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces. (English) Zbl 1117.65077
The author consider an abstract parabolic equation ${v}^{\text{'}}\left(t\right)+Av\left(t\right)=f\left(t\right)$ where the initial condition is replaced by the nonlocal condition $v\left(0\right)=v\left(\lambda \right)+\mu$. 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 ${\left\{exp\left(-At\right)\right\}}_{t\ge 0}$ 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 $min\left\{1/\tau ,|ln\parallel A{\parallel }_{E\to E}|\right\}$, where $\tau$ is the time step).
##### MSC:
 65J10 Equations with linear operators (numerical methods) 65M06 Finite difference methods (IVP of PDE) 65L05 Initial value problems for ODE (numerical methods) 47D06 One-parameter semigroups and linear evolution equations 34G10 Linear ODE in abstract spaces 35K90 Abstract parabolic equations