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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 $v'(t)+A v(t)=f(t)$ 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 $\min\{1/\tau,\vert \ln \Vert A\Vert _{E\to E}\vert \}$, where $\tau$ is the time step).

65J10Equations with linear operators (numerical methods)
65M06Finite difference methods (IVP of PDE)
65L05Initial value problems for ODE (numerical methods)
47D06One-parameter semigroups and linear evolution equations
34G10Linear ODE in abstract spaces
35K90Abstract parabolic equations
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