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Zbl 1117.65077
Ashyralyev, Allaberen
Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces.
(English)
[J] J. Evol. Equ. 6, No. 1, 1-28 (2006). ISSN 1424-3199; ISSN 1424-3202/e

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).
[Stefan Jakobsson (Göteborg)]
MSC 2000:
*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 evolution equations

Keywords: difference schemes; well-posedness; coercive inequalities; abstract parabolic equation; Hilbert space

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