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Zbl 1201.35064
Ashyralyev, Allaberen; Sözen, Yaşar
Well-posedness of parabolic differential and difference equations.
(English)
[J] Comput. Math. Appl. 60, No. 3, 792-802 (2010). ISSN 0898-1221

Summary: We consider the abstract parabolic differential equation $u'(t)+Au(t)=f(t)$, $-\infty <t<\infty$ in a Banach space $E$ with - $A$ the infinitesimal generator of an analytic, exponentially decreasing semigroup $\exp\{-tA\}$ $(t\ge 0)$. The main purpose of this paper is to establish the well-posedness of this equation in $C^{\beta}(\bbfR, E_{\alpha}$, $(\alpha, \beta\in [0,1])$, and the well-posedness of the corresponding Rothe difference scheme in $C^{\beta}(\bbfR_{\tau}, E_{\alpha}$, $(\alpha, \beta\in [0,1])$. Moreover, we apply our theoretical results to obtain new coercivity inequalities for the solution of parabolic difference equations.
MSC 2000:
*35B65 Smoothness of solutions of PDE
65M06 Finite difference methods (IVP of PDE)

Keywords: abstract parabolic equation; Banach spaces; fractional spaces; well-posedness; difference scheme

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