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Well-posedness of parabolic differential and difference equations. (English) Zbl 1201.35064
Summary: We consider the abstract parabolic differential equation u ' (t)+Au(t)=f(t), -<t< in a Banach space E with - A the infinitesimal generator of an analytic, exponentially decreasing semigroup exp{-tA} (t0). The main purpose of this paper is to establish the well-posedness of this equation in C β (,E α , (α,β[0,1]), and the well-posedness of the corresponding Rothe difference scheme in C β ( τ ,E α , (α,β[0,1]). Moreover, we apply our theoretical results to obtain new coercivity inequalities for the solution of parabolic difference equations.
35B65Smoothness and regularity of solutions of PDE
65M06Finite difference methods (IVP of PDE)
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