A note on the parabolic differential and difference equations

The differential equation u(t)+Au(t)=f(t)( -∞<t<∞) in a general Banach space E with the strongly positive operator A is ill-posed in the Banach space C(E)=C( ,E) with norm ||φ|| C(E) = sup - ∞<t<∞ || φ(t) || E . In the present paper, the well-posedness of this equation in the Hölder space Cα (E)=Cα ( ,E) with norm ||φ|| Cα (E) = sup -∞<t<∞ || φ(t) || E + sup -∞<t<t+s<∞ (|| φ( t+s)-φ(t) || E / sα), 0<α<1, is established. The almost coercivity inequality for solutions of the Rothe difference scheme in C( τ,E) spaces is proved. The well-posedness of this difference scheme in Cα ( τ,E) spaces is obtained.