A Note on the Parabolic Differential and Difference Equations

The differential equation u ' ( t ) + A u ( 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 C α ( E ) = C α ( ℠, E ) with norm ‖ ϕ ‖ C α ( E ) = sup − ∞ < t < ∞ ‖ ϕ ( t ) ‖ E + sup − ∞ < t < t + s < ∞ (‖ ϕ ( t + s ) − ϕ ( t ) ‖ E / s α ), 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 C α ( ℠τ , E ) spaces is obtained.