A Note on the Parabolic Differential and Difference Equations

Abstract

The differential equation $u\text{'}\left(t\right)+\mathbf{A}u\left(t\right)=f\left(t\right)\left(-\infty <t<\infty \right)$ in a general Banach space $\mathbf{E}$ with the strongly positive operator $\mathbf{A}$ is ill-posed in the Banach space $C\left(\mathbf{E}\right)=C\left(ℝ,\mathbf{E}\right)$ with norm ${\Vert \varphi \Vert }_{C\left(\mathbf{E}\right)}={sup}_{-\infty <t<\infty }{\Vert \varphi \left(t\right)\Vert }_{\mathbf{E}}$. In the present paper, the well-posedness of this equation in the Hölder space $C^{\alpha }\left(\mathbf{E}\right)=C^{\alpha }\left(ℝ,\mathbf{E}\right)$ with norm ${\Vert \varphi \Vert }_{C^{\alpha }\left(\mathbf{E}\right)}={sup}_{-\infty <t<\infty }{\Vert \varphi \left(t\right)\Vert }_{\mathbf{E}}+{sup}_{-\infty <t<t+s<\infty }{(\Vert \varphi \left(t+s\right)-\varphi \left(t\right)\Vert }_{\mathbf{E}}/s^{\alpha }$), $0<\alpha <1$, is established. The almost coercivity inequality for solutions of the Rothe difference scheme in $C\left({ℝ}_{\tau },\mathbf{E}\right)$ spaces is proved. The well-posedness of this difference scheme in $C^{\alpha }\left({ℝ}_{\tau },\mathbf{E}\right)$ spaces is obtained.