A Note on the Parabolic Differential and Difference Equations

Abstract

The differential equation $u^{\prime} (t) + \mathbf{A} u(t) = f(t) ( -\infty \lt \lt \infty ) $ in a general Banach space $\mathbf{E}$ with the strongly positive operator $\mathbf{A}$ is ill-posed in the Banach space $C (\mathbf{E}) = C (\mathbb{R}, \mathbf{E})$ with norm $\| \varphi \|_{C(\mathbf{E})} = {\sup}_{-\infty \lt t \lt \infty} \|\varphi (t)\|_{\mathbf{E}}$. In the present paper, the well-posedness of this equation in the Hölder space $C^{\alpha} (\mathbf{E}) = C^{\alpha} (\mathbb{R}, \mathbf{E} )$ with norm $\|\varphi\|_{C^{\alpha} (\mathbf{E})} = {\sup}_{-\infty \lt t \lt \infty} \|\varphi (t)\|_{\mathbf{E}} + {\sup}_{-\infty \lt t \lt t+s \lt \infty} (\|\varphi (t+s)- \varphi (t)\|_{\mathbf{E}} /s^{\alpha})$), $0 \lt \alpha \lt 1$, is established. The almost coercivity inequality for solutions of the Rothe difference scheme in $C(\mathbb{R}_{\tau}, \mathbf{E})$ spaces is proved. The well-posedness of this difference scheme in $C(\mathbb{R}_{\tau}, \mathbf{E})$ spaces is obtained.