Abstract and Applied Analysis

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.

Article information

Source
Abstr. Appl. Anal., Volume 2007 (2007), Article ID 61659, 16 pages.

Dates
First available in Project Euclid: 5 July 2007

https://projecteuclid.org/euclid.aaa/1183666871

Digital Object Identifier
doi:10.1155/2007/61659

Mathematical Reviews number (MathSciNet)
MR2302192

Zentralblatt MATH identifier
1153.35353

Citation

Ashyralyev, Allaberen; Sozen, Yasar; Sobolevskii, Pavel E. A Note on the Parabolic Differential and Difference Equations. Abstr. Appl. Anal. 2007 (2007), Article ID 61659, 16 pages. doi:10.1155/2007/61659. https://projecteuclid.org/euclid.aaa/1183666871

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