Continuous Interpolation Spaces and Spatial Regularity in Nonlinear Volterra Integrodifferential Equations

• Published in: Journal of integral equations 1983-10, Vol.5 (4), p.287-308
• Main Author: Sinestrari, Eugenio
• Format: Article
• Language: English
• Subjects: Abstract spaces; Approximation; Banach space; Continuous functions; Differential equations; Infinitesimals; Interpolation; Linear transformations; Mathematical functions; Semigroups
• ISSN: 0163-5549

Let E and F be Banach spaces and F continuously and densely embedded in E. We study the nonlinear integrodifferential equation $\mathrm{u}\prime \left(\mathrm{t}\right)=\mathrm{f}(\mathrm{t},\mathrm{u}(\mathrm{t}\left)\right)+{\int }_{0}^{\mathrm{t}}\mathrm{g}(\mathrm{t},\mathrm{s},\mathrm{u}(\mathrm{s}\left)\right)\mathrm{d}\mathrm{s}, \quad \mathrm{u}\left(0\right)=\mathrm{x}, \quad \mathrm{t}\ge 0$, where f(t, u) is continuous from ℝ+ × F to E and the partial Fréchet derivative fu is continuous and generates an analytic semigroup with domain F; moreover, g(t, s, u) is continuous and g(t, s, ·) is locally Lipschitz continuous from F to E. By means of a suitable class of interpolation spaces an application is given to the study of the equation $\begin{array}{c}{\mathrm{u}}_{\mathrm{t}}(\mathrm{t},\mathrm{x})=\mathrm{\varphi }(\mathrm{t},{\mathrm{u}}_{\mathrm{x}\mathrm{x}}(\mathrm{t},\mathrm{x}\left)\right)+{\int }_{0}^{\mathrm{t}}\mathrm{K}(\mathrm{t},\mathrm{s},{\mathrm{u}}_{\mathrm{x}\mathrm{x}}(\mathrm{s},\mathrm{x}\left)\right)\mathrm{d}\mathrm{s}, \quad \mathrm{t}\ge 0, \quad 0\le \mathrm{x}\le 1,\\ \mathrm{u}(\mathrm{t},0)=\mathrm{u}(\mathrm{t},1)=0, \quad \mathrm{t}\ge 0; \quad \mathrm{u}(0,\mathrm{x})={\mathrm{u}}_{0}\left(\mathrm{x}\right), \quad 0\le \mathrm{x}\le 1,\end{array}$ for which strict solutions are found such that uxx is little Hölder continuous with respect to the spatial variable x if the initial datum u0(x) verifies the same condition.