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Zeitschrift für Analysis und ihre Anwendungen

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Volume 19, Issue 1, 2000, pp. 159–201
DOI: 10.4171/ZAA/945

Published online: 2000-03-31

On a Class of Parabolic Integro-Differential Equations

W. Kohl[1]

(1) Wertheim, Germany

Existence and uniqueness results for the integro-differential equation $$u_1(x, t) - au_{xx} (x, t) = c(x, t)u(x, t) + \int^1_0 k(s, x)h(s, t, u(s, t)) ds + f(x, t)\\\ ((x,t) \in Q)$$ subject to the boundary condition $$u(x,t) = \varphi (x,t)\\\ ((x, t) \in R)$$ and, especially, for the linear case $h(s,t,u) = u$ are given. To this end, this equation is written as operator equation in a suitable Hölder space. The main tools are the calculation of the spectral radius in the linear case, and fixed point principles in the nonlinear case.

Keywords: Integro-differential equations, parabolic operators, multiplication operators, integral operators, Hölder spaces, heat potential, existence and uniqueness of solutions, Neumann series, fixed point principle

Kohl W.: On a Class of Parabolic Integro-Differential Equations. Z. Anal. Anwend. 19 (2000), 159-201. doi: 10.4171/ZAA/945