# On a Class of Parabolic Integro-Differential Equations

### W. Kohl

Wertheim, Germany

## Abstract

Existence and uniqueness results for the integro-differential equation

$u_{1}(x,t)−au_{xx}(x,t)=c(x,t)u(x,t)+∫_{0}k(s,x)h(s,t,u(s,t))ds+f(x,t)((x,t)∈Q)$

subject to the boundary condition

$u(x,t)=φ(x,t)((x,t)∈R)$

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.

## Cite this article

W. Kohl, On a Class of Parabolic Integro-Differential Equations. Z. Anal. Anwend. 19 (2000), no. 1, pp. 159–201

DOI 10.4171/ZAA/945