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.

On a Class of Parabolic Integro-Differential Equations

W. Kohl

Abstract

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