On a Class of Parabolic Integro-Differential Equations

W. Kohl

Abstract: Existence and uniqueness results for the integro-differential equation $$ u_t(x,t) - au_{xx}(x,t) = c(x,t)u(x,t) + \int_0^1 k(s,x)h(s,t,u(s,t))\,ds + f(x,t) \ \ ((x,t) \in Q) $$ subject to the boundary condition $$ u(x,t) = \varphi(x,t) \qquad ((x,t) \in R) $$ 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

Classification (MSC2000): 47G20, 47H10, 47H30, 45K05, 35K99, 26B35

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