JournalszaaVol. 19, No. 1pp. 159–201

On a Class of Parabolic Integro-Differential Equations

  • W. Kohl

    Wertheim, Germany
On a Class of Parabolic Integro-Differential Equations cover
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Abstract

Existence and uniqueness results for the integro-differential equation

u1(x,t)auxx(x,t)=c(x,t)u(x,t)+01k(s,x)h(s,t,u(s,t))ds+f(x,t) ((x,t)Q)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)=φ(x,t) ((x,t)R)u(x,t) = \varphi (x,t)\\\ ((x, t) \in R)

and, especially, for the linear case h(s,t,u)=uh(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