# Nonlocal Nonlinear Problems for One-Dimensional Parabolic System

### Eugeniusz M. Chrzanowski

Warsaw Technical University, Poland

## Abstract

In the paper two nonlocal, nonlinear problems for a system of parabolic equations are considered:

to find a solution of the system $u_{t}(x,t)=Du_{xx}(x,t)+f (x,t,u(x,t))$ subject to the conditions $u(0,t)=φ (t),t∈(0,T),$ $u(x,0)=ψ (x),x∈(0,1),$ $u(1,t)−u(x_{0},t)=h(x_{0},t,u(x_{0},t))$ or $∫_{0}u(x,t)dx=g (t).$ For this an operator $L:C(Ωˉ)→C(Ωˉ)$ being a sum of four potentials is constructed. It is shown that the operator $L$ has only one fixed point. Moreover it is proved that the fixed point is the only solution of the considered problem.

## Cite this article

Eugeniusz M. Chrzanowski, Nonlocal Nonlinear Problems for One-Dimensional Parabolic System. Z. Anal. Anwend. 3 (1984), no. 4, pp. 329–336

DOI 10.4171/ZAA/111