# 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

$\vec u_t (x,t) = D\vec u_{xx}(x, t) + \vec f (x, t, \vec u (x, t))$

subject to the conditions

$\vec u (0, t) = \vec \varphi (t), \quad t \in (0, T),$

$\vec u (x, 0) = \vec \psi (x), \quad x \in (0, 1),$

$\vec u (1, t) - \vec u (x_0, t) = \vec h (x_0, t, \vec u (x_0, t))$

or

$\int ^1_0 \vec u (x, t) dx = \vec g (t).$

For this an operator $L: C(\bar \Omega) \to C(\bar \Omega)$ 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