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Zeitschrift für Analysis und ihre Anwendungen


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Volume 3, Issue 4, 1984, pp. 329–336
DOI: 10.4171/ZAA/111

Published online: 1984-08-31

Nonlocal Nonlinear Problems for One-Dimensional Parabolic System

Eugeniusz M. Chrzanowski

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.

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Chrzanowski Eugeniusz: Nonlocal Nonlinear Problems for One-Dimensional Parabolic System. Z. Anal. Anwend. 3 (1984), 329-336. doi: 10.4171/ZAA/111