Nonlocal Nonlinear Problems for One-Dimensional Parabolic System

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 $\vec{u}(0,t)=\vec{\phi }(t),t\in (0,T),$ $\vec{u}(x,0)=\vec{\psi }(x),x\in (0,1),$ $\vec{u}(1,t)-\vec{u}(x_0,t)=\vec{h}(x_0,t,\vec{u}(x_0,t))$ or ${\int }_0^1\vec{u}(x,t)dx=\vec{g}(t).$ 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.