Nonlocal Nonlinear Problems for One-Dimensional Parabolic System

  • Eugeniusz M. Chrzanowski

    Warsaw Technical University, Poland


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

to find a solution of the system

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

subject to the conditions

u(0,t)=φ(t),t(0,T),\vec u (0, t) = \vec \varphi (t), \quad t \in (0, T),
u(x,0)=ψ(x),x(0,1),\vec u (x, 0) = \vec \psi (x), \quad x \in (0, 1),
u(1,t)u(x0,t)=h(x0,t,u(x0,t))\vec u (1, t) - \vec u (x_0, t) = \vec h (x_0, t, \vec u (x_0, t))


01u(x,t)dx=g(t).\int ^1_0 \vec u (x, t) dx = \vec g (t).

For this an operator L:C(Ωˉ)C(Ωˉ)L: C(\bar \Omega) \to C(\bar \Omega) being a sum of four potentials is constructed. It is shown that the operator LL 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