Seventh Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 17 (2009), pp. 39-49. 

Title: A boundary control problem with a nonlinear reaction term

Authors: John R. Cannon (Univ. of Central Florida, FL, USA)
         Mohamed Salman (Tuskegee Univ.,  AL, USA)
 
Abstract: 
  The authors study the problem $u_t=u_{xx}-au$,
 $0<x<1$, $t>0$; $u(x,0)=0$, and $-u_x(0,t)=u_x(1,t)=\phi(t)$,
 where $a=a(x,t,u)$, and $\phi(t)=1$ for $t_{2k} < t<t_{2k+1}$ and
 $\phi(t)=0$ for $t_{2k+1} <t<t_{2k+2}$, $k=0,1,2,\ldots$ with
 $t_0=0$ and the sequence $t_{k}$ is determined by the equations
 $\int_0^1 u(x,t_k)dx = M$, for $k=1,3,5,\dots$, and
 $\int_0^1 u(x,t_k)dx = m$, for $k=2,4,6,\dots$,  where $0<m<M$.
 Note that the switching points $t_k$, are unknown.
 A maximum principal argument has been used to prove that the solution
 is positive under certain conditions. Existence and uniqueness are
 demonstrated. Theoretical estimates of the $t_k$ and $t_{k+1}-t_k$
 are obtained and numerical verifications of the estimates are
 presented.

Published April 15, 2009.
Math Subject Classifications: 35K57, 35K55.
Key Words: Reaction-diffusion, Parabolic, Nonlocal boundary conditions.