Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 39-49.

A boundary control problem with a nonlinear reaction term

John R. Cannon, Mohamed Salman

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.

Show me the PDF file (321 KB), TEX file, and other files for this article.

John R. Cannon
University of Central Florida, Department of Mathematics
Orlando, FL 32816, USA
email: jcannon@pegasus.cc.ucf.edu
Mohamed Salman
Tuskegee University, Department of Mathematics
Tuskegee, AL 36088, USA
email: msalmanz@gmail.com

Return to the table of contents for this conference.
Return to the EJDE web page