Internal nonnegative stabilization for some parabolic equations

Pages: 491 - 512, Volume 7, Issue 3, May 2008

doi:10.3934/cpaa.2008.7.491       Abstract        Full Text (243.0K)       Related Articles

B. E. Ainseba - Mathématiques Appliquées de Bordeaux, UMR CNRS 5466, case 26, U.F.R. Sciences et Modélisation, Université Victor Segalen Bordeaux 2,33076 Bordeaux Cedex, France (email)
Sebastian Aniţa - Faculty of Mathematics, University “Al.I. Cuza” and, Institute of Mathematics “Octav Mayer”, Iaşi 700506, Romania (email)

Abstract: The internal zero-stabilization of the nonnegative solutions to some parabolic equations is investigated. We provide a necessary and a sufficient condition for nonnegative stabilizability in terms of the sign of the principal eigenvalue of a certain elliptic operator. This principal eigenvalue is related to the rate of the convergence of the solution. We give some evaluations of this principal eigenvalue with respect to the geometry of the domain and of the support of the control. A stabilization result for an age-dependent population dynamics with diffusion is also established.

Keywords:  Nonnegative stabilization, parabolic equations, Rayleigh's principle, comparison principle, age-dependent population dynamics.
Mathematics Subject Classification:  Primary: 35K05, 35P15, 93D15, 92D25; Secondary: 35B37.

Received: February 2007;      Revised: August 2007;      Published: February 2008.