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Null controllability of degenerate/singular parabolic equations

Abstract

The purpose of this paper is to provide a full analysis of the null controllability problem for the one dimensional degenerate/singular parabolic equation \( {u_t} - {\left( {a(x){u_x}} \right)_x} - \frac{\lambda }{{{x^\beta }}}u = 0 \), (t, x) ∈ (0, T) × (0, 1), where the diffusion coefficient a(∙) is degenerate at x = 0. Also the boundary conditions are considered to be Dirichlet or Neumann type related to the degeneracy rate of a(∙). Under some conditions on the function a(∙) and parameters β, λ, we prove global Carleman estimates. The proof is based on an improved Hardy-type inequality.

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Correspondence to M. Fotouhi.

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Fotouhi, M., Salimi, L. Null controllability of degenerate/singular parabolic equations. J Dyn Control Syst 18, 573–602 (2012). https://doi.org/10.1007/s10883-012-9160-5

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  • DOI: https://doi.org/10.1007/s10883-012-9160-5

Key words and phrases

  • Degenerate parabolic equations
  • singular potential
  • null controllability
  • Carleman estimates
  • improved Hardy inequality

2000 Mathematics Subject Classification

  • 35K65
  • 93B05
  • 93B07