Uniqueness for an elliptic-parabolic problem with Neumann boundary condition

Abstract

We consider the problem \(b(u) - \Delta u + div F(u) = f\) in a smooth boundary domain \(\Omega \subset \mathbb{R}^N\), as well as the corresponding evolution equation \(b(u)_t - \Delta u + div F(u) = f, b(u(0, .)) = b^0\). For the stationary equation we show existence results, then we adapt the techniques of doubling of variables to the case of the homogeneous Neumann boundary conditions and obtain the appropriate L 1 -contraction principle and uniqueness. Subsequently, we are able to apply the nonlinear semigroup theory and prove the L 1 -contraction principle for the associated evolution equation.

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Correspondence to Boris P. Andreianov or Fouzia Bouhsiss.

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Andreianov, B.P., Bouhsiss, F. Uniqueness for an elliptic-parabolic problem with Neumann boundary condition. J.evol.equ. 4, 273–295 (2004). https://doi.org/10.1007/s00028-004-0143-1

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Mathematics Subject Classification (2000):

  • 35J65
  • 35K60
  • 35K65
  • 47H06
  • 47H20

Key words:

  • Degenerate parabolic equations
  • Neumann boundary condition
  • doubling of variables
  • nonlinear semigroup theory