Stability properties and dynamics of solutions to viscous conservation laws with mean curvature operator

Abstract

In this paper we study the long time dynamics of the solutions to an initial-boundary value problem for a scalar conservation law with a saturating nonlinear diffusion. After discussing the existence of a unique stationary solution and its asymptotic stability, we focus our attention on the phenomenon of metastability, whereby the time-dependent solution develops into a layered function in a relatively short time and subsequently approaches a steady state in a very long time interval. Numerical simulations illustrate the results.

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Acknowledgements

We thank the anonymous referee for her/his comments that helped to improve the paper.

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Correspondence to Marta Strani.

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Folino, R., Garrione, M. & Strani, M. Stability properties and dynamics of solutions to viscous conservation laws with mean curvature operator. J. Evol. Equ. 20, 517–551 (2020). https://doi.org/10.1007/s00028-019-00528-2

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Keywords

  • Mean curvature operator
  • Steady states
  • Stability
  • Metastability

Mathematics Subject Classification

  • 35K20
  • 35B36
  • 35B40
  • 35P15