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


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. 1.

    Al-Gwaiz, M.A.; Sturm-Liouville Theory and its Applications, Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 2008.

    MATH  Google Scholar 

  2. 2.

    Alikakos, N.; Bates, P.W.; Fusco, G.; Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differential Equations 90 (1991), 81–135.

    MathSciNet  Article  Google Scholar 

  3. 3.

    Berestycki, H.; Kamin S.; Sivashinsky G.; Metastability in a flame front evolution equation, Interfaces Free Bound. 3 (2001), 361–392.

    MathSciNet  Article  Google Scholar 

  4. 4.

    Carr, J.; Pego, R. L.; Metastable patterns in solutions of \(u_t=\varepsilon ^2 u_{xx}+f(u)\), Comm. Pure Appl. Math. 42 (1989), 523–576.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chertock, A.; Kurganov, A.; Rosenau, P.; On degenerate saturated-diffusion equations with convection, Nonlinearity 18 (2005), 609–630.

    MathSciNet  Article  Google Scholar 

  6. 6.

    Folino, R.; Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension, J. Hyperbolic Differ. Equ. 14 (2017), 1–26.

    MathSciNet  Article  Google Scholar 

  7. 7.

    Folino, R.; Lattanzio, C.; Mascia, C.; Metastable dynamics for hyperbolic variations of the Allen-Cahn equation, Commun. Math. Sci. 15 (2017), 2055–2085.

    MathSciNet  Article  Google Scholar 

  8. 8.

    Folino, R.; Lattanzio, C.; Mascia, C.; Slow dynamics for the hyperbolic Cahn-Hilliard equation in one-space dimension, Math. Meth. Appl. Sci. 42 (2019), 2492–2512.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Folino, R.; Lattanzio, C.; Mascia, C.; Strani, M.; Metastability for nonlinear convection-diffusion equations, NODEA Nonlinear Differ. Equ. Appl. (2017), 24–35.

  10. 10.

    Fusco, G.; Hale, J. K.; Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations 1 (1989), 75–94.

    MathSciNet  Article  Google Scholar 

  11. 11.

    Garrione, M.; Sanchez, L.; Monotone traveling waves for reaction-diffusion equations involving the curvature operator, Bound. Value Probl. 45 (2015), 1–31.

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Garrione, M.; Strani, M.; Heteroclinic traveling fronts for a generalized Fisher-Burgers equation with saturating diffusion, Indiana Univ. Math. J., to appear. arXiv:1702.03782.

  13. 13.

    Goodman, J.; Kurganov, A.; Rosenau, P.; Breakdown in Burgers-type equations with saturating dissipation fluxes, Nonlinearity 12 (1999), 247–268.

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kurganov, A.; Levy, D.; Rosenau, P.; On Burgers-type equations with nonmonotonic dissipative fluxes, Comm. Pure Appl. Math. 51 (1998), 443–473.

    MathSciNet  Article  Google Scholar 

  15. 15.

    Kurganov, A.; Rosenau, P.; Effects of a saturating dissipation in Burgers-type equations, Comm. Pure Appl. Math. 50 (1997), 753–771.

    MathSciNet  Article  Google Scholar 

  16. 16.

    Laforgue, J. G. L.; O’Malley, R. E. Jr.; On the motion of viscous shocks and the supersensitivity of their steady-state limits, Methods Appl. Anal. 1 (1994), 465–487.

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Laforgue, J. G. L.; O’Malley, R. E. Jr.; Shock layer movement for Burgers’ equation, Perturbations methods in physical mathematics (Troy, NY, 1993). SIAM J. Appl. Math. 55 (1995), 332–347.

  18. 18.

    Lax, P. D.; Weak solutions of nonlinear hyperbolic equations and their numerical computations, Comm. Pure Appl. Math. 7 (1954), 159–193.

    MathSciNet  Article  Google Scholar 

  19. 19.

    Lieberman, G. M.; Second Order Parabolic Differential Equations, World Scientific Publishing, 1996.

  20. 20.

    Mascia, C.; Strani, M.; Metastability for nonlinear parabolic equations with application to scalar conservation laws, SIAM J. Math. Anal. 45 (2013), 3084–3113.

    MathSciNet  Article  Google Scholar 

  21. 21.

    Nessyahu, H.; Convergence rate of approximate solutions to weakly coupled nonlinear system, Math. Comput. 65 (1996), 575–586.

    MathSciNet  Article  Google Scholar 

  22. 22.

    Otto, F.; Reznikoff, M. G.; Slow motion of gradient flows, J. Differential Equations 237 (2006), 372–420.

    MathSciNet  Article  Google Scholar 

  23. 23.

    Pazy, A.; Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983.

  24. 24.

    Pego, R. L.; Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A 422 (1989), 261–278.

    MathSciNet  Article  Google Scholar 

  25. 25.

    Reyna, L. G.; Ward, M. J.; On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math. 48 (1995), 79–120.

    MathSciNet  Article  Google Scholar 

  26. 26.

    Rosenau, P.; Free-energy functionals at the high-gradient limit, Phys. Rev. A 41 (1990), 2227–2230.

    Article  Google Scholar 

  27. 27.

    Strani, M.; On the metastable behavior of solutions to a class of parabolic systems, Asymptot. Anal. 90 (2014), 325–344.

    MathSciNet  Article  Google Scholar 

  28. 28.

    Strani, M.; Metastable dynamics of internal interfaces for a convection-reaction-diffusion equation, Nonlinearity 28 (2015), 4331–4368.

    MathSciNet  Article  Google Scholar 

  29. 29.

    Strani, M.; Slow dynamics in reaction-diffusion systems, Asymptot. Anal. 98 (2016), 131–154.

    MathSciNet  Article  Google Scholar 

  30. 30.

    Strani, M.; Fast-slow dynamics in parabolic-hyperbolic systems, Adv. Nonlinear Anal. 7 (2018), 117–138.

    MathSciNet  Article  Google Scholar 

  31. 31.

    Sun, X.; Ward, M. J.; Metastability for a generalized Burgers equation with application to propagating flame fronts, European J. Appl. Math. 10 (1999), 27–53.

    MathSciNet  Article  Google Scholar 

  32. 32.

    Zhang, L.; Curvature flow with driving force on fixed boundary points, J. Geom. Anal. 28 (2018), 3491–3521.

    MathSciNet  Article  Google Scholar 

Download references


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

Author information



Corresponding author

Correspondence to Marta Strani.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation


  • Mean curvature operator
  • Steady states
  • Stability
  • Metastability

Mathematics Subject Classification

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