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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Al-Gwaiz, M.A.; Sturm-Liouville Theory and its Applications, Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 2008.
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.
Berestycki, H.; Kamin S.; Sivashinsky G.; Metastability in a flame front evolution equation, Interfaces Free Bound. 3 (2001), 361–392.
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.
Chertock, A.; Kurganov, A.; Rosenau, P.; On degenerate saturated-diffusion equations with convection, Nonlinearity 18 (2005), 609–630.
Folino, R.; Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension, J. Hyperbolic Differ. Equ. 14 (2017), 1–26.
Folino, R.; Lattanzio, C.; Mascia, C.; Metastable dynamics for hyperbolic variations of the Allen-Cahn equation, Commun. Math. Sci. 15 (2017), 2055–2085.
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.
Folino, R.; Lattanzio, C.; Mascia, C.; Strani, M.; Metastability for nonlinear convection-diffusion equations, NODEA Nonlinear Differ. Equ. Appl. (2017), 24–35.
Fusco, G.; Hale, J. K.; Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations 1 (1989), 75–94.
Garrione, M.; Sanchez, L.; Monotone traveling waves for reaction-diffusion equations involving the curvature operator, Bound. Value Probl. 45 (2015), 1–31.
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.
Goodman, J.; Kurganov, A.; Rosenau, P.; Breakdown in Burgers-type equations with saturating dissipation fluxes, Nonlinearity 12 (1999), 247–268.
Kurganov, A.; Levy, D.; Rosenau, P.; On Burgers-type equations with nonmonotonic dissipative fluxes, Comm. Pure Appl. Math. 51 (1998), 443–473.
Kurganov, A.; Rosenau, P.; Effects of a saturating dissipation in Burgers-type equations, Comm. Pure Appl. Math. 50 (1997), 753–771.
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.
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.
Lax, P. D.; Weak solutions of nonlinear hyperbolic equations and their numerical computations, Comm. Pure Appl. Math. 7 (1954), 159–193.
Lieberman, G. M.; Second Order Parabolic Differential Equations, World Scientific Publishing, 1996.
Mascia, C.; Strani, M.; Metastability for nonlinear parabolic equations with application to scalar conservation laws, SIAM J. Math. Anal. 45 (2013), 3084–3113.
Nessyahu, H.; Convergence rate of approximate solutions to weakly coupled nonlinear system, Math. Comput. 65 (1996), 575–586.
Otto, F.; Reznikoff, M. G.; Slow motion of gradient flows, J. Differential Equations 237 (2006), 372–420.
Pazy, A.; Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983.
Pego, R. L.; Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A 422 (1989), 261–278.
Reyna, L. G.; Ward, M. J.; On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math. 48 (1995), 79–120.
Rosenau, P.; Free-energy functionals at the high-gradient limit, Phys. Rev. A 41 (1990), 2227–2230.
Strani, M.; On the metastable behavior of solutions to a class of parabolic systems, Asymptot. Anal. 90 (2014), 325–344.
Strani, M.; Metastable dynamics of internal interfaces for a convection-reaction-diffusion equation, Nonlinearity 28 (2015), 4331–4368.
Strani, M.; Slow dynamics in reaction-diffusion systems, Asymptot. Anal. 98 (2016), 131–154.
Strani, M.; Fast-slow dynamics in parabolic-hyperbolic systems, Adv. Nonlinear Anal. 7 (2018), 117–138.
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.
Zhang, L.; Curvature flow with driving force on fixed boundary points, J. Geom. Anal. 28 (2018), 3491–3521.
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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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DOI: https://doi.org/10.1007/s00028-019-00528-2