Details

Linda Maria De Cave^[a] and Marta Strani^[b]

Asymptotic behavior of interface solutions to semilinear parabolic equations with nonlinear forcing terms

Pages: 85-131
Received: 12 March 2018
Accepted in revised form: 7 August 2018
Mathematics Subject Classification (2010): 35B25, 35B36, 35B40, 35K45.
Keywords: Metastability, slow motion, internal interfaces, asymptotic dynamics, semilinear diffusion.
Authors address:
[a]: Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
[b]: Università Ca' Foscari, Dipartimento di Scienze Molecolari e Nanosistemi, Via Torino 155, 30172, Venezia Mestre, Italy.

Abstract: We investigate the asymptotic behavior of solutions to semilinear parabolic equations in bounded intervals. In particular, we are concerned with a special class of solutions, called interface solutions, which exhibit a metastable behavior, meaning that their convergence towards the asymptotic configuration of the system is exponentially slow. The key of our analysis is a linearization around an approximation of the steady state of the problem, and the reduction of the dynamics to a one-dimensional motion, describing the slow convergence of the interfaces towards the equilibrium.

References

[1]: N. Alikakos, P. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differential Equations 90 (1991), 81-135. MR1094451
[2]: N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima, Indiana Univ. Math. J. 57 (2008), 1871-1906. MR2440884
[3]: C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), 1017-1034. MR0542510
[4]: M. Beck and C. E. Wayne, Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity, SIAM J. Appl. Dyn. Syst. 8 (2009), 1043-1065. MR2551255
[5]: H. Berestycki, S. Kamin and G. Sivashinsky, Metastability in a flame front evolution equation, Interfaces Free Bound. 3 (2001), 361-392. MR1869585
[6]: F. Bethuel, G. Orlandi and D. Smets, Slow motion for gradient systems with equal depth multiple-well potentials, J. Differential Equations 250 (2011), 53-94. MR2737835
[7]: J. Carr and R. L. Pego, Metastable patterns in solutions of \(u_t=\varepsilon^2 u_{xx} - f(u)\), Comm. Pure Appl. Math. 42 (1989), 523-576. MR0997567
[8]: S. Dipierro, G. Palatucci and E. Valdinoci, Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting, Comm. Math. Phys. 333 (2015), 1061-1105. MR3296170
[9]: P. P. N. de Groen and G. E. Karadzhov, Exponentially slow traveling waves on a finite interval for Burgers' type equation, Electron. J. Differential Equations 1998 (1998), No. 30, 38 pp. MR1657187
[10]: L. De Luca, M. Goldman and M. Strani, A gradient flow approach to relaxation rates for the multi-dimensional Cahn-Hilliard equation, arXiv:1802.08082, preprint, 2018.
[11]: R. Folino, C. Lattanzio, C. Mascia and M. Strani, Metastability for nonlinear convection-diffusion equations, NoDEA Nonlinear Differential Equations Appl. 24 (2017), Art. 35, 20 pp. MR3662481
[12]: G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations 1 (1989), 75-94. MR1010961
[13]: J. N. Zhao, Existence and nonexistence of solutions for \(u_t= {\rm div} \left( |\nabla u|^{p-2} \nabla u\right) + f (\nabla u, u , x, t)\), J. Math. Anal. Appl. 172 (1993), 130-146. MR1199500
[14]: Y. J. Kim and A. E. Tzavaras, Diffusive \(N\)-waves and metastability in the Burgers equation, SIAM J. Math. Anal. 33 (2001), 607-633. MR1871412
[15]: G. Kreiss and H.-O. Kreiss, Convergence to steady state of solutions of Burgers' equation, Appl. Numer. Math. 2 (1986), 161-179. MR0863984
[16]: G. Kreiss, H.-O. Kreiss and J. Lorenz, Stability of viscous shocks on finite intervals, Arch. Ration. Mech. Anal. 187 (2008), 157-183. MR2358338
[17]: S. N. Kruzkov, First order quasilinear equations in several independent variables (Russian), Mat. Sb. (N.S.) 81 (123) (1970), 228-255. English translation in: Math. USSR-Sb. 10 (1970), 217-243. MR0267257 | DOI
[18]: O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and quasi-linear equations of parabolic type (Russian), Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, 1968. MR0241822
[19]: J. G. L. Laforgue and R. E. O'Malley, Shock layer movement for Burgers' equation, Perturbations methods in physical mathematics (Troy, NY, 1993), SIAM J. Appl. Math. 55 (1995), 332-347. MR1322763
[20]: G. M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., River Edge, NJ, 1996. MR1465184
[21]: C. Mascia and A. Terracina, Large-time behavior for conservation laws with source in a bounded domain, J. Differential Equations 159 (1999), 485-514. MR1730729
[22]: C. Mascia and M. Strani, Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws, SIAM J. Math. Anal. 45 (2013), 3084-3113. MR3115459
[23]: G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl. (4) 192 (2013), 673-718. MR3081641
[24]: F. Otto and M. G. Reznikoff, Slow motion of gradient flows, J. Differential Equations 237 (2007), 372-420. MR2330952
[25]: A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Science, 44, Springer-Verlag, New York, 1983. MR0710486
[26]: R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A 422 (1989), 261-278. MR0997638
[27]: L. G. Reyna and M. J. Ward, On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math. 48 (1995), 79-120. MR1319697
[28]: E. Risler, Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 381-424. MR2400108
[29]: P. Sternberg Vector-valued local minimizers of nonconvex variational problems, Current directions in nonlinear partial differential equations, (Provo, UT, 1987), Rocky Mountain J. Math. 21 (1991), 799-807. MR1121542
[30]: M. Strani, On the metastable behavior of solutions to a class of parabolic systems, Asymptot. Anal. 90 (2014), 325-344. MR3323890
[31]: M. Strani, Slow motion of internal shock layers for the Jin-Xin system in one space dymension, J. Dynam. Differential Equations 27 (2015), 1-27. MR3317389
[32]: M. Strani, Slow dynamics in reaction-diffusion systems, Asymptot. Anal. 98 (2016), 131-154. MR3502375
[33]: M. Strani, Metastable dynamics of internal interfaces for a convection-reaction-diffusion equation, Nonlinearity 28 (2015), 4331-4368. MR3461580
[34]: M. Strani, Semigroup estimates and fast-slow dynamics in parabolic-hyperbolic systems, Adv. Nonlinear Anal. 7 (2018), 117-138. MR3757459
[35]: X. Sun and M. J. Ward, Metastability for a generalized Burgers equation with applications to propagating flame fronts, European J. Appl. Math. 10 (1999), 27-53. MR1685819