Riv. Mat. Univ. Parma, Vol. 9, No. 1, 2018

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.

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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.

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
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
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
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
H. Berestycki, S. Kamin and G. Sivashinsky, Metastability in a flame front evolution equation, Interfaces Free Bound. 3 (2001), 361-392. MR1869585
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
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
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
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
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.
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
G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations 1 (1989), 75-94. MR1010961
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
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
G. Kreiss and H.-O. Kreiss, Convergence to steady state of solutions of Burgers' equation, Appl. Numer. Math. 2 (1986), 161-179. MR0863984
G. Kreiss, H.-O. Kreiss and J. Lorenz, Stability of viscous shocks on finite intervals, Arch. Ration. Mech. Anal. 187 (2008), 157-183. MR2358338
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
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
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
G. M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., River Edge, NJ, 1996. MR1465184
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
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
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
F. Otto and M. G. Reznikoff, Slow motion of gradient flows, J. Differential Equations 237 (2007), 372-420. MR2330952
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Science, 44, Springer-Verlag, New York, 1983. MR0710486
R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A 422 (1989), 261-278. MR0997638
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
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
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
M. Strani, On the metastable behavior of solutions to a class of parabolic systems, Asymptot. Anal. 90 (2014), 325-344. MR3323890
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
M. Strani, Slow dynamics in reaction-diffusion systems, Asymptot. Anal. 98 (2016), 131-154. MR3502375
M. Strani, Metastable dynamics of internal interfaces for a convection-reaction-diffusion equation, Nonlinearity 28 (2015), 4331-4368. MR3461580
M. Strani, Semigroup estimates and fast-slow dynamics in parabolic-hyperbolic systems, Adv. Nonlinear Anal. 7 (2018), 117-138. MR3757459
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

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