Nonlinear Diﬀer. Equ. Appl.
2017 Springer International Publishing AG
Nonlinear Diﬀerential Equations
and Applications NoDEA
Metastability for nonlinear
Special issue in honor of Alberto Bressan.
Raﬀaele Folino, Corrado Lattanzio, Corrado Mascia and
Abstract. We study the metastable dynamics of solutions to nonlinear
evolutive equations of parabolic type, with a particular attention to the
case of the viscous scalar Burgers equation with small viscosity ε.Inorder
to describe rigorously such slow motion, we adapt the strategy ﬁrstly
proposed in Mascia and Strani (SIAM J Math Anal 45:3084–3113, 2013)
by linearizing the original equation around a metastable state and by
studying the system obtained for the couple (ξ, v), where ξ is the position
of the internal shock layer and v is a perturbative term. The main result
of this paper provides estimates for the speed of the shock layer and for
the error v; in particular, in the case of the viscous Burgers equation, we
prove they are exponentially small in ε. As a consequence, the time taken
for the solution to reach the unique stable steady state is exponentially
large, and we have exponentially slow motion.
Mathematics Subject Classiﬁcation. Primary 35K20; Secondary 35B25,
Keywords. Metastability, Slow motion, Viscous conservation laws.
In the analysis of PDEs, metastability is a broad term describing the per-
sistence of unsteady structures for a very long time. We refer to metastable
dynamics when, in a ﬁrst stage, the evolution of a (non-stationary) solution is
This work was partially supported by the Italian Project FIRB 2012 “Dispersive dynamics:
Fourier Analysis and Variational Methods”.
This article is part of the topical collection “Hyperbolic PDEs, Fluids, Transport and Appli-
cations: Dedicated to Alberto Bressan for his 60th birthday” guest edited by Fabio Ancona,
Stefano Bianchini, Pierangelo Marcati, Andrea Marson.