On the speed rate of convergence of solutions to conservation laws with nonlinear diffusions. (July 2020)

• Record Type: Journal Article
• Authors: Folino, Raffaele; Strani, Marta
• Is Part Of: Nonlinear analysis. Volume 196(2020)
• Publication Date: 2020-07
• Subjects: 35K20 -- 35B35 -- 35B36 -- 35B40 -- 35P15; Saturating diffusion -- Minkowski curvature operator -- Steady states -- Stability -- Metastability; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111762
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we analyze the long-time behavior of solutions to conservation laws with nonlinear diffusion terms of different types: saturating diffusions (monotone and non monotone) and singular nonlinear diffusions are considered. In particular, the cases of mean curvature-type diffusions both in the Euclidean space and in Lorentz–Minkowski space enter in our framework. After dealing with existence and stability of monotone steady states in a bounded interval of the real line with Dirichlet boundary conditions, we discuss the speed rate of convergence to the asymptotic limit as t → + ∞ . Finally, in the particular case of a Burgers flux function, we show that the solutions exhibit the phenomenon of metastability.