Opuscula Math. 38, no. 1 (2018), 41-80
https://doi.org/10.7494/OpMath.2018.38.1.41

Opuscula Mathematica

Asymptotic profiles for a class of perturbed Burgers equations in one space dimension

F. Dkhil
M. A. Hamza
B. Mannoubi

Abstract. In this article we consider the Burgers equation with some class of perturbations in one space dimension. Using various energy functionals in appropriate weighted Sobolev spaces rewritten in the variables $$\frac{\xi}{\sqrt\tau}$$ and $$\log\tau$$, we prove that the large time behavior of solutions is given by the self-similar solutions of the associated Burgers equation.

Keywords: Burgers equation, self-similar variables, asymptotic behavior, self-similar solutions.

Mathematics Subject Classification: 35B20, 35B40, 35C20, 35K55, 35L05, 35L60.

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• F. Dkhil
• Université Tunis El Manar, Département de Mathématiques, Institut Supérieur d'Informatique, 2 rue Abou Raihan Bayrouni, 2080 Ariana, Tunisia
• Université de Tunis El Manar, Faculté des Sciences de Tunis, UR13ES32 Analyse non linéaire et géometrie, 2092 Tunis, Tunisia
• M. A. Hamza
• Department of Basic Sciences, Deanship of Preparatory and Supporting Studies, Imam Abdulrahman Bin Faisal University, P.O. Box 1982 Dammam, Saudi Arabia
• B. Mannoubi
• Université de Tunis El Manar, Faculté des Sciences de Tunis, UR13ES32 Analyse non linéaire et géometrie, 2092 Tunis, Tunisia
• Communicated by Vicentiu D. Radulescu.
• Accepted: 2017-03-18.
• Published online: 2017-11-13.