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Asymptotics of solutions to the periodic problem for a Burgers type equation

Asymptotics of solutions to the periodic problem for a Burgers type equation We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear Burgers type equation $$ \left\{ \begin{array}{l} \psi_{t}=\psi_{xx}+\lambda \psi +\psi \psi_{x},\quad x\in \Omega, \quad t >0 , \\ \psi (0,x)=\widetilde{\psi}(x), \quad x\in \Omega, \end{array} \right. $$ where Ω = (− π , π ), λ < 1. We prove that if the initial data $${\widetilde{\psi}\in {\bf L}^{2}(\Omega)}$$ , then there exists a unique solution $${\psi (t,x) \in {\bf C}\left( ( 0,\infty ) ;{\bf L}^{2}(\Omega) \right) \cap {\bf C}^{\infty }\left( ( 0,\infty ) \times {\bf R}\right)}$$ of the periodic problem. Moreover, under some additional conditions we find the asymptotic expansion for the solutions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Asymptotics of solutions to the periodic problem for a Burgers type equation

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References (30)

Publisher
Springer Journals
Copyright
Copyright © 2011 by Springer Basel AG
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-010-0085-8
Publisher site
See Article on Publisher Site

Abstract

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear Burgers type equation $$ \left\{ \begin{array}{l} \psi_{t}=\psi_{xx}+\lambda \psi +\psi \psi_{x},\quad x\in \Omega, \quad t >0 , \\ \psi (0,x)=\widetilde{\psi}(x), \quad x\in \Omega, \end{array} \right. $$ where Ω = (− π , π ), λ < 1. We prove that if the initial data $${\widetilde{\psi}\in {\bf L}^{2}(\Omega)}$$ , then there exists a unique solution $${\psi (t,x) \in {\bf C}\left( ( 0,\infty ) ;{\bf L}^{2}(\Omega) \right) \cap {\bf C}^{\infty }\left( ( 0,\infty ) \times {\bf R}\right)}$$ of the periodic problem. Moreover, under some additional conditions we find the asymptotic expansion for the solutions.

Journal

Journal of Evolution Equations – Springer Journals

Published: Mar 1, 2011

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