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

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.

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Correspondence to Pavel I. Naumkin.

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Naumkin, P.I., Rojas-Milla, C.J. Asymptotics of solutions to the periodic problem for a Burgers type equation. J. Evol. Equ. 11, 107–119 (2011). https://doi.org/10.1007/s00028-010-0085-8

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Keywords and phrases

  • Asymptotics of solutions
  • Periodic problem
  • Burgers type equation