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On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

Part of: Parabolic equations and systems Partial differential equations Infinite-dimensional dissipative dynamical systems

Published online by Cambridge University Press:  14 March 2019

Piotr Kalita  and
Piotr Zgliczyński
Piotr Kalita
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348Kraków, Poland (piotr.kalita@ii.uj.edu.pl; piotr.zgliczynski@ii.uj.edu.pl)
Piotr Zgliczyński
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348Kraków, Poland (piotr.kalita@ii.uj.edu.pl; piotr.zgliczynski@ii.uj.edu.pl)
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Abstract

We study the non-autonomously forced Burgers equation

$$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$
on the space interval (0, 1) with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique H1 bounded trajectory of this equation defined for all t ∈ ℝ. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.

Keywords

Burgers equationnon-autonomous forcingasymptotic behaviourglobal stabilityeternal solutionenergy methodmaximum principlepullback attractor

MSC classification

Primary: 37L05: General theory, nonlinear semigroups, evolution equations
Secondary: 35B35: Stability 35B41: Attractors 35K55: Nonlinear parabolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 2025 - 2054
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Copyright
Copyright © Royal Society of Edinburgh 2019

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