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

Published online by Cambridge University Press:  14 March 2019

Piotr Kalita
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348Kraków, Poland ([email protected]; [email protected])
Piotr Zgliczyński
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348Kraków, Poland ([email protected]; [email protected])

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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