Final dynamics of systems of nonlinear parabolic equations on the circle

Aleksandr V. Romanov; Aleksandr V. Romanov

doi:10.3934/math.2021776

AIMS Mathematics

2021, Volume 6, Issue 12: 13407-13422. doi: 10.3934/math.2021776

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Research article

Final dynamics of systems of nonlinear parabolic equations on the circle

Aleksandr V. Romanov ^,

School of Applied Mathematics, National Research University Higher School of Economics, 34 Tallinskaya St., Moscow 123458, Russia

Received: 12 June 2021 Accepted: 18 August 2021 Published: 17 September 2021
MSC : 35B41, 35K57, 35K40, 35K90, 35K91

We consider the class of dissipative reaction-diffusion-convection systems on the circle and obtain conditions under which the final (at large times) phase dynamics of a system can be described by an ODE with Lipschitz vector field in $ \mathbb{R}^{N} $. Precisely in this class, the first example of a parabolic problem of mathematical physics without the indicated property was recently constructed.
- reaction-diffusion-convection equations,
- finite-dimensional dynamics on attractor,
- inertial manifold
Citation: Aleksandr V. Romanov. Final dynamics of systems of nonlinear parabolic equations on the circle[J]. AIMS Mathematics, 2021, 6(12): 13407-13422. doi: 10.3934/math.2021776

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Abstract

We consider the class of dissipative reaction-diffusion-convection systems on the circle and obtain conditions under which the final (at large times) phase dynamics of a system can be described by an ODE with Lipschitz vector field in $ \mathbb{R}^{N} $. Precisely in this class, the first example of a parabolic problem of mathematical physics without the indicated property was recently constructed.

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