Large time dynamics for non-autonomous weakly damped KdV-Burgers equations on the real line
Abstract
The large time dynamics of the weakly damped non-autonomous Korteweg-de Vries-Burgers equation ut + uxxx+ uux + λu − μuxx = g(x,t) in the phase space L2(ℝ) is investigated. Due to the low regularity of the initial data, the framework of Bourgain spaces is employed. We first obtain the existence of a unique pullback attractor {𝒜μ(t)}t∈ℝ in L2(ℝ) for each μ≥ 0. Then we show an optimal asymptotic smoothing effect of the equation, namely the pullback attractor is compact in H3(ℝ). These results are proved by using an energy equation method, splitting solutions into parts with low and high frequencies and exploiting a dispersive regularization. Furthermore, the attractors are shown to be upper semicontinuous as μ→ 0 and have finite fractal dimensions in H1(ℝ).
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BibTeX
@misc{anh2026large,
title = {{Large time dynamics for non-autonomous weakly damped KdV-Burgers equations on the real line}},
author = {Cung The Anh and Nesibe Ayhan and Bao Quoc Tang and Ingo Witt},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03905},
eprint = {2611.03905},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03905},
note = {CC0},
}
RIS
TY - UNPB
TI - Large time dynamics for non-autonomous weakly damped KdV-Burgers equations on the real line
AU - Cung The Anh
AU - Nesibe Ayhan
AU - Bao Quoc Tang
AU - Ingo Witt
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03905
UR - https://arxiv.org/abs/2611.03905
AB - The large time dynamics of the weakly damped non-autonomous Korteweg-de Vries-Burgers equation u_t + u_xxx+ uu_x + λu − μu_xx = g(x,t) in the phase space L^2(ℝ) is investigated. Due to the low regularity of the initial data, the framework of Bourgain spaces is employed. We first obtain the existence of a unique pullback attractor {𝒜_μ(t)}_t∈ℝ in L^2(ℝ) for each μ≥ 0. Then we show an optimal asymptotic smoothing effect of the equation, namely the pullback attractor is compact in H^3(ℝ). These results are proved by using an energy equation method, splitting solutions into parts with low and high frequencies and exploiting a dispersive regularization. Furthermore, the attractors are shown to be upper semicontinuous as μ→ 0 and have finite fractal dimensions in H^1(ℝ).
ER -