This research demonstrates global well-posedness and the existence of attractors in wave equations, indicating implications for solution behavior over time.
In this paper, we study the long-time dynamics for the autonomous wave equation with nonlocal weak damping and super-cubic nonlinearity in a bounded smooth domain of R3. Based on the Strichartz estimates for the case of bounded domains, we first prove the global well-posedness of the Shatah–Struwe solutions. Then we establish the existence of the global attractor for the Shatah–Struwe solution semigroup by the method of contractive function. Finally, we verify the existence of a polynomial attractor for this semigroup.
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Yan et al. (2025) studied this question.