This analysis establishes the existence of a global attractor in Balakrishnan–Taylor equations, suggesting stability. The approach leverages properties of gradient systems and nonlinear damping.
The main objective of this paper is to consider the long-time behavior of solutions for a Balakrishnan-Taylor extensible beam equations with nonlinear damping and critical nonlinearity. Due to the shortage of the compactness of Sobolev embedding theorem, the asymptotic compactness of the corresponding semigroup generated by problem (1.1) can not be obtained by the method of contraction function in the case that the exponent of the source term is critical. In this paper, we first prove the corresponding dynamical system is a gradient system. By the abstract results about the existence of global attractor, we will establish a quasi-stability inequality by using the global Lr-regularity (r ≥ 2) in time of the nonlocal Balakrishnan-Taylor term under some suitable assumptions on the nonlinear damping term, which is sufficient to show the existence of a finite dimensional global attractor for such system.
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Fang et al. (2025) studied this question.
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