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<p>Existence of the random attractors for a nonlocal weak damping wave equation driven by nonlinear colored noise is investigated in a bounded domain, where the nonlinear terms <i>f</i>(<i>u</i>) and <i>h</i>(<i>t</i>,&#xa0;<i>x</i>,&#xa0;<i>u</i>) in the equation are critical growth. First, the global well-posedness of solutions is established using the theory of monotone operators. Second, the existence of a random absorbing set is proved via energy estimates. Meanwhile, we extend the method of contraction functions verifying the pullback asymptotic compactness of non-autonomous hyperbolic systems from the deterministic case to the random case. With the aid of above theoretical findings, we further obtain the pullback asymptotic compactness of the random dynamical system associated with the problem. Ultimately, existence of the random attractors is shown. It’s worth mentioning that the abstract conclusions of [<CitationRef CitationID="CR39">39</CitationRef>] are extended from the deterministic systems to the random ones. Moreover, we employ the weaker nonlinearity conditions in this paper than in [<CitationRef CitationID="CR19">19</CitationRef>]. In order to deal with the critical growth of nonlinear function and nonlinear colored noise, we seek out a useful Bihari-type integral inequality introduced in [<CitationRef CitationID="CR16">16</CitationRef>], which helps us overcome the difficulty caused by the critical growth of two nonlinear terms.</p>

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Asymptotic Behavior of Wave Equations with Critical Nonlinearity, Nonlocal Weak Damping and Nonlinear Colored Noise

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Wenjuan Hao,
  • Qiaozhen Ma

摘要

Existence of the random attractors for a nonlocal weak damping wave equation driven by nonlinear colored noise is investigated in a bounded domain, where the nonlinear terms f(u) and h(t, x, u) in the equation are critical growth. First, the global well-posedness of solutions is established using the theory of monotone operators. Second, the existence of a random absorbing set is proved via energy estimates. Meanwhile, we extend the method of contraction functions verifying the pullback asymptotic compactness of non-autonomous hyperbolic systems from the deterministic case to the random case. With the aid of above theoretical findings, we further obtain the pullback asymptotic compactness of the random dynamical system associated with the problem. Ultimately, existence of the random attractors is shown. It’s worth mentioning that the abstract conclusions of [39] are extended from the deterministic systems to the random ones. Moreover, we employ the weaker nonlinearity conditions in this paper than in [19]. In order to deal with the critical growth of nonlinear function and nonlinear colored noise, we seek out a useful Bihari-type integral inequality introduced in [16], which helps us overcome the difficulty caused by the critical growth of two nonlinear terms.