Random attractors for the stochastic coupled suspension bridge equations of Kirchhoff type

Abstract

Abstract This paper is devoted to the dynamical behavior of stochastic coupled suspension bridge equations of Kirchhoff type. For the deterministic cases, there are many classical results such as existence and uniqueness of a solution and long-term behavior of solutions. To the best of our knowledge, the existence of random attractors for the stochastic coupled suspension bridge equations of Kirchhoff type is not yet considered. We intend to investigate these problems. We first obtain the dissipativeness of a solution in higher-energy spaces H3(U)×H01(U)×(H2(U)∩H01(U))×H01(U) $H^{3}(U)\times H_{0}^{1}(U)\times (H^{2}(U)\cap H_{0}^{1}(U))\times H_{0}^{1}(U)$. This implies that the random dynamical system generated by the equation has a random attractor in (H2(U)∩H01(U))×L2(U)×H01(U)×L2(U) $(H^{2}(U)\cap H_{0}^{1}(U))\times L^{2}(U) \times H_{0}^{1}(U)\times L^{2}(U)$, which is a tempered random set in the space in H3(U)×H01(U)×(H2(U)∩H01(U))×H01(U) $H^{3}(U)\times H_{0}^{1}(U)\times (H^{2}(U)\cap H_{0} ^{1}(U))\times H_{0}^{1}(U)$.

Keywords

• Coupled suspension bridge equations
• Random dynamical system
• Random attractors
• Kirchhoff-type