Long-time behavior of coupled beam models with fractional energy damping

The paper investigates the well-posedness and the long-time dynamical behavior of a class of coupled beam systems with fractional energy damping. We find new critical exponents: $p^{\alpha _{1}}_{1}\equiv \frac{N+4-4(\frac{1}{\alpha _{10}}-1)\alpha _{1}}{(N-4)^{+}}(\text{ with }p^{\alpha _{1}}_{1}>p^{*}=\frac{N+2}{(N-4)^{+}})$ , $q^{\alpha _{1},\alpha _{2}}_{1}\equiv \frac{N+2\alpha _{1}+4-2(\frac{2}{\alpha _{20}}-1)\alpha _{2}}{(N-4)^{+}}(\text{ with }q^{\alpha _{1},\alpha _{2}}_{1}>q^{*}=\frac{N+2}{(N-4)^{+}})$ , $p^{\alpha _{1},\alpha _{2}}_{2}\equiv \frac{N+2\alpha _{2}+4-2(\frac{2}{\alpha _{10}}-1)\alpha _{1}}{(N-4)^{+}}(\text{ with }p^{\alpha _{1},\alpha _{2}}_{2}>p^{*}=\frac{N+2}{(N-4)^{+}})$ , $q^{\alpha _{2}}_{2}\equiv \frac{N+4-4(\frac{1}{\alpha _{20}}-1)\alpha _{2}}{(N-4)^{+}}(\text{ with }q^{\alpha _{2}}_{2} >q^{*}=\frac{N+2}{(N-4)^{+}})$ . These exponents depending on $\alpha _{i}\in [\alpha _{i0},1]$ are constants, where $0<\alpha _{i0}<1$ , $(i=1,2)$ . We demonstrate that when $1\le p_{1}< p^{\alpha _{1}}_{1}$ , $1\le q_{1}< q^{\alpha _{1},\alpha _{2}}_{1}$ , $1\le p_{2}< p^{\alpha _{1},\alpha _{2}}_{2}$ , $1\le q_{2}< q^{\alpha _{2}}_{2}$ : (i) The initial-boundary value problem (IBVP) of the equations admits a unique solution; (ii) the related solution semigroup possesses a family of global attractors. We systematically propose the definition and proof process for the family of global attractors, thereby enriching the theoretical framework for coupled beam models. The conclusions lay a theoretical foundation for future practical applications.