Well-posedness and global attractor for wave equation with displacement dependent damping and super-cubic nonlinearity

This work investigates the semilinear wave equation featuring the displacement dependent term σ(u)∂ₜ u and nonlinearity $f(u)$. By developing refined space-time a priori estimates under extended ranges of the nonlinearity exponents with σ(u) and $f(u)$, the well-posedness of the weak solution is established. Furthermore, the existence of a global attractor in the naturally phase space H¹₀(Ω)× L²(Ω) is obtained. Moreover, the regularity of the global attractor is established, implying that it is a bounded subset of (H²(Ω)∩ H¹₀(Ω))× H¹₀(Ω).