Global Asymptotical Behavior of Solutions to a Mixed Pseudo-Parabolic \(r\left( x\right) \)-Laplacian Equation

This paper concerns the asymptotic behavior of global solutions to the initial-boundary value problem of a mixed pseudo-parabolic \(r\left( x\right) \) $r\left(x\right)$ -Laplacian equation. By employing the potential well method and Lagrange multiplier method, we prove the existence of ground state solutions for the stationary problem. Then by applying the Sobolev imbedding inequality and the H \(\ddot{\textrm{o}}\) $\ddot{\text{o}}$ lder inequality, we establish a convergence relationship between the global solutions of the evolution problem and the ground state solutions of the corresponding stationary problem.