Asymptotic behavior and ground-state solutions for a semilinear pseudo-parabolic equation with a nonlocal source term

Asymptotic behavior and ground-state solutions for a semilinear pseudo-parabolic equation with a nonlocal source term are investigated under the null Dirichlet boundary condition. By using the Faedo–Galerkin approximation and contraction mapping principle, we give the local well-posedness. Furthermore, based on the variational structure of the equation, we obtain decay estimates for the global solution and its energy functionals. Meantime, we establish the convergent relationship between the global solutions and the ground-state solutions of the corresponding stationary problem.