Asymptotic behavior of the solution to the non-isothermal phase separation

Moreover, the ω-limit set associated with the trajectory of the unique global solution is non-empty, connected, and compact in some suitable space; as well as being composed of solutions to the steady state problem. For the stability of stationary solutions, we show that the dynamically stable solutions to the steady state problem are characterized by linearized stable solutions to the elliptic problem with a non-local term, which is equivalent to our steady state problem.