Longtime behavior of solutions for the fourth-order degenerate hyperbolic equations involving subelliptic operators in RN${\mathbb{R}}^{N}$

Abstract In this paper, we study the long-term behavior of a fourth-order hyperbolic equation involving a subelliptic operator on the whole space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{N}$ . We first establish the well-posedness of the problem and then define the time-dependent evolution process associated with the problem. Next, by introducing a new family of cut-off functions together with several novel analytical techniques, we derive uniform tail-estimates of the solution, which play a key role in proving the asymptotic compactness of the process. Furthermore, by applying the theory of processes in time-dependent spaces, we prove the existence of a time-dependent global attractor for our problem in strong topological spaces. Several essential difficulties arise from the unboundedness of the domain and the strong degeneracy of the subelliptic operator, which must be carefully overcome in our analysis. The results obtained in this paper are significant and very important for further studies of the long-time behavior of solutions to fourth-order hyperbolic equations, and contribute to a deeper understanding of the subelliptic operator.