Global Attractors and Well-Posedness for Degenerate Coupled Reaction-Diffusion Systems in Unbounded Domains

This paper investigates the long-time dynamical behavior and well-posedness of a class of degenerate coupled reaction-diffusion systems defined on unbounded spatial domains \(\mathbb{R}^N\). The presence of space-dependent degenerate diffusion coefficients and the non-compactness of Sobolev embeddings on unbounded domains pose significant mathematical challenges to classical dynamical systems theory. By introducing suitable weighted Sobolev spaces and employing a parabolic regularization technique combined with a Galerkin approximation scheme, we first establish the global existence, uniqueness, and continuous dependence of weak solutions, thereby generating a continuous nonlinear solution semigroup. To overcome the lack of spatial compactness on \(\mathbb{R}^N\), we derive uniform energy estimates and establish asymptotic compactness of the semigroup via a tail-cut estimate technique in weighted spaces. Consequently, we prove the existence of a compact global attractor in the phase space. Furthermore, we provide high-order split-step spectral numerical simulations on truncated domains with absorbing boundary conditions to confirm our theoretical predictions regarding asymptotic stabilization and pattern formation under degenerate diffusion.