On the Existence and Long-Time Behavior of Solutions to Swelling Models With Weak Logarithmic Damping Mechanism: Optimal Polynomial Decay Rate

In this work, we investigate a nonlinear swelling problem subject to only one weak logarithmic damping of the form (Formula presented.) acting on the fluid equation. We establish the global existence of strong solutions by employing the Galerkin method in conjunction with compactness arguments, thereby providing a rigorous mathematical framework for the well-posedness of the model. Furthermore, by applying the multiplier method, we derive a general decay estimate for the associated energy functional. The obtained decay rate is shown to be faster than any polynomial rate yet slower than the exponential one, which represents the optimal decay behavior achievable under this weak logarithmic dissipation. These results not only shed light on the intricate interplay between swelling dynamics and logarithmic feedback mechanisms but also open new avenues for future investigations of related systems governed by such unconventional damping laws.