| Abstract: |
| In this study, we consider an inhomogeneous PDE with time-dependent damping coefficients, derived as the pseudo-parabolic gradient flow of a total variation energy with perturbation. Such perturbed total variation energies arise in applications including grain boundary motion and image denoising, and their stationary points play a fundamental role in physical and engineering contexts. In our PDE, the Euler--Lagrange equation for the perturbed total variation energy appears as the associated steady-state problem. In contrast, this Euler--Lagrange equation is formulated as a singular diffusion type equation, which admits many discontinuous solutions. Hence, it involves significant analytical difficulties. The aim of this study is to establish a method for asymptotically approaching the singular Euler--Lagrange equation through a smooth framework provided by our PDE. Moreover, with time-dependent damping coefficients, this framework enables control of the rate of time evolution. Based on these, we focus on the relationship between the set of steady-state solutions and the $\omega$-limit set of solutions to our PDE as time tends to $\infty$. In particular, by analyzing the decay rate of the time-dependent damping coefficients, we show that the decay rate essentially determines the relationship between the steady-state solutions and the asymptotic behavior of solutions. |
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