Well-Posedness of Pseudo-Parabolic Gradient Systems with State-Dependent Dynamics

TL;DR

This paper introduces a comprehensive mathematical framework for pseudo-parabolic gradient systems with state-dependent dynamics, proving existence, uniqueness, and continuous dependence of solutions, applicable to models in materials science and image processing.

Contribution

It develops a general well-posedness theory for nonlinear pseudo-parabolic systems with state-dependent operators, including new models in grain boundary motion and image denoising.

Findings

Proved existence of energy-dissipating solutions.

Established uniqueness and continuous dependence on initial data.

Applied framework to models in materials science and image processing.

Abstract

This paper develops a general mathematical framework for pseudo-parabolic gradient systems with state-dependent dynamics. The state dependence is induced by variable coefficient fields in the governing energy functional. Such coefficients arise naturally in scientific and technological models, including state-dependent mobilities in KWC-type grain boundary motion and variable orientation-adaptation operators in anisotropic image denoising. We establish two main results: the existence of energy-dissipating solutions, and the uniqueness and continuous dependence on initial data. The proposed framework yields a general well-posedness theory for a broad class of nonlinear evolutionary systems driven by state-dependent operators. As illustrative applications, we present an anisotropic image-denoising model and a new pseudo-parabolic KWC-type model for anisotropic grain boundary motion, and prove that both fit naturally within the abstract structure of $(\mathrm{S})_\nu$.