A class of thin-film equations with space-time dependent gradient nonlinearity and its application to image sharpening

TL;DR

We study a class of thin-film type PDEs with a space-time dependent gradient nonlinearity and a mixed local-nonlocal biharmonic operator $\mathcal{L}_{\alpha}$, formulated as $u_t + \mathcal{L}_{\alpha}u = \operatorname{div}(F(t,x,\nabla u))$ with $F = \nabla u - k(t)|\nabla u|^{p(x)-2}\nabla u$, and apply it to image sharpening. A modified potential-well framework accommodates the moving Nehari manifold induced by the time-dependent coefficient $k(t)$ and variable exponent $p(\cdot)$, enabling a complete classification of global existence versus finite-time blow-up, along with decay rates for global solutions and lifespan bounds for blow-up. The results are complemented by a Galerkin construction, differential-inequality methods, and Levine’s concavity arguments, yielding explicit thresholds $\underline{d}$ and $\overline{d}$ that govern the dynamics. Numerical schemes and experiments demonstrate effective edge enhancement and texture preservation, validating the model as a robust image-sharpening tool with controlled diffusion in smooth regions.

Abstract

We introduce a class of thin-film equations with space-time dependent gradient nonlinearity and apply them to image sharpening. By modifying the potential well method, we overcome the challenges arising from variable exponents and the "moving" Nehari manifold, thereby providing a classification of global existence and finite time blow-up of solutions under different initial conditions. Decay rates for global solutions and lifespan estimates for blow-up solutions are also established. Numerical experiments demonstrate the effectiveness of this class of equations as image sharpening models.

Figures (8)

• Figure 1: Time evolution of the numerical solution in Example \ref{['exm:example1']}. Blow-up occurs at approximately $t=0.064$.
• Figure 2: The choice of the variable exponent and the evolution of $F_1(t) = \frac{1}{2} \|u(t)\|_2^2$ in time, both corresponding to Example \ref{['exm:example1']}.
• Figure 3: Time evolution of the numerical solution in Example \ref{['exm:example2']}.
• Figure 4: The choice of the variable exponent and the evolution of $F_1(t) = \frac{1}{2} \|u(t)\|_2^2$ in time, both corresponding to Example \ref{['exm:example2']}.
• Figure 5: (a) Blurry and noisy synthetic image. (b) Sharpening result using linear backward diffusion equation \ref{['eqn:bh-bwd']} with $\varepsilon=0.001$ and $t=0.2$. (c) Sharpening result using shock filter with $t=0.5$. (d) Sharpening result using the proposed model \ref{['eqn:main']} with $t=0.025$.

Theorems & Definitions (53)

• Lemma 2.1: DIAZ19941085
• Lemma 2.2: DIENING2011RADULESCU2015
• Lemma 2.3: DIENING2011RADULESCU2015
• Lemma 2.4: Hölder's inequality DIENING2011
• Lemma 2.5
• proof
• Definition 2.6: Weak solution
• Lemma 2.7
• proof
• Lemma 2.8