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

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.