An improved nonlinear anisotropic PDE with p(x)-growth conditions applied to image restoration and enhancement

This work proposes a novel nonlinear parabolic equation with p(x)-growth conditions for image restoration and enhancement. Based on the generalized Lebesgue and Sobolev spaces with variable exponent, we demonstrate the well-posedness of the proposed model. As a first result, we prove the existence of a weak solution to our model when the reaction term is bounded by a suitable function. Secondly, we use the approximations method to establish the existence of a nonnegative weak SOLA solution (Solution Obtained as Limit of Approximations) to the proposed model. Finally, numerical experiments illustrate that the proposed model performs better for image enhancement and denoising.