Integral Solution for a Parabolic Equation Driven by the p(x)-Laplacian Operator with Nonlinear Boundary Conditions and $$L^{1}$$ Data

Hathout, Mohamed El

doi:10.1007/s00009-023-02446-7

Cited by 4 publications

(1 citation statement)

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“…As well-known, theoretical analysis of PDEs involving p(x)-growth conditions need the use of some complex spaces called Lebesgue and Sobolev spaces with variable exponents (see, e.g., earlier studies [37][38][39][40][41][42][43][44][45][46][47]). Therefore, the variable exponent p(•) that appears in problem (1) requires the consideration of these types of spaces.…”

Section: Mathematical Backgrounds and Assumptionsmentioning

confidence: 99%

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

Alaa,

Eddine Alaa,

Bouchriti

et al. 2024

Math Methods in App Sciences

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This work proposes a novel nonlinear parabolic equation with ‐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 Obtained as Limit of Approximations) solution to the proposed model. We illustrate our theoretical results in the context of image denoising and enhancement. Specifically, we present various numerical implementations on a set of grayscale images. To further enrich these simulations, we assess the efficiency of the proposed model on several color images. The obtained numerical results strongly suggest that our model outperforms existing state‐of‐the‐art methods in terms of both visual and quantitative comparison, demonstrating superior efficiency and robustness in noise removal and contrast enhancement.

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