Periodic parabolic equation involving singular nonlinearity with variable exponent

Charkaoui, Abderrahim; Taourirte, Laila; Alaa, Nour Eddine

doi:10.1007/s11587-021-00609-w

Cited by 10 publications

(6 citation statements)

References 20 publications

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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 [37–47]). Therefore, the variable exponent

p\left(\cdotp \right)

that appears in problem () 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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“…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 [37–47]). Therefore, the variable exponent

p\left(\cdotp \right)

that appears in problem () 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

Self Cite

View full text Add to dashboard Cite

show abstract

“…As well known, theoretical analysis of PDEs involving ( )-growth conditions need the use of some complex spaces called Lebesgue and Sobolev spaces with variable exponents (see for example 3,7,16,17,24,32,40,52,53 ). Therefore, the variable exponent (⋅) appears in problem (1) requires the consideration of these types of spaces.…”

Section: Mathematical Backgrounds and Assumptionsmentioning

confidence: 99%

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

Alaa

Bouchriti

et al. 2022

Preprint

Self Cite

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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.

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“…At the same time, the literature of periodic PDEs is very vast and for more clarification, we refer the readers to see the works 6,11,13,14,15,16,20,25 . Particularly, in 6 Amann studied a periodic parabolic equation with an additionnal nonlinearity and subquadratic growth on the gradient.…”

Section: Introductionmentioning

confidence: 99%

Time periodic solutions for strongly nonlinear parabolic systems with p(x)-growth conditions

CHARKAOUI,

Alaa,

Alaa

2024

Preprint

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We study a class of nonlinear periodic systems driven by general differential operators with variable exponent. We assume that the reactions contains p(x)-growth nonlinearities with respect to the gradients. By using Leray Schauder’s topological degree combined with the sub- and super-solutions method, we establish the existence and uniqueness results of weak periodic solutions to the studied systems.

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Periodic parabolic equation involving singular nonlinearity with variable exponent

Cited by 10 publications

References 20 publications

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

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

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

Time periodic solutions for strongly nonlinear parabolic systems with p(x)-growth conditions

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