Non-Lipschitz heterogeneous reaction with a p-Laplacian operator

Palencia, José Luis Díaz

doi:10.3934/math.2022189

Cited by 7 publications

(1 citation statement)

References 21 publications

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“…The fact of introducing heterogeneous diffusion shall be contemplated in accordance with the problem particularities to predict. Along this analysis, a fourth order operator has been admitted, nonetheless other possible diffusion types can be assumed depending of the reality to model (for instance, refer to the p-Laplacian approach in 28 together with the p-Laplacian non-Lipschitz 29 ). Finally and in relation with the advection term introduced, some interesting results to show well-posedness of solutions have been proved by Montaru in 30 .…”

Section: Introductionmentioning

confidence: 99%

A mathematical analysis of an extended MHD Darcy–Forchheimer type fluid

Palencia

2022

Sci Rep

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The presented analysis has the aim of introducing general properties of solutions to an Extended Darcy–Forchheimer flow. The Extended Darcy–Forchheimer set of equations are introduced based on mathematical principles. Firstly, the diffusion is formulated with a non-homogeneous operator, and is supported by the addition of a non-linear advection together with a non-uniform reaction term. The involved analysis is given in generalized Hilbert–Sobolev spaces to account for regularity, existence and uniqueness of solutions supported by the semi-group theory. Afterwards, oscillating patterns of Travelling wave solutions are analyzed inspired by a set of Lemmas focused on solutions instability. Based on this, the Geometric Perturbation Theory provides linearized flows for which the eigenvalues are provided in an homotopy representation, and hence, any exponential bundles of solutions by direct linear combination. In addition, a numerical exploration is developed to find exact Travelling waves profiles and to study zones where solutions are positive. It is shown that, in general, solutions are oscillating in the proximity of the null critical state. In addition, an inner region (inner as a contrast to an outer region where solutions oscillate) of positive solutions is shown to hold locally in time.

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Section: Introductionmentioning

confidence: 99%

A mathematical analysis of an extended MHD Darcy–Forchheimer type fluid

Palencia

2022

Sci Rep

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Instabilities of standing waves and positivity in traveling waves to a higher‐order Schrödinger equation

Díaz Palencia

2024

Math Methods in App Sciences

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The aim of this paper is to explore a Schrödinger equation that incorporates a higher‐order operator. Traditional models for electron dynamics have utilized a second‐order diffusion Schrödinger equation, where oscillatory behavior is achieved through complex domain formulations. Incorporating a higher‐order operator enables the induction of oscillatory spatial patterns in solutions. Our analysis initiates with a variational formulation within generalized spaces, facilitating the examination of solution boundedness. Subsequently, we delve into the oscillatory characteristics of solutions, drawing upon a series of lemmas originally applied to the Kuramoto–Sivashinsky equation, the Cahn–Hilliard equation, and other equations that employ higher‐order operators. Specific solution types, such as standing waves, are numerically investigated to illustrate the oscillatory spatial patterns. The discussion then extends to the theory of traveling waves to establish general conditions for positive solutions. A contribution of this work is the precise evaluation of a critical traveling wave speed, denoted as , above which the first minimum remains positive. For values of the traveling wave speed significantly greater than , the solutions can be entirely positive.

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The study of nonlinear fractional boundary value problems involving the p-Laplacian operator

Khan,

Ali

et al. 2024

Phys. Scr.

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The $p$-Laplacian has attracted considerable attention in numerous fields such as mechanics, image processing and game theory. It is a nonlinear operator which has been used in the modelling and qualitative aspects in numerous problems. In this research work, we propose a new nonlinear fractional differential equation involving the $p$-Laplacian, which include the generalized Caputo fractional derivatives. We investigate the existence and uniqueness of solutions to our proposed problem through the application using the Banach and Schauder's fixed-point theorems. Additionally, we illustrate the practical applicability of our findings by applying them to a specific example, thereby validating their efficacy.

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Non-Lipschitz heterogeneous reaction with a p-Laplacian operator

Cited by 7 publications

References 21 publications

A mathematical analysis of an extended MHD Darcy–Forchheimer type fluid

A mathematical analysis of an extended MHD Darcy–Forchheimer type fluid

Instabilities of standing waves and positivity in traveling waves to a higher‐order Schrödinger equation

The study of nonlinear fractional boundary value problems involving the p-Laplacian operator

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