Department of Mathematics

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In this article, we study the following fractional p-Laplacian equation with critical growth and singular non-linearity: (equation presented). where ω is a bounded domain in ℝ n with smooth boundary δω, n &gt; sp, s ϵ (0, 1), λ &gt; 0, 0 &lt; q ≥ 1 and 1 &lt; p &lt; α + 1 ≥ p ∗ s . We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ. © 2019 Walter de Gruyter GmbH, Berlin/Boston Fractional p-Laplacian.

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Advances in Nonlinear Analysis

On Dirichlet problem for fractional p-Laplacian with singular non-linearity

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity (Pt s) {ut + (−∆)su = u−q + f(x, u), u &gt; 0 in (0, T) × Ω, u = 0 in (0, T) × (Rn \ Ω), u(0, x) = u0(x) in Rn, where Ω is a bounded domain in Rn with smooth boundary ∂Ω, n &gt; 2s, s ∈ (0, 1), q &gt; 0, q(2s − 1) &lt; (2s + 1), u0 ∈ L∞(Ω) ∩ X0(Ω) and T &gt; 0. We suppose that the map (x, y) ∈ Ω × R+ → f(x, y) is a bounded from below Carathéodary function, locally Lipschitz with respect to the second variable and uniformly for x ∈ Ω and it satisfies lim supf(x, y) &lt; λs 1(Ω), y→+∞ y (0.1) where λs 1(Ω) is the first eigenvalue of (−∆)s in Ω with homogeneous Dirichlet boundary condition in Rn\Ω. We prove the existence and uniqueness of a weak solution to (Pt s) on assuming u0 satisfies an appropriate cone condition. We use the semi-discretization in time with implicit Euler method and study the stationary problem to prove our results. We also show additional regularity on the solution of (Pt s) when we regularize our initial function u0. © American Institute of Mathematical Sciences. All rights reserved.}, author_keywords={Fractional Laplacian; Non-local operator; Parabolic equation; Singular nonlinearity; Sub-super solution}, funding_details={Indo-French Centre for Applied MathematicsIndo-French Centre for Applied Mathematics, IFCAM}, funding_details={College of Natural Resources and Sciences, Humboldt State UniversityCollege of Natural Resources and Sciences, Humboldt State University, CNRS, 3494}, funding_text_1={The authors were funded by IFCAM (Indo-French Centre for Applied Mathematics) UMI CNRS 3494 under the project “Singular phenomena in reaction diffusion equations and in conservation laws”}, funding_text_2={Acknowledgments. The authors were funded by IFCAM (Indo-French Centre for Applied Mathematics) UMI CNRS 3494 under the project “Singular phenomena in reaction diffusion equations and in conservation laws”}, references={Abdellaoui, B., Medina, M., Peral, I., Primo, A., Optimal results for the fractional heat equation involving the hardy potential (2016) Nonlinear Anal, 140, pp. 166-207; Adimurthi, J.G., Santra, S., Positive solutions to a fractional equation with singular nonlinearity (2018) J. Differential Equations, 265, pp. 1191-1226; Alibaud, N., Imbert, C., Fractional semi-linear parabolic equations with unbounded data (2009) Transactions of The American Mathematical Society, 361, pp. 2527-2566; Amghibech, S., On the discrete version of picone's identity (2008) Discrete Applied Mathematics, 156, pp. 1-10; Avelin, B., Gianazza, U., Salsa, S., Boundary estimates for certain degenerate and singular parabolic equations (2016) Journal of The European Mathematical Society, 18, pp. 381-424; Badra, M., Bal, K., Giacomoni, J., A singular parabolic equation: Existence, stabilization (2012) J. Differential Equations, 252, pp. 5042-5075; Barbu, V., (2010) Nonlinear Differential Equations of Monotone Types in Banach Spaces, , 1st edition, Springer Monogr. 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Nonlinear Anal., 6, pp. 61-84; Frank, R.L., Seiringer, R., Non-linear ground state representations and sharp hardy inequalities (2008) Journal of Functional Analysis, 255, pp. 3407-3430; Giacomoni, J., Mukherjee, T., Sreenadh, K., Positive solutions of fractional elliptic equation with critical and singular nonlinearity (2016) Adv. Nonlinear Anal., 6, pp. 327-354; Kim, S., Lee, K.-A., Hölder estimates for singular non-local parabolic equations (2011) Journal of Functional Analysis, 261, pp. 3482-3518; Leonori, T., Peral, I., Primo, A., Soria, F., Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations (2015) Discrete Contin. Dyn. Syst., 35, pp. 6031-6068; Mukherjee, T., Sreenadh, K., Fractional elliptic equations with critical growth and singular nonlinearities (2016) Electronic Journal of Differential Equations, 54, pp. 1-23; Ros-Oton, X., Serra, J., The dirichlet problem for the fractional laplacian: Regularity up to the boundary (2014) Journal De Mathématiques Pures Et Appliquées, 101, pp. 275-302; Servadei, R., Valdinoci, E., The brezis-nirenberg result for the fractional laplacian (2015) Transactions of The American Mathematical Society, 367, pp. 67-102; Servadei, R., Valdinoci, E., Variational methods for non-local operators of elliptic type (2013) Discrete Contin. Dyn. Syst., 33, pp. 2105-2137; Silvestre, L., Regularity of the obstacle problem for a fractional power of the laplace operator (2007) Comm. Pure Appl. Math., 60, pp. 67-112; Simon, J., Compact sets in the space Lp(0, T; B) (1987) Ann. Mat. Pura Appl., 146, pp. 65-96; Vázquez, J.L., Nonlinear diffusion with fractional laplacian operators (2012) Nonlinear Partial Differential Equations, 7, pp. 271-298. , Holden, Helge and Karlsen, Kenneth H. eds., Springer; Vázquez, J.L., Recent progress in the theory of nonlinear diffusion with fractional laplacian operators (2014) Discrete and Continuous Dynamical Systems - Series S, 7, pp. 857-885}, correspondence_address1={Giacomoni, J.; Université de Pau et des Pays de l'Adour, avenue de l'université, France; email: jacques.giacomoni@univ-pau.fr}, publisher={American Institute of Mathematical Sciences}, issn={19371632}, language={English}, abbrev_source_title={Discrete Contin. Dyn. Syst. Ser. S}, document_type={Article}, source={Scopus},

Discrete and Continuous Dynamical Systems - Series S

Existence and stabilization results for a singular parabolic equation involving the fractional laplacian

We consider a question of global existence for two component volume-surface reaction-diffusion systems. The first of the components diffuses in a region, and then reacts on the boundary with the second component, which diffuses on the boundary. We show that if the first component is bounded a priori on any time interval, and the kinetic terms satisfy a generalized balancing condition, then both solutions exist globally. We also pose an open question in the opposite direction, and give some a priori estimates for associated m component systems. © Springer International Publishing AG, part of Springer Nature 2019.

Computational Methods in Applied Sciences

Martin’s problem for volume-surface reaction-diffusion systems

In this paper, the absorption properties of resonant patch absorbers are analyzed using the quality factor approach. The total quality factor contributing from both the dissipative quality factor and radiative quality factor is deduced. The resonant circular patch absorbers are studied under the influence of position randomness with increasing filling factor. It is observed that by introducing position disorder, it is possible to enhance the absorption bandwidth of the otherwise periodic circular patch resonant absorbers. This enhancement in absorption bandwidth is obtained due to the coupling among neighboring resonators, which results in broadening of collective resonances. The designed disordered resonant patch absorbers also exhibit polarization insensitivity for oblique incidences for both polarizations. In addition, square patch absorbers are also investigated under the periodic and disordered setting. It is found that the enhancement in the bandwidth of the disordered absorber structures is independent of the shape of the patch resonators, and depends solely on the coupling among the neighboring resonators. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

Applied Physics A: Materials Science and Processing

Numerical investigation of disordered patch resonator absorbers

A power-law fluid flowing down a slippery inclined plane under the action of gravity is deliberated in this research work. A Newtonian layer at a small strain rate is introduced to take care of the divergence of the viscosity at a zero strain rate. A low-dimensional two-equation model is formulated using a weighted-residual approach in terms of two coupled evolution equations for the film thickness h and a local velocity amplitude or the flow rate q within the framework of lubrication theory. Moreover, a long-wave instability is shown in detail. Linear stability analysis of the proposed two-equation model reveals good agreement with the spatial Orr-Sommerfeld analysis. The influence of a wall-slip on the primary instability has been found to be non-trivial. It has the stabilizing effect at larger values of the Reynolds number, whereas at the onset of the instability, the role is destabilizing which may be because of the increase in dynamic wave speed by the wall slip. Competing impressions of shear-thinning/shear-thickening and wall slip velocity on the primary instability are captured. The impact of slip velocity on the traveling-wave solutions is discussed using the bifurcation diagram. An increasing value of the slip shows a significant effect on the traveling wave and free surface amplitude. Slip velocity controls both the kinematic and dynamic waves of the system, and thus, it has the profound passive impact on the instability. © 2019 Author(s).

Physics of Fluids

Dynamics and stability of a power-law film flowing down a slippery slope

This paper presents a new blind image watermarking scheme using binary decimal sequence (d-sequence) and lifting wavelet transform (LWT) for copyright protection. The core idea is to produce a d- sequence based on random number generator (RNG) algorithm and secret keys. A reference set is then generated using the d-sequence for embedding purpose. For embedding, the host image is decomposed into different frequency bands using the LWT and watermark bits are then embedded in selected band, considering the reference set. The extensive experimental results, comparative and security analysis demonstrate the better robustness of the proposed scheme against different kind of attacks. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Multimedia Tools and Applications

A simplified watermarking algorithm based on lifting wavelet transform

In this paper, we discuss the dynamics of alterations and rearrangements of a non-autonomous dynamical system generated by the family F. We prove that while insertion/deletion of a map in the family F can disturb the dynamics of a system, the dynamics of the system does not change if the map inserted/deleted is feeble open. In the process, we prove that if the inserted/deleted map is feeble open, the altered system exhibits any form of mixing/sensitivity if and only if the original system exhibits the same. We extend our investigations to properties like equicontinuity, minimality, and proximality for the two systems. We prove that any finite rearrangement of a non-autonomous dynamical system preserves the dynamics of original system if the family F is feeble open. We also give examples to show that the dynamical behavior of a system need be not be preserved under infinite rearrangement. © 2019, Iranian Mathematical Society.

Bulletin of the Iranian Mathematical Society

Alterations and Rearrangements of a Non-autonomous Dynamical System

In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem: (Equation Presented), where (-Δ) s denotes the fractional Laplace operator for s ∈ (0, 1), n &gt; 2s, q ∈ (0, 1), λ &gt; 0 and Ω is a smooth bounded domain in ℝ n . Here f : [0, ∞) → [0, ∞) is a continuous nondecreasing map satisfying (Equation Presented). We show that under certain additional assumptions on f, the above problem possesses at least three distinct solutions for a certain range of λ. We use the method of sub-supersolutions and a critical point theorem by Amann [H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620.709] to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest. © 2019 Walter de Gruyter GmbH.

Advanced Nonlinear Studies

Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

In this article we show the global multiplicity result for the following nonlocal singular problem where ω is a bounded domain in ℝn with smooth boundary ∂ω, n &gt; 2s, s ε (0; 1), λ &gt; 0, q &gt; 0 satisfies q(2s - 1) &lt; (2s + 1) and 2* s = 2n/(n - 2s). Employing the variational method, we show the existence of at least two distinct weak positive solutions for (Pλ) in X0 when λ ε (0, Λ) and no solution when λ &gt; Λ, where Λ &gt; 0 is appropriately chosen. We also prove a result of independent interest that any weak solution to (Pλ) is in Cα(ℝn) with α = α(s, q) ε (0, 1). The asymptotic behaviour of weak solutions reveals that this result is sharp. © 2019 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University in Toruń.

Topological Methods in Nonlinear Analysis

A global multiplicity result for a very singular critical nonlocal equation

Stability analysis of stratified multiphase flow for a spanwise system of rotation plays a pivotal role in micromixing and micromachines. In several such systems, centrifugal actuation is the driving force, which creates a pressure gradient in a rotating channel and Coriolis force enhances mixing in a short span by destabilizing the flow. Here, we focus on the impact of the Coriolis force on a rotating two-fluid flow through a microchannel, which is miscible in nature, having small viscosity difference and thereby forming a thin diffusive interface between fluids due to viscosity stratification. Modal stability analysis is used to estimate the critical flow parameters which are, in turn, responsible for regulating the instability mechanism for different viscosity contrasts and mixed layer thicknesses. Usually, viscosity stratified flow with respect to streamwise disturbance becomes more unstable for a thinner mixed layer. On the contrary, our numerical computation confirms a completely discrepant scenario by considering Coriolis force-driven instability of a miscible flow system on account of spanwise disturbances. Possible physical mechanisms for the same are discussed in terms of base flow pattern and the energy fluctuation between the perturbed and base flow. Comparison of three-dimensional disturbances of the flow field, in both clockwise and anticlockwise directions (for two different viscosity ratios), is executed to provide an insight into the dynamics of the flow system. Distributions of the velocity perturbations display a critical bonding between the vortices near and away from the mixed layer. These vortices are, in turn, responsible for the variation in instability mechanism with respect to different viscosity ratios and rotational directions. © 2019 American Physical Society.