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Journal articles on the topic 'Degenerate equation'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Trudinger, Neil S. "On degenerate fully nonlinear elliptic equations in balls." Bulletin of the Australian Mathematical Society 35, no. 2 (April 1987): 299–307. http://dx.doi.org/10.1017/s0004972700013253.

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We establish derivative estimates and existence theorems for the Dirichlet and Neumann problems for nonlinear, degenerate elliptic equations of the form F (D2u) = g in balls. The degeneracy arises through the possible vanishing of the function g and the degenerate Monge-Ampère equation is covered as a special case.

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2

PERTHAME, BENOÎT, and ALEXANDRE POULAIN. "Relaxation of the Cahn–Hilliard equation with singular single-well potential and degenerate mobility." European Journal of Applied Mathematics 32, no. 1 (March 24, 2020): 89–112. http://dx.doi.org/10.1017/s0956792520000054.

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The degenerate Cahn–Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn–Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second-order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system: global existence and convergence of the relaxed system to the degenerate Cahn–Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.

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3

Agosti, A. "Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 3 (May 2018): 827–67. http://dx.doi.org/10.1051/m2an/2018018.

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This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type, motivated by increasing interest in diffuse interface modelling of solid tumors. The degeneracy set of the mobility and the singularity set of the potential do not coincide, and the zero of the potential is an unstable equilibrium configuration. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. In particular, the singularities of the potential do not compensate the degeneracy of the mobility by constraining the solution to be strictly separated from the degeneracy values. The error analysis of a well posed continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality, is developed. Whilst in previous works the error analysis of suitable finite element approximations has been studied for second order degenerate and fourth order non degenerate parabolic equations, in this work the a priori estimates of the error between the discrete solution and the weak solution to which it converges are obtained for a degenerate fourth order parabolic equation. The theoretical error estimates obtained in the present case state that the norms of the approximation errors, calculated on the support of the solution in the proper functional spaces, are bounded by power laws of the discretization parameters with exponent 1/2, while in the case of the classical Cahn-Hilliard equation with constant mobility the exponent is 1. The estimates are finally succesfully validated by simulation results in one and two space dimensions.

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4

Igisinov, S. Zh, L. D. Zhumaliyeva, A. O. Suleimbekova, and Ye N. Bayandiyev. "Estimates of singular numbers (s-numbers) for a class of degenerate elliptic operators." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 107, no. 3 (September 30, 2022): 51–58. http://dx.doi.org/10.31489/2022m3/51-58.

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In this paper we study a class of degenerate elliptic equations with an arbitrary power degeneracy on the line. Based on the research carried out in the course of the work, the authors propose methods to overcome various difficulties associated with the behavior of functions from the definition domain for a differential operator with piecewise continuous coefficients in a bounded domain, which affect the spectral characteristics of boundary value problems for degenerate elliptic equations. It is shown the conditions imposed on the coefficients at the lowest terms of the equation, which ensure the existence and uniqueness of the solution. The existence, uniqueness, and smoothness of a solution are proved, and estimates are found for singular numbers (s-numbers) and eigenvalues of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration.

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5

Nazarova, K. "ON ONE METHOD FOR OBTAINING UNIQUE SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR AN INTEGRO-DIFFERENTIAL EQUATION." Q A Iasaýı atyndaǵy Halyqaralyq qazaq-túrіk ýnıversıtetіnіń habarlary (fızıka matematıka ınformatıka serııasy), no. 1 (March 15, 2022): 42–54. http://dx.doi.org/10.47526/2022-2/2524-0080.04.

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The modified method of parametrization is used to study a linear Fredholm integro-differential equation with a degenerate kernel. Using the fundamental matrix, the conditions are established for the existence of a solution to the special Cauchy problem for the Fredholm integro-differential equation with a degenerate kernel. A system of linear algebraic equations is constructed with respect to the introduced additional parameters. Conditions for the unique solvability of a linear boundary value problem for the Fredholm integro-differential equation with a degenerate kernel are obtained.

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6

Christodoulou, Dimitris M., Eric Kehoe, and Qutaibeh D. Katatbeh. "Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations." Axioms 10, no. 2 (May 19, 2021): 94. http://dx.doi.org/10.3390/axioms10020094.

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For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to possess analytic solutions, otherwise their properties and the properties of their solutions are de facto known as they are comparable to those already deduced for the fundamental equation. We analyze several particular cases of new families related to some of the famous differential equations applied to physical problems, and the degenerate eigenstates of the radial Schrödinger equation for the hydrogen atom in N dimensions.

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7

Koilyshov, U. K., K. A. Beisenbaeva, and S. D. Zhapparova. "A priori estimate of the solution of the Cauchy problem in the Sobolev classes for discontinuous coefficients of degenerate heat equations." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 107, no. 3 (September 30, 2022): 59–69. http://dx.doi.org/10.31489/2022m3/59-69.

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Partial differential equations of the parabolic type with discontinuous coefficients and the heat equation degenerating in time, each separately, have been well studied by many authors. Conjugation problems for time-degenerate equations of the parabolic type with discontinuous coefficients are practically not studied. In this work, in an n-dimensional space, a conjugation problem is considered for a heat equation with discontinuous coefficients which degenerates at the initial moment of time. A fundamental solution to the set problem has been constructed and estimates of its derivatives have been found. With the help of these estimates, in the Sobolev classes, the estimate of the solution to the set problem was obtained.

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8

Gutlyanskiĭ, V., O. Martio, T. Sugawa, and M. Vuorinen. "On the degenerate Beltrami equation." Transactions of the American Mathematical Society 357, no. 3 (October 19, 2004): 875–900. http://dx.doi.org/10.1090/s0002-9947-04-03708-0.

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9

Henriques, Eurica, and Vincenzo Vespri. "On the double degenerate equation." Nonlinear Analysis: Theory, Methods & Applications 75, no. 4 (March 2012): 2304–25. http://dx.doi.org/10.1016/j.na.2011.10.030.

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10

Rubinstein, Yanir A., and Jake P. Solomon. "The degenerate special Lagrangian equation." Advances in Mathematics 310 (April 2017): 889–939. http://dx.doi.org/10.1016/j.aim.2017.02.008.

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11

Xu, Xiangsheng. "A nonlinear degenerate parabolic equation." Nonlinear Analysis: Theory, Methods & Applications 14, no. 2 (January 1990): 141–57. http://dx.doi.org/10.1016/0362-546x(90)90020-h.

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12

Ryoo, Cheon-Seoung, and Jung-Yoog Kang. "Some Identities Involving Degenerate q-Hermite Polynomials Arising from Differential Equations and Distribution of Their Zeros." Symmetry 14, no. 4 (March 31, 2022): 706. http://dx.doi.org/10.3390/sym14040706.

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This paper intends to define degenerate q-Hermite polynomials, namely degenerate q-Hermite polynomials by means of generating function. Some significant properties of degenerate q-Hermite polynomials such as recurrence relations, explicit identities and differential equations are established. Many mathematicians have been studying the differential equations arising from the generating functions of special numbers and polynomials. Based on the results so far, we find the differential equations for the degenerate q-Hermite polynomials. We also provide some identities for the degenerate q-Hermite polynomials using the coefficients of this differential equation. Finally, we use a computer to view the location of the zeros in degenerate q-Hermite equations. Numerical experiments have confirmed that the roots of the degenerate q-Hermit equations are not symmetric with respect to the imaginary axis.

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13

Rocca, Elisabetta, and Riccarda Rossi. "A degenerating PDE system for phase transitions and damage." Mathematical Models and Methods in Applied Sciences 24, no. 07 (April 14, 2014): 1265–341. http://dx.doi.org/10.1142/s021820251450002x.

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In this paper, we analyze a PDE system arising in the modeling of phase transition and damage phenomena in thermoviscoelastic materials. The resulting evolution equations in the unknowns ϑ (absolute temperature), u (displacement), and χ (phase/damage parameter) are strongly nonlinearly coupled. Moreover, the momentum equation for u contains χ-dependent elliptic operators, which degenerate at the pure phases (corresponding to the values χ = 0 and χ = 1), making the whole system degenerate. That is why, we have to resort to a suitable weak solvability notion for the analysis of the problem: it consists of the weak formulations of the heat and momentum equation, and, for the phase/damage parameter χ, of a generalization of the principle of virtual powers, partially mutuated from the theory of rate-independent damage processes. To prove an existence result for this weak formulation, an approximating problem is introduced, where the elliptic degeneracy of the displacement equation is ruled out: in the framework of damage models, this corresponds to allowing for partial damage only. For such an approximate system, global-in-time existence and well-posedness results are established in various cases. Then, the passage to the limit to the degenerate system is performed via suitable variational techniques.

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14

Baleanu, Dumitru, Vladimir E. Fedorov, Dmitriy M. Gordievskikh, and Kenan Taş. "Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case." Mathematics 7, no. 8 (August 12, 2019): 735. http://dx.doi.org/10.3390/math7080735.

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We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter—Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input.

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15

Yimamu, Yilihamujiang. "Determining the Volatility in Option Pricing From Degenerate Parabolic Equation." WSEAS TRANSACTIONS ON MATHEMATICS 21 (September 13, 2022): 629–34. http://dx.doi.org/10.37394/23206.2022.21.73.

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This contribution deals with the inverse volatility problem for a degenerate parabolic equation from numerical perspective. Being different from other inverse volatility problem in classical parabolic equations, the model in this paper is degenerate parabolic equation. Due to solve the deficiencies caused by artificial truncation and control the volatility risk with precision, the linearization method and variable substitutions are applied to transformed the inverse principal term coefficient problem for classical parabolic equation into the inverse source problem for degenerate parabolic equation in bounded region. An iteration algorithm of Landweber type is designed to obtain the numerical solution of the inverse problem. Some numerical experiments are performed to validate that the proposed algorithm is robust and the unknown coefficient is recovered quite well.

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16

Sánchez-Garduño, Faustino, and Judith Pérez-Velázquez. "Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations." Scientific World Journal 2016 (2016): 1–21. http://dx.doi.org/10.1155/2016/5620839.

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This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (atD(0)=0) and advection-degenerate (ath′(0)=0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection termh(u):(1) h′(u)is constantk,(2) h′(u)=kuwithk>0, and(3)it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, wherek=0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE.

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17

Angelopoulos, Yannis, Stefanos Aretakis, and Dejan Gajic. "A Non-degenerate Scattering Theory for the Wave Equation on Extremal Reissner–Nordström." Communications in Mathematical Physics 380, no. 1 (September 23, 2020): 323–408. http://dx.doi.org/10.1007/s00220-020-03857-3.

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Abstract It is known that sub-extremal black hole backgrounds do not admit a (bijective) non-degenerate scattering theory in the exterior region due to the fact that the redshift effect at the event horizon acts as an unstable blueshift mechanism in the backwards direction in time. In the extremal case, however, the redshift effect degenerates and hence yields a much milder blueshift effect when viewed in the backwards direction. In this paper, we construct a definitive (bijective) non-degenerate scattering theory for the wave equation on extremal Reissner–Nordström backgrounds. We make use of physical-space energy norms which are non-degenerate both at the event horizon and at null infinity. As an application of our theory we present a construction of a large class of smooth, exponentially decaying modes. We also derive scattering results in the black hole interior region.

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18

FUCHS, F., and F. POUPAUD. "ASYMPTOTICAL AND NUMERICAL ANALYSIS OF DEGENERACY EFFECTS ON THE DRIFT-DIFFUSION EQUATIONS FOR SEMICONDUCTORS." Mathematical Models and Methods in Applied Sciences 05, no. 08 (December 1995): 1093–111. http://dx.doi.org/10.1142/s0218202595000577.

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A current approximation for modeling electron transport in semiconductor devices is to assume small electron density. Through this method nondegenerate models are obtained. Here we present an asymptotical analysis of that approximation on the drift-diffusion equation. The numerical approximations of the degenerate and nondegenerate equations are then compared. A modified Scharfetter-Gummel scheme which integrates the degenerate drift-diffusion equation is proposed for comparison.

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19

Wu, Bin, Qun Chen, Tingchun Wang, and Zewen Wang. "Null controllability of a coupled degenerate system with the first and zero order terms by a single control." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 107. http://dx.doi.org/10.1051/cocv/2020042.

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This paper concerns the null controllability of a system of m linear degenerate parabolic equations with coupling terms of first and zero order, and only one control force localized in some arbitrary nonempty open subset ω of Ω. The key ingredient for proving the null controllability is to obtain the observability inequality for the corresponding adjoint system. Due to the degeneracy, we transfer to study an approximate nondegenerate adjoint system. In order to deal with the coupling first order terms, we first prove a new Carleman estimate for a degenerate parabolic equation in Sobolev spaces of negative order. Based on this Carleman estimate, we obtain a uniform Carleman estimate and then an observation inequality for this approximate adjoint system.

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20

Zhan, Huashui. "On the Weak Characteristic Function Method for a Degenerate Parabolic Equation." Journal of Function Spaces 2019 (August 26, 2019): 1–11. http://dx.doi.org/10.1155/2019/9040284.

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For a nonlinear degenerate parabolic equation, how to impose a suitable boundary value condition to ensure the well-posedness of weak solutions is a very important problem. It is well known that the classical Fichera-Oleinik theory has perfectly solved the problem for the linear case, and the optimal boundary value condition matching up with a linear degenerate parabolic equation can be depicted out by Fechira function. In this paper, a new method, which is called the weak characteristic function method, is introduced. By this new method, the partial boundary condition matching up with a nonlinear degenerate parabolic equation can be depicted out by an inequality from the diffusion function, the convection function, and the geometry of the boundary ∂Ω itself. Though, by choosing different weak characteristic function, one may obtain the differential partial boundary value conditions, an optimal partial boundary value condition can be prophetic. Moreover, the new method works well in any kind of the degenerate parabolic equations.

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21

Gutlyanskii, Vladimir, Vladimir Ryazanov, Evgeny Sevost’yanov, and Eduard Yakubov. "BMO and Asymptotic Homogeneity." Axioms 11, no. 4 (April 12, 2022): 171. http://dx.doi.org/10.3390/axioms11040171.

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First, we prove that the BMO condition by John–Nirenberg leads in the natural way to the asymptotic homogeneity at the origin of regular homeomorphic solutions of the degenerate Beltrami equations. Then, on this basis we establish a series of criteria for the existence of regular homeomorphic solutions of the degenerate Beltrami equations in the whole complex plane with asymptotic homogeneity at infinity. These results can be applied to the fluid mechanics in strongly anisotropic and inhomogeneous media because the Beltrami equation is a complex form of the main equation of hydromechanics.

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22

POP, IULIU SORIN, and BEN SCHWEIZER. "REGULARIZATION SCHEMES FOR DEGENERATE RICHARDS EQUATIONS AND OUTFLOW CONDITIONS." Mathematical Models and Methods in Applied Sciences 21, no. 08 (August 2011): 1685–712. http://dx.doi.org/10.1142/s0218202511005532.

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We analyze regularization schemes for the Richards equation and a time discrete numerical approximation. The original equations can be doubly degenerate, therefore they may exhibit fast and slow diffusion. In addition, we treat outflow conditions that model an interface separating the porous medium from a free flow domain. In both situations we provide a regularization with a non-degenerate equation and standard boundary conditions, and discuss the convergence rates of the approximations.

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23

Le, Nam Q. "On the Harnack inequality for degenerate and singular elliptic equations with unbounded lower order terms via sliding paraboloids." Communications in Contemporary Mathematics 20, no. 01 (October 23, 2017): 1750012. http://dx.doi.org/10.1142/s0219199717500122.

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We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized Monge–Ampère equations. Our argument allows us to prove the doubling estimate for functions which, at points of large gradient, are solutions of (degenerate and singular) elliptic equations with unbounded drift.

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24

Balkizov, G. A. "Boundary value problems with data on opposite characteristics for a second-order mixed-hyperbolic equation." REPORTS ADYGE (CIRCASSIAN) INTERNATIONAL ACADEMY OF SCIENCES 20, no. 3 (2020): 6–13. http://dx.doi.org/10.47928/1726-9946-2020-20-3-6-13.

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Within the framework of this work, solutions of boundary value problems with data on “opposite” (“parallel”) characteristics are found for one mixed-hyperbolic equation consisting of a wave operator in one part of the domain and a degenerate hyperbolic Gellerstedt operator in the other part. It is known that problems with data on opposite (parallel) characteristics for the wave equation in the characteristic quadrangle are posed incorrectly. However, as shown in this paper, the solution of similar problems for a mixed-hyperbolic equation consisting of a wave operator in one part of the domain and a degenerate hyperbolic Gellerstedt operator with an order of degeneracy in the other part of the domain, under certain conditions on the given functions, exists, is unique and is written explicitly.

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25

ABDELSALAM, U. M., and M. M. SELIM. "Ion-acoustic waves in a degenerate multicomponent magnetoplasma." Journal of Plasma Physics 79, no. 2 (September 4, 2012): 163–68. http://dx.doi.org/10.1017/s0022377812000803.

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AbstractThe hydrodynamic equations of positive and negative ions, degenerate electrons, and the Poisson equation are used along with the reductive perturbation method to derive the three-dimensional Zakharov–Kuznetsov (ZK) equation. The G′/G-expansion method is used to obtain a new class of solutions for the ZK equation. At certain condition, these solutions can describe the solitary waves that propagate in our plasma. The effects of negative ion concentrations, the positive/negative ion cyclotron frequency, as well as positive-to-negative ion mass ratio on solitary pulses are examined. Finally, the present study might be helpful to understand the propagation of nonlinear ion-acoustic solitary waves in a dense plasma, such as in astrophysical objects.

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26

Kbiri Alaoui, Mohammed. "On Degenerate Parabolic Equations." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–7. http://dx.doi.org/10.1155/2011/506857.

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27

Bandyopadhyay, Saugata, Bernard Dacorogna, and Olivier Kneuss. "The Pullback equation for degenerate forms." Discrete & Continuous Dynamical Systems - A 27, no. 2 (2010): 657–91. http://dx.doi.org/10.3934/dcds.2010.27.657.

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28

Hirosawa, Fumihiko. "Degenerate Kirchhoff equation in ultradifferentiable class." Nonlinear Analysis: Theory, Methods & Applications 48, no. 1 (January 2002): 77–94. http://dx.doi.org/10.1016/s0362-546x(00)00174-7.

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29

Andreu, F., V. Caselles, and J. M. Mazón. "A strongly degenerate quasilinear elliptic equation." Nonlinear Analysis: Theory, Methods & Applications 61, no. 4 (May 2005): 637–69. http://dx.doi.org/10.1016/j.na.2004.11.020.

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30

Xinhua, JI. "Möbius transformation and degenerate hyperbolic equation." Advances in Applied Clifford Algebras 11, S2 (June 2001): 155–75. http://dx.doi.org/10.1007/bf03219129.

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31

Fujisaki, Masatoshi. "Degenerate Bellman equation and its applications." Stochastic Processes and their Applications 26 (1987): 195. http://dx.doi.org/10.1016/0304-4149(87)90089-5.

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32

Betancourt, F., R. Bürger, and K. H. Karlsen. "A strongly degenerate parabolic aggregation equation." Communications in Mathematical Sciences 9, no. 3 (2011): 711–42. http://dx.doi.org/10.4310/cms.2011.v9.n3.a4.

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33

SCHULZ, RAPHAEL. "Degenerate equations in a diffusion–precipitation model for clogging porous media." European Journal of Applied Mathematics 31, no. 6 (December 18, 2019): 1050–69. http://dx.doi.org/10.1017/s0956792519000391.

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In this article, we consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity. Thereby, the evolution of the microstructure is caused by precipitation of the transported substance. We are particularly interested in analysing the model when the equations degenerate due to clogging. Introducing an appropriate weighted function space, we are able to handle the degeneracy and obtain analytical results for the transport equation. Also the decay behaviour of this solution with respect to the porosity is investigated. There a restriction on the decay order is assumed, that is, besides low initial concentration also dense precipitation leads to possible high decay. We obtain nonnegativity and boundedness for the weak solution to the transport equation. Moreover, we study an ordinary differential equation (ODE) describing the change of porosity. Thereby, the control of an appropriate weighted norm of the gradient of the porosity is crucial for the analysis of the transport equation. In order to obtain global in time solutions to the overall coupled system, we apply a fixed point argument. The problem is solved for substantially degenerating hydrodynamic parameters.

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34

Floridia, G., C. Nitsch, and C. Trombetti. "Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 18. http://dx.doi.org/10.1051/cocv/2019066.

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In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied in this paper: the weakly degenerate case, that is, if the reciprocal of the diffusion coefficient is summable, and the strongly degenerate case, that is, if that reciprocal isn’t summable. In our main result we show that the above systems can be steered from an initial continuous state that admits a finite number of points of sign change to a target state with the same number of changes of sign in the same order. Our method uses a recent technique introduced for uniformly parabolic equations employing the shifting of the points of sign change by making use of a finite sequence of initial-value pure diffusion problems. Our interest in degenerate reaction-diffusion equations is motivated by the study of some energy balance models in climatology (see, e.g., the Budyko-Sellers model) and some models in population genetics (see, e.g., the Fleming-Viot model).

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35

Holzegel, Gustav, Jonathan Luk, Jacques Smulevici, and Claude Warnick. "Asymptotic Properties of Linear Field Equations in Anti-de Sitter Space." Communications in Mathematical Physics 374, no. 2 (November 4, 2019): 1125–78. http://dx.doi.org/10.1007/s00220-019-03601-6.

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Abstract We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

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36

Krasovitskii, T. I. "Degenerate elliptic equations and nonuniqueness of solutions to the Kolmogorov equation." Доклады Академии наук 487, no. 4 (August 27, 2019): 361–64. http://dx.doi.org/10.31857/s0869-56524874361-364.

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In this paper we propose a new method of constructing examples of nonuniqueness of probability solutions by reducing the stationary Fokker-Planck-Kolmogorov equation to a degenerate elliptic equation on a bounded domain.

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37

Pyo, Sung-Soo, Taekyun Kim, and Seog-Hoon Rim. "Degenerate Daehee Numbers of the Third Kind." Mathematics 6, no. 11 (November 6, 2018): 239. http://dx.doi.org/10.3390/math6110239.

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In this paper, we define new Daehee numbers, the degenerate Daehee numbers of the third kind, using the degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee, and degenerate Daehee numbers of the second kind. In addition, we derive a differential equation associated with the degenerate log function. We deduce some identities from the differential equation.

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38

Tanirbergen, Aisulu K. "A MIXED PROBLEM FOR A DEGENERATE MULTIDIMENSIONAL ELLIPTIC EQUATION." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 3 (211) (September 30, 2021): 37–41. http://dx.doi.org/10.18522/1026-2237-2021-3-37-41.

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This article shows the unique solvability and obtains an explicit form of the classical solution of the mixed prob-lem in a cylindrical domain for a model degenerate multidimensional elliptic equation. The correctness of boundary value problems in the plane for elliptic equations by the method of the theory of ana-lytic functions of a complex variable has been well studied. The first boundary value problem or the Dirichlet problem for multidimensional elliptic equations with degeneration on the boundary has been sufficiently analyzed. However, as we know, the mixed problem for the indicated equations has been studied very little.

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Rani, Neelam, and Manikant Yadav. "The Nonlinear Magnetosonic Waves in Magnetized Dense Plasma for Quantum Effects of Degenerate Electrons." 4, no. 4 (December 10, 2021): 180–88. http://dx.doi.org/10.26565/2312-4334-2021-4-24.

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The nonlinear magnetosonic solitons are investigated in magnetized dense plasma for quantum effects of degenerate electrons in this research work. After reviewing the basic introduction of quantum plasma, we described the nonlinear phenomenon of magnetosonic wave. The reductive perturbation technique is employed for low frequency nonlinear magnetosonic waves in magnetized quantum plasma. In this paper, we have derived the Korteweg-de Vries (KdV) equation of magnetosonic solitons in a magnetized quantum plasma with degenerate electrons having arbitrary electron temperature. It is observed that the propagation of magnetosonic solitons in a magnetized dense plasma with the quantum effects of degenerate electrons and Bohm diffraction. The quantum or degeneracy effects become relevant in plasmas when fermi temperature and thermodynamic temperatures of degenerate electrons have same order.

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Sarıaydın-Filibelioğlu, Ayşe, Bülent Karasözen, and Murat Uzunca. "Energy Stable Interior Penalty Discontinuous Galerkin Finite Element Method for Cahn–Hilliard Equation." International Journal of Nonlinear Sciences and Numerical Simulation 18, no. 5 (July 26, 2017): 303–14. http://dx.doi.org/10.1515/ijnsns-2016-0024.

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AbstractAn energy stable conservative method is developed for the Cahn–Hilliard (CH) equation with the degenerate mobility. The CH equation is discretized in space with the mass conserving symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting semi-discrete nonlinear system of ordinary differential equations are solved in time by the unconditionally energy stable average vector field (AVF) method. We prove that the AVF method preserves the energy decreasing property of the fully discretized CH equation. Numerical results for the quartic double-well and the logarithmic potential functions with constant and degenerate mobility confirm the theoretical convergence rates, accuracy and the performance of the proposed approach.

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Chouaou, Fatiha, Chahira Aichi, and Abbes Benaissa. "Decay estimates for a degenerate wave equation with a dynamic fractional feedback acting on the degenerate boundary." Filomat 35, no. 10 (2021): 3219–39. http://dx.doi.org/10.2298/fil2110219c.

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In this paper, we consider a one-dimensional weakly degenerate wave equation with a dynamic nonlocal boundary feedback of fractional type acting at a degenerate point. First We show well-posedness by using the semigroup theory. Next, we show that our system is not uniformly stable by spectral analysis. Hence, we look for a polynomial decay rate for a smooth initial data by using a result due Borichev and Tomilov which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the generator associated with the semigroup. This analysis proves that the degeneracy affect the energy decay rates.

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Xiao, Stanley Yao, and Shuntaro Yamagishi. "Zeroes of Polynomials With Prime Inputs and Schmidt’s -invariant." Canadian Journal of Mathematics 72, no. 3 (February 7, 2019): 805–33. http://dx.doi.org/10.4153/s0008414x19000026.

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AbstractIn this paper we show that a polynomial equation admits infinitely many prime-tuple solutions, assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related to the $h$-invariant introduced by W. M. Schmidt. Our results prove a conjecture by B. Cook and Á. Magyar for hypersurfaces of degree 3.

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CID, CARLOS, and PATRICIO FELMER. "ORBITAL STABILITY OF STANDING WAVES FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH POTENTIAL." Reviews in Mathematical Physics 13, no. 12 (December 2001): 1529–46. http://dx.doi.org/10.1142/s0129055x01001095.

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We prove existence and orbital stability of standing waves for the nonlinear Schrödinger equation [Formula: see text] concentrating near a possibly degenerate local minimum of the potential V, when the Plank's constant ℏ is small enough. Our method applies to general nonlinearities, including f(s)=sp - 1 with p ∈ (1,1 + 4/N), but does not require uniqueness nor non-degeneracy of the limiting equation.

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Wang, Yulan, Xiaojun Song, and Chao Ye. "Fujita Exponent for a Nonlinear Degenerate Parabolic Equation with Localized Source." Advances in Mathematical Physics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/301747.

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This paper is devoted to understand the blow-up properties of reaction-diffusion equations which combine a localized reaction term with nonlinear diffusion. In particular, we study the critical exponent of ap-Laplacian equation with a localized reaction. We obtain the Fujita exponentqcof the equation.

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MI, YONG-SHENG, CHUN-LAI MU, and DENG-MING LIU. "GLOBAL EXISTENCE AND BLOW-UP FOR A DOUBLY DEGENERATE PARABOLIC EQUATION SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS." Glasgow Mathematical Journal 54, no. 2 (December 12, 2011): 309–24. http://dx.doi.org/10.1017/s0017089511000619.

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AbstractIn this paper, we deal with the global existence and blow-up of solutions to a doubly degenerative parabolic system with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of non-negative solutions, which extend the recent results of Zheng, Song and Jiang (S. N. Zheng, X. F. Song and Z. X. Jiang, Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), 308–324), Xiang, Chen and Mu (Z. Y. Xiang, Q. Chen, C. L. Mu, Critical curves for degenerate parabolic equations coupled via nonlinear boundary flux, Appl. Math. Comput. 189 (2007), 549–559) and Zhou and Mu (J. Zhou and C. L Mu, On critical Fujita exponents for degenerate parabolic system coupled via nonlinear boundary flux, Pro. Edinb. Math. Soc. 51 (2008), 785–805) to more general equations.

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46

Su, Ning, and Li Zhang. "Existence for Nonlinear Evolution Equations and Application to Degenerate Parabolic Equation." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/567241.

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We consider an abstract Cauchy problem for a doubly nonlinear evolution equation of the formd/dt𝒜u+ℬu∋ftinV′,t∈0, T, whereVis a real reflexive Banach space,𝒜andℬare maximal monotone operators (possibly multivalued) fromVto its dualV′. In view of some practical applications, we assume that𝒜andℬare subdifferentials. By using the back difference approximation, existence is established, and our proof relies on the continuity of𝒜and the coerciveness ofℬ. As an application, we give the existence for a nonlinear degenerate parabolic equation.

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Krasovitskii, T. I. "Degenerate Elliptic Equations and Nonuniqueness of Solutions to the Kolmogorov Equation." Doklady Mathematics 100, no. 1 (July 2019): 354–57. http://dx.doi.org/10.1134/s1064562419040112.

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Benci, Vieri, and Donato Fortunato. "A strongly degenerate elliptic equation arising from the semilinear Maxwell equations." Comptes Rendus Mathematique 339, no. 12 (December 2004): 839–42. http://dx.doi.org/10.1016/j.crma.2004.07.029.

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Soldatova, Е. А., and A. V. Keller. "ALGORITHMS AND INFORMATION PROCESSING IN NUMERICAL RESEARCH OF THE BARENBLATT–ZHELTOV–KOCHINA STOCHASTIC MODEL." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 13, no. 4 (2021): 29–36. http://dx.doi.org/10.14529/mmph210404.

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The paper investigates a model of pressure dynamics of a liquid filtered in a fractured-porous medium with random external action. It is based on the Cauchy–Dirichlet problem for the BarenblattZheltov–Kochina stochastic equation. An algorithm for numerical research and information processing is presented, which provides for obtaining both degenerate and non-degenerate equations. The article describes an algorithm for the numerical solution of the Cauchy–Dirichlet problem for the Barenblatt–Zheltov–Kochina stochastic equation, which is based on the Galerkin method. Numerical study of the stochastic model implies obtaining and processing the results of n experiments at various values of a random variable, including those related to rare events. The main theoretical results that have made it possible to conduct this numerical study are the methods of the theory of degenerate groups of operators and the theory of Sobolev-type equations. Algorithms are represented by schemes that allow to build flowcharts of programs on their basis, for conducting computational experiments.

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Plekhanova, Marina, and Guzel Baybulatova. "Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems." Mathematics 8, no. 4 (April 1, 2020): 483. http://dx.doi.org/10.3390/math8040483.

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A theorem on unique solvability in the sense of the strong solutions is proved for a class of degenerate multi-term fractional equations in Banach spaces. It applies to the deriving of the conditions on unique solution existence for an optimal control problem to the corresponding equation. Obtained results are used to an optimal control problem study for a model system which is described by an initial-boundary value problem for a partial differential equation.

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