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Journal articles on the topic 'Degenerate equation'

Author: Grafiati

Published: 9 March 2023

Last updated: 31 July 2025

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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 (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 (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 ener

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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 (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 constrain

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4

Oza, Priyank. "Fully nonlinear degenerate equations with applications to Grad equations." Electronic Journal of Qualitative Theory of Differential Equations, no. 26 (2024): 1–13. http://dx.doi.org/10.14232/ejqtde.2024.1.26.

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We consider a class of degenerate elliptic fully nonlinear equations with applications to Grad equations: { | D u | γ M λ , Λ + ( D 2 u ( x ) ) = f ( | u ≥ u ( x ) | ) in Ω u = g on ∂ Ω , where γ ≥ 1 is a constant, Ω is a bounded domain in R N with C 1 , 1 boundary. We prove the existence of a W 2 , p -viscosity solution to the above equation, which degenerates when the gradient of the solution vanishes.

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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 (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

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6

Mai, La-Su, and Suriguga. "Local well-posedness of 1D degenerate drift diffusion equation." Mathematics in Engineering 6, no. 1 (2024): 155–72. http://dx.doi.org/10.3934/mine.2024007.

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<abstract><p>This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates.</p></abstract>

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7

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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8

Pasichnyk, H. "SOLUTIONS OF SOME INTEGRAL EQUATIONS OF THE SECOND KIND." Bukovinian Mathematical Journal 12, no. 1 (2024): 84–93. http://dx.doi.org/10.31861/bmj2024.01.08.

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The article examines the solutions of some integral equations of the second kind. Such equations arise when using Levi's method to construct a fundamental solution of the Cauchy problem for a degenerate equation of the Kolmogorov type. The equation may also contain a degeneracy on the initial hyperplane. The coefficients of this equation are bounded in the group of principal terms and ones are increasing functions in the group of lowest terms. The considered classes of kernels of integral equations make it possible to preserve the function that determines the growth of the coefficients of the

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9

Christodoulou, Dimitris M., Eric Kehoe, and Qutaibeh D. Katatbeh. "Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations." Axioms 10, no. 2 (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 a

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10

Shakhmurov, V. B., and Н. К. Musaev. "Nonlocal separable elliptic and parabolic equations and applications." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 1 (February 5, 2025): 66–92. https://doi.org/10.26907/0021-3446-2025-1-66-92.

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The regularity properties of nonlocal anisotropic elliptic equations with parameters are investigated in abstract weighted Lp spaces. The equations include the variable coefficients and abstract operator function A = A (x) in a Banach space E in leading part. We find the sufficient growth assumptions on A and appropriate symbol polynomial functions that guarantee the uniformly separability of the linear problem. It is proved that the corresponding anisotropic elliptic operator is sectorial and is also the negative generator of an analytic semigroup. Byusing these results, the existence and uni

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11

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 (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

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12

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 (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 b

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13

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 (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 equation

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14

Rocca, Elisabetta, and Riccarda Rossi. "A degenerating PDE system for phase transitions and damage." Mathematical Models and Methods in Applied Sciences 24, no. 07 (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 c

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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 equa

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16

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 (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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17

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 equatio

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18

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

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19

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

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20

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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21

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

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22

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 (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

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23

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

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24

Suleimbekova, A. O., and B. M. Musilimov. "On the existence and coercive estimates of solutions to the Dirichlet problem for a class of third-order differential equations." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 114, no. 2 (2024): 178–85. http://dx.doi.org/10.31489/2024m2/178-185.

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As you know, the third order partial differential equation is one of the basic equations of wave theory. For example, in particular, a linearized Korteweg-de Vries type equation with variable coefficients models ion-acoustic waves into plasma and acoustic waves on a crystal lattice. In this paper, the properties of solutions of а class of the third order degenerate partial differential equations with variable coefficients given in a rectangle were studied. Sufficient conditions for the existence and uniqueness of a strong solution have been established. Note that the solution of the degenerate

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25

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 nonli

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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 (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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27

Gutlyanskii, Vladimir, Vladimir Ryazanov, Evgeny Sevost’yanov, and Eduard Yakubov. "BMO and Asymptotic Homogeneity." Axioms 11, no. 4 (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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28

Vikulov, A. G. "REGULARIZATION OF THE SOLUTION OF DEGENERATE SYSTEMS OF ALGEBRAIC EQUATIONS BY THE EXAMPLE OF IDENTIFICATION OF THE VIRIAL EQUATION OF STATE OF A REAL GAS." Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki 64, no. 7 (2024): 1112–27. https://doi.org/10.31857/s0044466924070024.

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To carry out the thermodynamic calculation of the cycle in the two-phase region, an equation of state of the working fluid is necessary, as which a virial equation with unknown temperature functions is used. A degenerate system of algebraic equations is constructed with respect to unknown coefficients, which are the values of virial functions on a discrete temperature grid. Based on the regularization method, a variational iterative algorithm for solving a degenerate system of equations has been developed. A computational experiment has been conducted to confirm the efficiency of the method.

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29

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 (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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30

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

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31

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

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32

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 (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

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33

Krasovitskii, T. I. "Degenerate elliptic equations and nonuniqueness of solutions to the Kolmogorov equation." Доклады Академии наук 487, no. 4 (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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Pyo, Sung-Soo, Taekyun Kim, and Seog-Hoon Rim. "Degenerate Daehee Numbers of the Third Kind." Mathematics 6, no. 11 (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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35

Otarova, J. A. "A boundary value problem for the fourth-order degenerate equation of the mixed type." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 113, no. 1 (2024): 140–48. http://dx.doi.org/10.31489/2024m1/140-148.

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Many problems in mechanics, physics, and geophysics lead to solving partial differential equations that are not included in the known classes of elliptic, parabolic or hyperbolic equations. Such equations, as a rule, began to be called non-classical equations of mathematical physics. The theory of degenerate equations is one of the central branches of the modern theory of partial differential equations. This is primarily due to the identification of a variety of applied problems, the mathematical modeling of which serves the study of various types of degenerate equations. The study of boundary

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36

ABDELSALAM, U. M., and M. M. SELIM. "Ion-acoustic waves in a degenerate multicomponent magnetoplasma." Journal of Plasma Physics 79, no. 2 (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. Finall

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37

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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38

SCHULZ, RAPHAEL. "Degenerate equations in a diffusion–precipitation model for clogging porous media." European Journal of Applied Mathematics 31, no. 6 (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 deca

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39

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

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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 (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 p

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41

Bednarczuk, Ewa, Olga Brezhneva, Krzysztof Leśniewski, Agnieszka Prusińska, and Alexey A. Tret’yakov. "Towards Nonlinearity: The p-Regularity Theory." Entropy 27, no. 5 (2025): 518. https://doi.org/10.3390/e27050518.

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We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality condition

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42

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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43

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

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44

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

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45

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

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46

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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47

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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48

Gaillard, Pierre. "From Algebro Geometric Solutions of the Toda Equation to Sato Formulas." AppliedMath 4, no. 3 (2024): 856–67. http://dx.doi.org/10.3390/appliedmath4030046.

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We know that the degeneracy of solutions to PDEs, given in terms of theta functions on Riemann surfaces, provides important results about particular solutions, as in the case of the NLS equation. Here, we degenerate the so called finite gap solutions of the Toda lattice equation from the general formulation in terms of abelian functions when the gaps tend to points. This degeneracy allows us to recover the Sato formulas without using inverse scattering theory or geometric or representation theoretic methods.

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49

Abdullaev, A. A., N. M. Safarbayeva, and B. Z. Usmonov. "On the unique solvability of a nonlocal boundary value problem with the poincaré condition." E3S Web of Conferences 401 (2023): 03048. http://dx.doi.org/10.1051/e3sconf/202340103048.

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As is known, it is customary in the literature to divide degenerate equations of mixed type into equations of the first and second kind. In the case of an equation of the second kind, in contrast to the first, the degeneracy line is simultaneously the envelope of a family of characteristics, i.e. is itself a characteristic, which causes additional difficulties in the study of boundary value problems for equations of the second kind. In this paper, in order to establish the unique solvability of one nonlocal problem with the Poincaré condition for an elliptic-hyperbolic equation of the second k

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50

Akil, Mohammad, Genni Fragnelli, and Amine Sbai. "Exponential Stability for a Degenerate/Singular Beam-Type Equation in Non-Divergence Form." Axioms 14, no. 3 (2025): 159. https://doi.org/10.3390/axioms14030159.

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The paper deals with the stability of a degenerate/singular beam equation in non-divergence form. In particular, we assume that the degeneracy and the singularity are at the same boundary point and we impose clamped conditions where the degeneracy occurs and dissipative conditions at the other endpoint. Using the energy method, we provide some conditions to obtain the stability for the considered problem.

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