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Journal articles on the topic 'Degenerate PDE'

Author: Grafiati

Published: 10 August 2024

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1

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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2

Nytrebych, Zinovii, and Oksana Malanchuk. "The conditions of existence of a solution of the degenerate two-point in time problem for PDE." Asian-European Journal of Mathematics 12, no. 03 (May 27, 2019): 1950037. http://dx.doi.org/10.1142/s1793557119500372.

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The problem with local nonhomogeneous two-point in time conditions for homogeneous PDE of the second order in time and, generally, infinite order in spatial variables is investigated. This problem is degenerated namely its characteristic determinant is identically zero. The condition of existence of a solution of the degenerate problem is established. Also, we proposed the differential-symbol method of constructing the solution of the problem in the classes of entire functions. Some examples of solving the degenerate two-point in time problems are presented.

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3

Kubo, Masahiro, and Quqin Lu. "Evolution equations for nonlinear degenerate parabolic PDE." Nonlinear Analysis: Theory, Methods & Applications 64, no. 8 (April 2006): 1849–59. http://dx.doi.org/10.1016/j.na.2005.07.027.

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4

Spence, S., G. Rena, G. Sweeney, and M. D. Houslay. "Induction of Ca2+/calmodulin-stimulated cyclic AMP phosphodiesterase (PDE1) activity in Chinese hamster ovary cells (CHO) by phorbol 12-myristate 13-acetate and by the selective overexpression of protein kinase C isoforms." Biochemical Journal 310, no. 3 (September 15, 1995): 975–82. http://dx.doi.org/10.1042/bj3100975.

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The cAMP phosphodiesterase (PDE) activity of CHO cells was unaffected by the addition of Ca2+ +calmodulin (CaM), indicating the absence of any PDE1 (Ca2+/CaM-stimulated PDE) activity. Treatment with the tumour promoting phorbol ester phorbol 12-myristate 13-acetate (PMA) led to the rapid transient induction of PDE1 activity which attained a maximum value after about 13 h before slowly decreasing. Such induction was attenuated by actinomycin D. PCR primers were designed to hybridize with two regions identified as being characteristic of PDE1 forms found in various species and predicted to amplify a 601 bp fragment. RT-PCR using degenerate primers allowed an approx. 600 bp fragment to be amplified from RNA preparations of rat brain but not from CHO cells unless they had been treated with PMA. CHO cells transfected to overexpress protein kinase C (PKC)-alpha and PKC-epsilon, but not those transfected to overexpress PKC-beta I or PKC-gamma, exhibited a twofold higher PDE activity. They also expressed a PDE1 activity, with Ca2+/CaM effecting a 1.8-2.8-fold increase in total PDE activity. RT-PCR, with PDE1-specific primers, identified an approx. 600 bp product in CHO cells transfected to overexpress PKC-alpha and PKC-epsilon, but not in those overexpressing PKC-beta I or PKC-gamma. Treatment of PKC-alpha transfected cells with PMA caused a rapid, albeit transient, increase in PDE1 activity, which reached a maximum some 1 h after PMA challenge, before returning to resting levels some 2 h later. The residual isobutylmethylxanthine (IBMX)-insensitive PDE activity was dramatically reduced (approx. 4-fold) in the PKC-gamma transfectants, suggesting that the activity of the cyclic AMP-specific IBMX-insensitive PDE7 activity was selectively reduced by overexpression of this particular PKC isoform. These data identify a novel point of ‘cross-talk’ between the lipid and cyclic AMP signalling systems where the action of specific PKC isoforms is shown to cause the induction of Ca2+/CaM-stimulated PDE (PDE1) activity. It is suggested that this protein kinase C-mediated process might involve regulation of PDE1 gene expression by the AP-1 (fos/jun) system.

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5

PARDOUX, ÉTIENNE, and AHMADOU BAMBA SOW. "HOMOGENIZATION OF A PERIODIC DEGENERATE SEMILINEAR ELLIPTIC PDE." Stochastics and Dynamics 11, no. 02n03 (September 2011): 475–93. http://dx.doi.org/10.1142/s0219493711003401.

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In this paper, a semilinear elliptic PDE with rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of ℝd. Our fully probabilistic method is based on the connection between PDEs and BSDEs with random terminal time and the weak convergence of a class of diffusion processes.

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6

Elshegmani, Zieneb Ali, and Rokiah Rozita Ahmad. "Fourier Transform of the Continuous Arithmetic Asian Options PDE." ISRN Applied Mathematics 2011 (September 26, 2011): 1–8. http://dx.doi.org/10.5402/2011/643749.

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Price of the arithmetic Asian options is not known in a closed-form solution, since arithmetic Asian option PDE is a degenerate partial differential equation in three dimensions. In this work we provide a new method for computing the continuous arithmetic Asian option price by means of partial differential equations. Using Fourier transform and changing some variables of the PDE we get a new direct method for solving the governing PDE without reducing the dimensionality of the PDE as most authors have done. We transform the second-order PDE with nonconstant coefficients to the first order with constant coefficients, which can be solved analytically.

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7

Katzourakis, Nikos. "On linear degenerate elliptic PDE systems with constant coefficients." Advances in Calculus of Variations 9, no. 3 (July 1, 2016): 283–91. http://dx.doi.org/10.1515/acv-2015-0004.

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AbstractLet ${\mathbf{A}}$ be a symmetric convex quadratic form on ${\mathbb{R}^{Nn}}$ and Ω $\subset$$\mathbb{R}^{n}$ a bounded convex domain. We consider the problem of existence of solutions u: Ω $\subset$$\mathbb{R}^{n}$$\to$$\mathbb{R}^{N}$ to the problem${}\left\{\begin{aligned} \displaystyle\sum_{\beta=1}^{N}\sum_{i,j=1}^{n}% \mathbf{A}_{\alpha i\beta j}D^{2}_{ij}u_{\beta}&\displaystyle=f_{\alpha}&&% \displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.\phantom{\}}$when ${f\in L^{2}(\Omega,\mathbb{R}^{N})}$. Problem (1) is degenerate elliptic and it has not been considered before without the assumption of strict rank-one convexity. In general, it may not have even distributional solutions. By introducing an extension of distributions adapted to (1), we prove existence, partial regularity and by imposing an extra condition uniqueness as well. The satisfaction of the boundary condition is also an issue due to the low regularity of the solution. The motivation to study (1) and the method of the proof arose from recent work of the author [10] on generalised solutions for fully nonlinear systems.

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8

Biswas, Imran H., Espen R. Jakobsen, and Kenneth H. Karlsen. "Difference-Quadrature Schemes for Nonlinear Degenerate Parabolic Integro-PDE." SIAM Journal on Numerical Analysis 48, no. 3 (January 2010): 1110–35. http://dx.doi.org/10.1137/090761501.

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9

Antil, Harbir, and Shawn W. Walker. "Optimal Control of a Degenerate PDE for Surface Shape." Applied Mathematics & Optimization 78, no. 2 (March 16, 2017): 297–328. http://dx.doi.org/10.1007/s00245-017-9407-3.

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10

Ivochkina, Nina, Neil Trudinger, and Xu-Jia Wang. "The Dirichlet Problem for Degenerate Hessian Equations." Communications in Partial Differential Equations 29, no. 1-2 (January 2005): 219–35. http://dx.doi.org/10.1081/pde-120028851.

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11

Kallel-Jallouli, Saoussen. "The Dirichlet Problem for Degenerate Elliptic Darboux Equation." Communications in Partial Differential Equations 29, no. 7-8 (January 11, 2004): 1097–125. http://dx.doi.org/10.1081/pde-200033756.

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12

MANOEL, MIRIAM, and IAN STEWART. "DEGENERATE BIFURCATIONS WITH Z2⊕Z2-SYMMETRY." International Journal of Bifurcation and Chaos 09, no. 08 (August 1999): 1653–67. http://dx.doi.org/10.1142/s0218127499001140.

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Bifurcation problems with the symmetry group Z2⊕Z2 of the rectangle are common in applied science, for example, whenever a Euclidean invariant PDE is posed on a rectangular domain. In this work we derive normal forms for one-parameter bifurcations of steady states with symmetry of the group Z2⊕Z2. We study degeneracies of Z2⊕Z2-codimension 3 and modality 1. We also deduce persistent bifurcation diagrams when the system is subject to symmetry-preserving perturbations.

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13

Mikayelyan, Hayk, and Henrik Shahgholian. "Hopf's lemma for a class of singular/degenerate PDE-s." Annales Academiae Scientiarum Fennicae Mathematica 40 (January 2015): 475–84. http://dx.doi.org/10.5186/aasfm.2015.4033.

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14

Biset, Tesfa, and Ahmed Mohammed. "A singular boundary value problem for a degenerate elliptic PDE." Nonlinear Analysis: Theory, Methods & Applications 119 (June 2015): 222–34. http://dx.doi.org/10.1016/j.na.2014.09.028.

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15

Surdo, C. Lo. "Global magnetofluidostatic fields (an unsolved PDE problem)." International Journal of Mathematics and Mathematical Sciences 9, no. 1 (1986): 123–30. http://dx.doi.org/10.1155/s0161171286000157.

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A satisfactory theory of the Global MagnetoFluidoStatic (GMFS) Fields, where symmetric and non-symmetric configurations can be dealt with on the same footing, has not yet been developed. However the formulation of the Nowhere-Force-Free, Local-Global MFS problem about a given smooth isobaric toroidal surface𝒮0(actually, a degenerate initial-value problem) can be weakened so as to include certain generalized solutions as formal power series in a natural transverse coordinate. lt is reasonable to conjecture that these series converge, for sufficiently smooth data on𝒮0. in the same function space which their coefficients belong to (in essence, a complete linear space over the2-torus).

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16

Igbida, N. "LARGE TIME BEHAVIOR OF SOLUTIONS TO SOME DEGENERATE PARABOLIC EQUATIONS." Communications in Partial Differential Equations 26, no. 7-8 (June 30, 2001): 1385–408. http://dx.doi.org/10.1081/pde-100106138.

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17

Laux, Tim, and J. Miguel Villas-Boas. "Well-posedness for degenerate elliptic PDE arising in optimal learning strategies." Interfaces and Free Boundaries 22, no. 1 (April 15, 2020): 119–29. http://dx.doi.org/10.4171/ifb/434.

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18

Diedhiou, Alassane. "Limit of the Solution of a PDE in the Degenerate Case." Applied Mathematics 04, no. 02 (2013): 338–42. http://dx.doi.org/10.4236/am.2013.42051.

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19

Battaglia, Erika, Stefano Biagi, and Giulio Tralli. "On the regularity of vector fields underlying a degenerate-elliptic PDE." Proceedings of the American Mathematical Society 146, no. 4 (November 13, 2017): 1651–64. http://dx.doi.org/10.1090/proc/13866.

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20

Pinnau, René, Claudia Totzeck, Oliver Tse, and Stephan Martin. "A consensus-based model for global optimization and its mean-field limit." Mathematical Models and Methods in Applied Sciences 27, no. 01 (January 2017): 183–204. http://dx.doi.org/10.1142/s0218202517400061.

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We introduce a novel first-order stochastic swarm intelligence (SI) model in the spirit of consensus formation models, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions. The CBO algorithm allows for passage to the mean-field limit, which results in a nonstandard, nonlocal, degenerate parabolic partial differential equation (PDE). Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the SI model. We further present numerical investigations underlining the feasibility of our approach.

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21

Lauter, Robert, and Sergiu Moroianu. "FREDHOLM THEORY FOR DEGENERATE PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBERED BOUNDARIES." Communications in Partial Differential Equations 26, no. 1-2 (January 31, 2001): 233–83. http://dx.doi.org/10.1081/pde-100001754.

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22

BERESTYCKI, HENRI, and FRANCOIS HAMEL. "Gradient Estimates for Elliptic Regularizations of Semilinear Parabolic and Degenerate Elliptic Equations." Communications in Partial Differential Equations 30, no. 1-2 (April 2005): 139–56. http://dx.doi.org/10.1081/pde-200044478.

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23

Novak, Andrej, and Jela Susic. "On a regularity of biharmonic approximations to a nonlinear degenerate elliptic PDE." Filomat 31, no. 7 (2017): 1835–42. http://dx.doi.org/10.2298/fil1707835n.

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Under appropriate assumption on the coefficients, we prove that a sequence of biharmonic regularization to a nonlinear degenerate elliptic equation with possibly rough coefficients preserves certain regularity as the approximation parameter tends to zero. In order to obtain the result, we introduce a generalization of the Chebyshev inequality. We also present numerical example.

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24

Donatelli, Marco, Matteo Semplice, and Stefano Serra-Capizzano. "Analysis of Multigrid Preconditioning for Implicit PDE Solvers for Degenerate Parabolic Equations." SIAM Journal on Matrix Analysis and Applications 32, no. 4 (October 2011): 1125–48. http://dx.doi.org/10.1137/100807880.

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25

Ndaw, Cheikhou Oumar. "On radial solutions for a Fully Non Linear degenerate or singular PDE." Nonlinear Analysis: Real World Applications 73 (October 2023): 103921. http://dx.doi.org/10.1016/j.nonrwa.2023.103921.

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26

Barles, G., and Jérôme Busca. "EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM1*." Communications in Partial Differential Equations 26, no. 11-12 (November 1, 2001): 2323–37. http://dx.doi.org/10.1081/pde-100107824.

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27

Baran, Gizem, Zeynep Özat, Bayram Çekim, and Mehmet Özarslan. "Some properties of degenerate Hermite appell polynomials in three variables." Filomat 37, no. 19 (2023): 6537–67. http://dx.doi.org/10.2298/fil2319537b.

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This study is about degenerate Hermite Appell polynomials in three variables or ?h-Hermite Appell polynomials which include both discrete and degenerate cases. After we recall the definition of these polynomials and special cases, we investigate some properties of them such as recurrence relation, lowering operators (LO), raising operators (RO), difference equation (DE), integro-difference equation (IDE) and partial difference equation (PDE).We also obtain the explicit expression in terms of the Stirling numbers of the first kind. Moreover, we introduce 3D- ?h-Hermite ?-Charlier polynomials, 3D-?h-Hermite degenerate Apostol-Bernoulli polynomials, 3D-?h-Hermite degenerate Apostol-Euler polynomials and 3D-?h-Hermite ?-Boole polynomials as special cases of ?h-Hermite Appell polynomials. Furthermore, wederive the explicit representation, determinantal form, recurrence relation, LO, RO and DE for these special cases. Finally, we introduce new approximating operators based on h-Hermite polynomials in three variables and examine the weighted Korovkin theorem. The error of approximation is also calculated in terms of the modulus of continuity and Peetre?s K-functional

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28

Daskalopoulos, Georgios, and Chikako Mese. "On the singular set of a nonlinear degenerate PDE arising in Teichmüller theory." Proceedings of the American Mathematical Society 150, no. 01 (October 25, 2021): 411–22. http://dx.doi.org/10.1090/proc/15573.

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29

Biswas, Imran H., Ananta K. Majee, and Guy Vallet. "On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise." Advances in Nonlinear Analysis 8, no. 1 (September 12, 2017): 809–44. http://dx.doi.org/10.1515/anona-2017-0113.

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Abstract In this article, we deal with the stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasis is on analyzing the effects of a multiplicative Lévy noise on such problems and on establishing a well-posedness theory by developing a suitable weak entropy solution framework. The proof of the existence of a solution is based on the vanishing viscosity technique. The uniqueness of the solution is settled by interpreting Kruzhkov’s doubling technique in the presence of a noise.

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30

Eckmann, J. P., and M. Hairer. "Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise." Communications in Mathematical Physics 219, no. 3 (June 1, 2001): 523–65. http://dx.doi.org/10.1007/s002200100424.

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31

Hartland, Tucker, and Ravi Shankar. "A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations." Axioms 12, no. 11 (November 18, 2023): 1059. http://dx.doi.org/10.3390/axioms12111059.

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We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we can formulate a strong maximum principle for nonlinearities satisfying a geometric condition related to the flux operator of the equation. In our formulation of the strong maximum principle, we find a physical re-interpretation and generalization of the standard PDE conclusion of the principle: we replace constant solutions with solutions of zero flux. We also consider nonlinearities outside the scope of our principle. For highly degenerate conductivities, we demonstrate the invalidity of the strong maximum principle. We also consider intermediate, inconclusive examples, and provide numerical evidence that the strong maximum principle is valid. This suggests that our geometric condition is sharp.

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32

Chen, Xia, Chen Tang, and Xiusheng Yan. "Switching degenerate diffusion PDE filter based on impulselike probability for universal impulse noise removal." AEU - International Journal of Electronics and Communications 68, no. 9 (September 2014): 851–57. http://dx.doi.org/10.1016/j.aeue.2014.03.012.

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33

Porzio, Maria Michaela. "A new approach to decay estimates – Application to a nonlinear and degenerate parabolic PDE." Rendiconti Lincei - Matematica e Applicazioni 29, no. 4 (December 28, 2018): 635–59. http://dx.doi.org/10.4171/rlm/826.

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34

Lions, Pierre-Louis, and Panagiotis E. Souganidis. "Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 22, no. 5 (September 2005): 667–77. http://dx.doi.org/10.1016/j.anihpc.2004.10.009.

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35

Li, Ben, and M. K. Stenstrom. "Dynamic one-dimensional modeling of secondary settling tanks and system robustness evaluation." Water Science and Technology 69, no. 11 (March 28, 2014): 2339–49. http://dx.doi.org/10.2166/wst.2014.155.

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One-dimensional secondary settling tank models are widely used in current engineering practice for design and optimization, and usually can be expressed as a nonlinear hyperbolic or nonlinear strongly degenerate parabolic partial differential equation (PDE). Reliable numerical methods are needed to produce approximate solutions that converge to the exact analytical solutions. In this study, we introduced a reliable numerical technique, the Yee–Roe–Davis (YRD) method as the governing PDE solver, and compared its reliability with the prevalent Stenstrom–Vitasovic–Takács (SVT) method by assessing their simulation results at various operating conditions. The YRD method also produced a similar solution to the previously developed Method G and Enquist–Osher method. The YRD and SVT methods were also used for a time-to-failure evaluation, and the results show that the choice of numerical method can greatly impact the solution. Reliable numerical methods, such as the YRD method, are strongly recommended.

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36

Di Marino, Simone, and Lénaïc Chizat. "A tumor growth model of Hele-Shaw type as a gradient flow." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 103. http://dx.doi.org/10.1051/cocv/2020019.

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In this paper, we characterize a degenerate PDE as the gradient flow in the space of nonnegative measures endowed with an optimal transport-growth metric. The PDE of concern, of Hele-Shaw type, was introduced by Perthame et. al. as a mechanical model for tumor growth and the metric was introduced recently in several articles as the analogue of the Wasserstein metric for nonnegative measures. We show existence of solutions using minimizing movements and show uniqueness of solutions on convex domains by proving the Evolutional Variational Inequality. Our analysis does not require any regularity assumption on the initial condition. We also derive a numerical scheme based on the discretization of the gradient flow and the idea of entropic regularization. We assess the convergence of the scheme on explicit solutions. In doing this analysis, we prove several new properties of the optimal transport-growth metric, which generally have a known counterpart for the Wasserstein metric.

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37

Lindstrom, Michael R., and Andrea L. Bertozzi. "Qualitative features of a nonlinear, nonlocal, agent-based PDE model with applications to homelessness." Mathematical Models and Methods in Applied Sciences 30, no. 10 (September 2020): 1863–91. http://dx.doi.org/10.1142/s0218202520400114.

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In this paper, we develop a continuum model for the movement of agents on a lattice, taking into account location desirability, local and far-range migration, and localized entry and exit rates. Specifically, our motivation is to qualitatively describe the homeless population in Los Angeles. The model takes the form of a fully nonlinear, nonlocal, non-degenerate parabolic partial differential equation. We derive the model and prove useful properties of smooth solutions, including uniqueness and [Formula: see text]-stability under certain hypotheses. We also illustrate numerical solutions to the model and find that a simple model can be qualitatively similar in behavior to observed homeless encampments.

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38

Hussein, M. S., Daniel Lesnic, Vitaly L. Kamynin, and Andrey B. Kostin. "Direct and inverse source problems for degenerate parabolic equations." Journal of Inverse and Ill-posed Problems 28, no. 3 (June 1, 2020): 425–48. http://dx.doi.org/10.1515/jiip-2019-0046.

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AbstractDegenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating newly formulated direct and inverse problems associated with non-uniform parabolic PDEs where the leading space- and time-dependent coefficient is allowed to vanish on a non-empty, but zero measure, kernel set. In the context of inverse analysis, we consider the linear but ill-posed identification of a space-dependent source from a time-integral observation of the weighted main dependent variable. For both, this inverse source problem as well as its corresponding direct formulation, we rigorously investigate the question of well-posedness. We also give examples of inverse problems for which sufficient conditions guaranteeing the unique solvability are fulfilled, and present the results of numerical simulations. It is hoped that the analysis initiated in this study will open up new avenues for research in the field of direct and inverse problems for degenerate parabolic equations with applications.

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39

Klein, Markus, and Andreas Prohl. "Optimal Control for the Thin Film Equation: Convergence of a Multi-Parameter Approach to Track State Constraints Avoiding Degeneracies." Computational Methods in Applied Mathematics 16, no. 4 (October 1, 2016): 685–702. http://dx.doi.org/10.1515/cmam-2016-0025.

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AbstractWe consider an optimal control problem subject to the thin-film equation. The PDE constraint lacks well-posedness for general right-hand sides due to possible degeneracies; state constraints are used to circumvent this problematic issue and to ensure well-posedness. Necessary optimality conditions for the optimal control problem are then derived. A convergent multi-parameter regularization is considered which addresses both, the possibly degenerate term in the equation and the state constraint. Some computational studies are then reported which evidence the relevant role of the state constraint, and motivate proper scalings of involved regularization and numerical parameters.

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40

WITELSKI, T. P., A. J. BERNOFF, and A. L. BERTOZZI. "Blowup and dissipation in a critical-case unstable thin film equation." European Journal of Applied Mathematics 15, no. 2 (April 2004): 223–56. http://dx.doi.org/10.1017/s0956792504005418.

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We study the dynamics of dissipation and blow-up in a critical-case unstable thin film equation. The governing equation is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for lubrication flows of thin viscous fluid layers on solid surfaces. %For a special balance between %destabilizing second-order terms and regularizing fourth-order terms, There is a critical mass for blow-up and a rich set of dynamics including families of similarity solutions for finite-time blow-up and infinite-time spreading. The structure and stability of the steady-states and the compactly-supported similarity solutions is studied.

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41

Efendiev, Messoud A., Mitsuharu Otani, and Hermann J. Eberl. "Mathematical Analysis of a PDE-ODE Coupled Model of Mitochondrial Swelling with Degenerate Calcium Ion Diffusion." SIAM Journal on Mathematical Analysis 52, no. 1 (January 2020): 543–69. http://dx.doi.org/10.1137/18m1227421.

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42

GALAKTIONOV, VICTOR A. "Shock waves and compactons for fifth-order non-linear dispersion equations." European Journal of Applied Mathematics 21, no. 1 (November 12, 2009): 1–50. http://dx.doi.org/10.1017/s0956792509990118.

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The followingfirst problemis posed:is a correct ‘entropy solution’ of the Cauchy problem for the fifth-order degenerate non-linear dispersion equations (NDEs), same as for the classic Euler oneut+uux= 0,These two quasi-linear degenerate partial differential equations (PDEs) are chosen as typical representatives; so other (2m+ 1)th-order NDEs of non-divergent form admit such shocks waves. As a relatedsecond problem, the opposite initial shockS+(x) = −S−(x) = signxis shown to be a non-entropy solution creating ararefaction wave, which becomesC∞for anyt> 0. Formation of shocks leads to non-uniqueness of any ‘entropy solutions’. Similar phenomena are studied for afifth-order in timeNDEuttttt= (uux)xxxxinnormal form.On the other hand, related NDEs, such asare shown to admit smoothcompactons, as oscillatorytravelling wavesolutions with compact support. The well-known non-negative compactons, which appeared in various applications (first examples by Dey, 1998,Phys. Rev.E, vol. 57, pp. 4733–4738, and Rosenau and Levy, 1999,Phys. Lett.A, vol. 252, pp. 297–306), are non-existent in general and are not robust relative to small perturbations of parameters of the PDE.

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Delgado, Joaquín, and Patricia Saavedra. "Global Bifurcation Diagram for the Kerner–Konhäuser Traffic Flow Model." International Journal of Bifurcation and Chaos 25, no. 05 (May 2015): 1550064. http://dx.doi.org/10.1142/s0218127415500649.

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We study traveling wave solutions of the Kerner–Konhäuser PDE for traffic flow. By a standard change of variables, the problem is reduced to a dynamical system in the plane with three parameters. In a previous paper [Carrillo et al., 2010] it was shown that under general hypotheses on the fundamental diagram, the dynamical system has a surface of critical points showing either a fold or cusp catastrophe when projected under a two-dimensional plane of parameters named qg–vg. In either case, a one parameter family of Takens–Bogdanov (TB) bifurcation takes place, and therefore local families of Hopf and homoclinic bifurcation arising from each TB point exist. Here, we prove the existence of a degenerate Takens–Bogdanov bifurcation (DTB) which in turn implies the existence of Generalized Hopf or Bautin bifurcations (GH). We describe numerically the global lines of bifurcations continued from the local ones, inside a cuspidal region of the parameter space. In particular, we compute the first Lyapunov exponent, and compare with the GH bifurcation curve. We present some families of stable limit cycles which are taken as initial conditions in the PDE leading to stable traveling waves.

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Kommonen, Bertel, Tarja Kylmä, Robert J. Cohen, John S. Penn, Lars Paulin, Mary Y. Hurwitz, and Richard L. Hurwitz. "Elevation of cGMP with Normal Expression and Activity of Rod cGMP-PDE in Photoreceptor Degenerate Labrador Retrievers." Ophthalmic Research 28, no. 1 (1996): 19–28. http://dx.doi.org/10.1159/000267869.

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Mary TANG, K., Elliott K. JANG, and Richard J. HASLAM. "Expression and mutagenesis of the catalytic domain of cGMP-inhibited phosphodiesterase (PDE3) cloned from human platelets." Biochemical Journal 323, no. 1 (April 1, 1997): 217–24. http://dx.doi.org/10.1042/bj3230217.

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We have used reverse transcriptase PCR, platelet mRNA and degenerate primers based on platelet peptide sequences, to amplify a fragment of platelet cGMP-inhibited phosphodiesterase (cGI-PDE; PDE3). Sequence analysis of this clone established that both the platelet and the cardiac forms of PDE3 were derived from the same gene (PDE3A). A RT-PCR product representing the C-terminal half of platelet PDE3 cDNA and corresponding to amino acid residues 560-1141 of the cardiac enzyme, was cloned and expressed in Escherichia coli cGI-PDEΔ1. Further deletion mutants were constructed by removing either an additional 100 amino acids from the N-terminus (cGI-PDEΔ2) or the 44-amino-acid insert characteristic of the PDE3 family, from the catalytic domain (cGI-PDEΔ1Δi). In addition, site-directed mutagenesis was performed to explore the function of the 44-amino-acid insert. All mutants were evaluated for their ability to hydrolyse cAMP and cGMP, their ability to be photolabelled by [32P]cGMP and for the effects of PDE3 inhibitors. The Km values for hydrolysis of cAMP and cGMP by immunoprecipitates of cGI-PDEΔ1 (182±12 nM and 153±12 nM respectively) and cGI-PDEΔ2 (131±17 nM and 99±1 nM respectively) were significantly lower than those for immunoprecipitates of intact platelet PDE3 (398±50 nM and 252±16 nM respectively). Moreover, N-terminal truncations of platelet enzyme increased the ratio of Vmax for cGMP/Vmax for cAMP from 0.16±0.01 in intact platelet enzyme, to 0.37±0.05 in cGI-PDEΔ1 and to 0.49±0.04 in cGI-PDEΔ2. Thus deletion of the N-terminus enhanced hydrolysis of cGMP relative to cAMP, suggesting that N-terminal sequences may exert selective effects on enzyme activity. Removal of the 44-amino-acid insert generated a mutant with a catalytic domain closely resembling those of other PDE gene families but despite a limited ability to be photolabelled by [32P]cGMP, no cyclic nucleotide hydrolytic activities of the mutant were detectable. Mutation of amino acid residues in putative β-turns at the beginning and end of the 44-amino-acid insert to alanine residues markedly reduced the ability of the enzyme to hydrolyse cyclic nucleotides. The PDE3 inhibitor, lixazinone, retained the ability to inhibit cAMP hydrolysis and [32P]cGMP binding by the N-terminal deletion mutants and the site-directed mutants, suggesting that PDE3 inhibitors may interact exclusively with the catalytic domain of the enzyme.

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ZHU, RONGCHAN. "BSDE AND GENERALIZED DIRICHLET FORMS: THE FINITE-DIMENSIONAL CASE." Infinite Dimensional Analysis, Quantum Probability and Related Topics 15, no. 04 (December 2012): 1250022. http://dx.doi.org/10.1142/s0219025712500221.

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We consider the following quasi-linear parabolic system of backward partial differential equations [Formula: see text] where L is a possibly degenerate second-order differential operator with merely measurable coefficients. We solve this system in the framework of generalized Dirichlet forms and employ the stochastic calculus associated to the Markov process with generator L to obtain a probabilistic representation of the solution u by solving the corresponding backward stochastic differential equation. The solution satisfies the corresponding mild equation which is equivalent to being a generalized solution of the PDE. A further main result is the generalization of the martingale representation theorem using the stochastic calculus associated to the generalized Dirichlet form given by L. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to ∇u.

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Tamayo, Rita, Stefan Schild, Jason T. Pratt, and Andrew Camilli. "Role of Cyclic Di-GMP during El Tor Biotype Vibrio cholerae Infection: Characterization of the In Vivo-Induced Cyclic Di-GMP Phosphodiesterase CdpA." Infection and Immunity 76, no. 4 (January 28, 2008): 1617–27. http://dx.doi.org/10.1128/iai.01337-07.

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ABSTRACT In Vibrio cholerae, the second messenger cyclic di-GMP (c-di-GMP) positively regulates biofilm formation and negatively regulates virulence and is proposed to play an important role in the transition from persistence in the environment to survival in the host. Herein we describe a characterization of the infection-induced gene cdpA, which encodes both GGDEF and EAL domains, which are known to mediate diguanylate cyclase and c-di-GMP phosphodiesterase (PDE) activities, respectively. CdpA is shown to possess PDE activity, and this activity is regulated by its inactive degenerate GGDEF domain. CdpA inhibits biofilm formation but has no effect on colonization of the infant mouse small intestine. Consistent with these observations, cdpA is expressed during in vitro growth in a biofilm but is not expressed in vivo until the late stage of infection, after colonization has occurred. To test for a role of c-di-GMP in the early stages of infection, we artificially increased c-di-GMP and observed reduced colonization. This was attributed to a significant reduction in toxT transcription during infection. Cumulatively, these results support a model of the V. cholerae life cycle in which c-di-GMP must be down-regulated early after entering the small intestine and maintained at a low level to allow virulence gene expression, colonization, and motility at appropriate stages of infection.

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JI, HANGJIE, and THOMAS P. WITELSKI. "Steady states and dynamics of a thin-film-type equation with non-conserved mass." European Journal of Applied Mathematics 31, no. 6 (November 22, 2019): 968–1001. http://dx.doi.org/10.1017/s0956792519000330.

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We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.

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LEVY, RACHEL, MICHAEL SHEARER, and THOMAS P. WITELSKI. "Gravity-driven thin liquid films with insoluble surfactant: smooth traveling waves." European Journal of Applied Mathematics 18, no. 6 (December 2007): 679–708. http://dx.doi.org/10.1017/s0956792507007218.

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The flow of a thin layer of fluid down an inclined plane is modified by the presence of insoluble surfactant. For any finite surfactant mass, traveling waves are constructed for a system of lubrication equations describing the evolution of the free-surface fluid height and the surfactant concentration. The one-parameter family of solutions is investigated using perturbation theory with three small parameters: the coefficient of surface tension, the surfactant diffusivity, and the coefficient of the gravity-driven diffusive spreading of the fluid. When all three parameters are zero, the nonlinear PDE system is hyperbolic/degenerate-parabolic, and admits traveling wave solutions in which the free-surface height is piecewise constant, and the surfactant concentration is piecewise linear and continuous. The jumps and corners in the traveling waves are regularized when the small parameters are nonzero; their structure is revealed through a combination of analysis and numerical simulation.

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Hengge, Regine, Michael Y. Galperin, Jean-Marc Ghigo, Mark Gomelsky, Jeffrey Green, Kelly T. Hughes, Urs Jenal, and Paolo Landini. "Systematic Nomenclature for GGDEF and EAL Domain-Containing Cyclic Di-GMP Turnover Proteins of Escherichia coli: TABLE 1." Journal of Bacteriology 198, no. 1 (July 6, 2015): 7–11. http://dx.doi.org/10.1128/jb.00424-15.

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In recent years,Escherichia colihas served as one of a few model bacterial species for studying cyclic di-GMP (c-di-GMP) signaling. The widely usedE. coliK-12 laboratory strains possess 29 genes encoding proteins with GGDEF and/or EAL domains, which include 12 diguanylate cyclases (DGC), 13 c-di-GMP-specific phosphodiesterases (PDE), and 4 “degenerate” enzymatically inactive proteins. In addition, six new GGDEF and EAL (GGDEF/EAL) domain-encoding genes, which encode two DGCs and four PDEs, have recently been found in genomic analyses of commensal and pathogenicE. colistrains. As a group of researchers who have been studying the molecular mechanisms and the genomic basis of c-di-GMP signaling inE. coli, we now propose a general and systematicdgcandpdenomenclature for the enzymatically active GGDEF/EAL domain-encoding genes of this model species. This nomenclature is intuitive and easy to memorize, and it can also be applied to additional genes and proteins that might be discovered in various strains ofE. coliin future studies.

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