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Journal articles on the topic 'Degenerate parabolic systems'

Author: Grafiati

Published: 4 June 2021

Last updated: 3 February 2022

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1

Kačur, J., and S. Luckhaus. "Approximation of degenerate parabolic systems by nondegenerate elliptic and parabolic systems." Applied Numerical Mathematics 26, no. 3 (1998): 307–26. http://dx.doi.org/10.1016/s0168-9274(97)00073-1.

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2

LIANG, FENG, and MAOAN HAN. "DEGENERATE HOPF BIFURCATION IN NONSMOOTH PLANAR SYSTEMS." International Journal of Bifurcation and Chaos 22, no. 03 (2012): 1250057. http://dx.doi.org/10.1142/s0218127412500575.

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In this paper, we mainly discuss Hopf bifurcation for planar nonsmooth general systems and Liénard systems with foci of parabolic–parabolic (PP) or focus–parabolic (FP) type. For the bifurcation near a focus, when the focus is kept fixed under perturbations we prove that there are at most k limit cycles which can be produced from an elementary weak focus of order 2k + 2 ( resp. k + 1)(k ≥ 1) if the focus is of PP (resp. FP) type, and we present the conditions to ensure these upper bounds are achievable. For the bifurcation near a center, the Hopf cyclicicy is studied for these systems. Some in

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3

Xingming, Guo. "Degenerate parabolic equation and unilateral constraint systems." Applied Mathematics and Mechanics 17, no. 10 (1996): 987–92. http://dx.doi.org/10.1007/bf00147136.

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4

Demetriou, M. A., and I. G. Rosen. "Adaptive Parameter Estimation for Degenerate Parabolic Systems." Journal of Mathematical Analysis and Applications 189, no. 3 (1995): 815–47. http://dx.doi.org/10.1006/jmaa.1995.1053.

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5

Schwarzacher, Sebastian. "Hölder–Zygmund estimates for degenerate parabolic systems." Journal of Differential Equations 256, no. 7 (2014): 2423–48. http://dx.doi.org/10.1016/j.jde.2014.01.009.

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6

Qi, Yuan-Wei, and H. A. Levine. "The critical exponent of degenerate parabolic systems." ZAMP Zeitschrift f�r angewandte Mathematik und Physik 44, no. 2 (1993): 249–65. http://dx.doi.org/10.1007/bf00914283.

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7

Wei, Na, Xiangyu Ge, Yonghong Wu, and Leina Zhao. "Lp Estimates for Weak Solutions to Nonlinear Degenerate Parabolic Systems." Discrete Dynamics in Nature and Society 2017 (2017): 1–12. http://dx.doi.org/10.1155/2017/2741326.

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This paper is devoted to the Lp estimates for weak solutions to nonlinear degenerate parabolic systems related to Hörmander’s vector fields. The reverse Hölder inequalities for degenerate parabolic system under the controllable growth conditions and natural growth conditions are established, respectively, and an important multiplicative inequality is proved; finally, we obtain the Lp estimates for the weak solutions by combining the results of Gianazza and the Caccioppoli inequality.

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8

L. Hollingsworth, Brooke, and R. E. Showalter. "Semilinear degenerate parabolic systems and distributed capacitance models." Discrete & Continuous Dynamical Systems - A 1, no. 1 (1995): 59–76. http://dx.doi.org/10.3934/dcds.1995.1.59.

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9

Kačur, Jozef. "Solution of degenerate parabolic systems by relaxation schemes." Nonlinear Analysis: Theory, Methods & Applications 30, no. 7 (1997): 4629–36. http://dx.doi.org/10.1016/s0362-546x(97)00463-x.

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10

Murakawa, H. "Reaction–diffusion system approximation to degenerate parabolic systems." Nonlinearity 20, no. 10 (2007): 2319–32. http://dx.doi.org/10.1088/0951-7715/20/10/003.

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11

Li, Yuxiang, Weibing Deng, and Chunhong Xie. "Global existence and nonexistence for degenerate parabolic systems." Proceedings of the American Mathematical Society 130, no. 12 (2002): 3661–70. http://dx.doi.org/10.1090/s0002-9939-02-06630-3.

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12

SHELUKHIN, V. V., and C. I. KONDO. "Non-local parabolic systems: applications in the three-phase capillary fluid filtration." European Journal of Applied Mathematics 16, no. 4 (2005): 493–517. http://dx.doi.org/10.1017/s0956792505006364.

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Non-local degenerate parabolic systems arise in three-phase capillary flows in porous media under a pressure control at the inflow- and outflow-boundaries. A mathematical study of such systems is performed for a class of capillarity pressure functions corresponding to triangular capillarity-diffusion tensors. To this end a theory of non-degenerate parabolic approximations is developed: the unique global solvability of initial boundary-value problems is proved.

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13

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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14

Amar, Micol, Daniele Andreucci, Roberto Gianni, and Claudia Timofte. "A degenerate pseudo-parabolic equation with memory." Communications in Applied and Industrial Mathematics 10, no. 1 (2019): 71–77. http://dx.doi.org/10.2478/caim-2019-0013.

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Abstract We prove the existence and uniqueness for a degenerate pseudo-parabolic problem with memory. This kind of problem arises in the study of the homogenization of some differential systems involving the Laplace-Beltrami operator and describes the effective behaviour of the electrical conduction in some composite materials.

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15

Kim, Sunghoon, and Ki-Ahm Lee. "Local continuity and asymptotic behaviour of degenerate parabolic systems." Nonlinear Analysis 192 (March 2020): 111702. http://dx.doi.org/10.1016/j.na.2019.111702.

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16

Sango, M. "Local boundedness for doubly degenerate quasi-linear parabolic systems." Applied Mathematics Letters 16, no. 4 (2003): 465–68. http://dx.doi.org/10.1016/s0893-9659(03)00021-1.

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17

Bögelein, Verena, Frank Duzaar, and Giuseppe Mingione. "The regularity of general parabolic systems with degenerate diffusion." Memoirs of the American Mathematical Society 221, no. 1041 (2012): 1. http://dx.doi.org/10.1090/s0065-9266-2012-00664-2.

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18

Aregba-Driollet, D., R. Natalini, and S. Tang. "Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems." Mathematics of Computation 73, no. 245 (2003): 63–94. http://dx.doi.org/10.1090/s0025-5718-03-01549-7.

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19

Litovchenko, V. A., and E. B. Nastasiĭ. "Degenerate parabolic systems of vector order Kolmogorov-type equations." Siberian Mathematical Journal 53, no. 1 (2012): 119–33. http://dx.doi.org/10.1134/s0037446612010107.

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20

Ivanov, A. V. "Hölder estimates for second-order quasilinear degenerate parabolic systems." Journal of Soviet Mathematics 49, no. 5 (1990): 1148–59. http://dx.doi.org/10.1007/bf02208711.

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21

Malytska, H. P., and I. V. Burtnyak. "Degenerate Parabolic Systems of the Diffusion Type with Inertia." Journal of Mathematical Sciences 249, no. 3 (2020): 355–68. http://dx.doi.org/10.1007/s10958-020-04947-2.

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22

Nambu, Takao. "Stabilization of parabolic systems via a degenerate nonnegative feedback." Journal of Dynamics and Differential Equations 3, no. 3 (1991): 399–422. http://dx.doi.org/10.1007/bf01049739.

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23

Jun Choe, Hi. "Hölder continuity for solutions of certain degenerate parabolic systems." Nonlinear Analysis: Theory, Methods & Applications 18, no. 3 (1992): 235–43. http://dx.doi.org/10.1016/0362-546x(92)90061-i.

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24

Matiichuk, M. I. "Cauchy problem for a class of degenerate parabolic systems." Ukrainian Mathematical Journal 36, no. 3 (1985): 288–93. http://dx.doi.org/10.1007/bf01077463.

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25

Bolsinov, Alexey, Lorenzo Guglielmi, and Elena Kudryavtseva. "Symplectic invariants for parabolic orbits and cusp singularities of integrable systems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2131 (2018): 20170424. http://dx.doi.org/10.1098/rsta.2017.0424.

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We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

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26

Cannarsa, P., P. Martinez, and J. Vancostenoble. "The cost of controlling strongly degenerate parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 2. http://dx.doi.org/10.1051/cocv/2018007.

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We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut − (xαux)x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as eC/((2−α)2T), when α → 2− and/or T → 0+. Our analysis builds on earlier results and methods (based on functional analysis and

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27

Hashira, Takahiro, Sachiko Ishida, and Tomomi Yokota. "Finite-time blow-up for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type." Journal of Differential Equations 264, no. 10 (2018): 6459–85. http://dx.doi.org/10.1016/j.jde.2018.01.038.

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28

Favini, A., A. Lorenzi, and H. Tanabe. "First-Order Regular and Degenerate Identification Differential Problems." Abstract and Applied Analysis 2015 (2015): 1–42. http://dx.doi.org/10.1155/2015/393624.

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We are concerned with both regular and degenerate first-order identification problems related to systems of differential equations of weakly parabolic type in Banach spaces. Several applications to partial differential equations and systems will be given in a subsequent paper to show the fullness of our abstract results.

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29

Le, Dung. "Higher integrability for gradients of solutions to degenerate parabolic systems." Discrete & Continuous Dynamical Systems - A 26, no. 2 (2010): 597–608. http://dx.doi.org/10.3934/dcds.2010.26.597.

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30

Wang, Mingxin. "Some degenerate and quasilinear parabolic systems not in divergence form." Journal of Mathematical Analysis and Applications 274, no. 1 (2002): 424–36. http://dx.doi.org/10.1016/s0022-247x(02)00347-5.

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31

Wang, Shu. "Doubly Nonlinear Degenerate Parabolic Systems with Coupled Nonlinear Boundary Conditions." Journal of Differential Equations 182, no. 2 (2002): 431–69. http://dx.doi.org/10.1006/jdeq.2001.4101.

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32

Tory, Elmer M., Kenneth H. Karlsen, Raimund Bürger, and Stefan Berres. "Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression." SIAM Journal on Applied Mathematics 64, no. 1 (2003): 41–80. http://dx.doi.org/10.1137/s0036139902408163.

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33

Frehse, Jens, and Sebastian Schwarzacher. "On Regularity of the Time Derivative for Degenerate Parabolic Systems." SIAM Journal on Mathematical Analysis 47, no. 5 (2015): 3917–43. http://dx.doi.org/10.1137/141000725.

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34

Luong, Vu Trong, Duc Hiep Pham, and Hien Anh Vu Thi. "Liouville type theorems for degenerate parabolic systems with advection terms." Journal of Elliptic and Parabolic Equations 6, no. 2 (2020): 871–82. http://dx.doi.org/10.1007/s41808-020-00086-6.

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35

Amann, Herbert. "Global existence for a class of highly degenerate parabolic systems." Japan Journal of Industrial and Applied Mathematics 8, no. 1 (1991): 143–51. http://dx.doi.org/10.1007/bf03167189.

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36

Ishida, Sachiko, and Tomomi Yokota. "Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type." Journal of Differential Equations 252, no. 2 (2012): 1421–40. http://dx.doi.org/10.1016/j.jde.2011.02.012.

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37

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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38

Deng, Weibing, Li Yuxiang, and Xie Chunhong. "Global existence and nonexistence for a class of degenerate parabolic systems." Nonlinear Analysis: Theory, Methods & Applications 55, no. 3 (2003): 233–44. http://dx.doi.org/10.1016/s0362-546x(03)00226-8.

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39

Dung, Le. "Ultimately uniform boundedness of solutions and gradients for degenerate parabolic systems." Nonlinear Analysis: Theory, Methods & Applications 39, no. 2 (2000): 157–71. http://dx.doi.org/10.1016/s0362-546x(98)00172-2.

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40

Bögelein, Verena, and Qifan Li. "Very weak solutions of degenerate parabolic systems with non-standard -growth." Nonlinear Analysis: Theory, Methods & Applications 98 (March 2014): 190–225. http://dx.doi.org/10.1016/j.na.2013.12.009.

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41

Duan, Zhi-wen, and Li Zhou. "Global and blow-up solutions for non-linear degenerate parabolic systems." Mathematical Methods in the Applied Sciences 26, no. 7 (2003): 557–87. http://dx.doi.org/10.1002/mma.367.

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42

Fuchs, M. "Existence of solutions of nonlinear degenerate systems of parabolic variational inequalities." Journal of Mathematical Sciences 87, no. 2 (1997): 3434–40. http://dx.doi.org/10.1007/bf02355594.

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43

Chen, Shaohua. "Global existence and nonexistence for some degenerate and quasilinear parabolic systems." Journal of Differential Equations 245, no. 4 (2008): 1112–36. http://dx.doi.org/10.1016/j.jde.2007.11.008.

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44

ANTONTSEV, S. N., and J. I. DÍAZ. "NEW L1-GRADIENT TYPE ESTIMATES OF SOLUTIONS TO ONE-DIMENSIONAL QUASILINEAR PARABOLIC SYSTEMS." Communications in Contemporary Mathematics 12, no. 01 (2010): 85–106. http://dx.doi.org/10.1142/s0219199710003725.

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We consider a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of the degenerate type. We derive some new L1-gradient type estimates for its solutions which are uniform in the sense that they do not depend on the coefficients nor on the size of the spatial domain. We also give some applications of such estimates to gas dynamics, filtration problems, a p-Laplacian parabolic type equation and some first order systems of Hamilton–Jacobi or conservation laws type.

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45

Vasil'eva, Adelaida B., and Leonid V. Kalachev. "Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions." Abstract and Applied Analysis 2006 (2006): 1–21. http://dx.doi.org/10.1155/aaa/2006/52856.

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We consider a class of singularly perturbed parabolic equations for which the degenerate equations obtained by setting the small parameter equal to zero are algebraic equations that have several roots. We study boundary layer type solutions that, as time increases, periodically go through two fairly long lasting stages with extremely fast transitions in between. During one of these stages the solution outside the boundary layer is close to one of the roots of the degenerate (reduced) equation, while during the other stage the solution is close to the other root. Such equations may be used as m

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46

Bögelein, V., and F. Duzaar. "Higher integrability for parabolic systems with non-standard growth and degenerate diffusions." Publicacions Matemàtiques 55 (January 1, 2011): 201–50. http://dx.doi.org/10.5565/publmat_55111_10.

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47

Dong, Yan. "Hölder regularity for weak solutions to divergence form degenerate quasilinear parabolic systems." Journal of Mathematical Analysis and Applications 410, no. 1 (2014): 375–90. http://dx.doi.org/10.1016/j.jmaa.2013.08.027.

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48

Liu, Bingchen, and Changcheng Zhang. "Blow-up analysis in degenerate parabolic systems coupled via norm-type reactions." Applicable Analysis 95, no. 3 (2015): 668–89. http://dx.doi.org/10.1080/00036811.2015.1026810.

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49

Caristi, Gabriella. "Existence and nonexistence of global solutions of degenerate and singular parabolic systems." Abstract and Applied Analysis 5, no. 4 (2000): 265–84. http://dx.doi.org/10.1155/s1085337501000380.

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50

Samusenko, P. F. "Asymptotic integration of degenerate singularly perturbed systems of parabolic partial differential equations." Journal of Mathematical Sciences 189, no. 5 (2013): 834–47. http://dx.doi.org/10.1007/s10958-013-1223-y.

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