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Journal articles on the topic 'Parabolic Variational Inequalities'

Author: Grafiati

Published: 11 September 2021

Last updated: 2 February 2022

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1

Ženíšek, Alexander. "Approximations of parabolic variational inequalities." Applications of Mathematics 30, no. 1 (1985): 11–35. http://dx.doi.org/10.21136/am.1985.104124.

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2

Eldessouky, A. T. "Strongly Nonlinear Parabolic Variational Inequalities." Journal of Mathematical Analysis and Applications 181, no. 2 (January 1994): 498–504. http://dx.doi.org/10.1006/jmaa.1994.1039.

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3

Ito, Kazufumi, and Karl Kunisch. "Optimal control of parabolic variational inequalities." Journal de Mathématiques Pures et Appliquées 93, no. 4 (April 2010): 329–60. http://dx.doi.org/10.1016/j.matpur.2009.10.005.

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4

Friedman, Avner. "Optimal Control for Parabolic Variational Inequalities." SIAM Journal on Control and Optimization 25, no. 2 (March 1987): 482–97. http://dx.doi.org/10.1137/0325027.

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5

Haussmann, U. G., and E. Pardoux. "Stochastic variational inequalities of parabolic type." Applied Mathematics & Optimization 20, no. 1 (July 1989): 163–92. http://dx.doi.org/10.1007/bf01447653.

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6

Kulieva, Gulchehra, and Komil Kuliev. "ON EXTENDED ROTHE’S METHOD FOR NONLINEAR PARABOLIC VARIATIONAL INEQUALITIES IN NONCYLINDRICAL DOMAINS." Eurasian Mathematical Journal 11, no. 3 (2020): 51–65. http://dx.doi.org/10.32523/2077-9879-2020-11-3-51-65.

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7

Quittner, Pavol. "A remark on the stability of stationary solutions of parabolic variational inequalities." Czechoslovak Mathematical Journal 40, no. 3 (1990): 472–74. http://dx.doi.org/10.21136/cmj.1990.102400.

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8

Boulaaras, Salah, and Mohamed Haiour. "A New Approach to Asymptotic Behavior for a Finite Element Approximation in Parabolic Variational Inequalities." ISRN Mathematical Analysis 2011 (July 7, 2011): 1–15. http://dx.doi.org/10.5402/2011/703670.

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The paper deals with the theta time scheme combined with a finite element spatial approximation of parabolic variational inequalities. The parabolic variational inequalities are transformed into noncoercive elliptic variational inequalities. A simple result to time energy behavior is proved, and a new iterative discrete algorithm is proposed to show the existence and uniqueness. Moreover, its convergence is established. Furthermore, a simple proof to asymptotic behavior in uniform norm is given.

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9

Jeong, Jin-Mun, and Jong-Yeoul Park. "Nonlinear variational inequalities of semilinear parabolic type." Journal of Inequalities and Applications 2001, no. 2 (2001): 896837. http://dx.doi.org/10.1155/s1025583401000133.

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10

Rudd, Matthew, and Klaus Schmitt. "VARIATIONAL INEQUALITIES OF ELLIPTIC AND PARABOLIC TYPE." Taiwanese Journal of Mathematics 6, no. 3 (September 2002): 287–322. http://dx.doi.org/10.11650/twjm/1500558298.

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11

Wang, GengshengBB. "Optimal control problem for parabolic variational inequalities." Acta Mathematica Scientia 21, no. 4 (October 2001): 509–25. http://dx.doi.org/10.1016/s0252-9602(17)30440-x.

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12

Ito, Kazufumi, and Karl Kunisch. "Parabolic variational inequalities: The Lagrange multiplier approach." Journal de Mathématiques Pures et Appliquées 85, no. 3 (March 2006): 415–49. http://dx.doi.org/10.1016/j.matpur.2005.08.005.

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13

Moon, Kyoung-Sook, Ricardo H. Nochetto, Tobias von Petersdorff, and Chen-song Zhang. "A posteriorierror analysis for parabolic variational inequalities." ESAIM: Mathematical Modelling and Numerical Analysis 41, no. 3 (May 2007): 485–511. http://dx.doi.org/10.1051/m2an:2007029.

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14

Bartosz, Krzysztof, Xiaoliang Cheng, Piotr Kalita, Yuanjie Yu, and Cong Zheng. "Rothe method for parabolic variational–hemivariational inequalities." Journal of Mathematical Analysis and Applications 423, no. 2 (March 2015): 841–62. http://dx.doi.org/10.1016/j.jmaa.2014.09.078.

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15

Liu, Gui-fang, and Yi-liang Liu. "On double degenerate quasilinear parabolic variational inequalities." Acta Mathematicae Applicatae Sinica, English Series 29, no. 4 (October 2013): 861–68. http://dx.doi.org/10.1007/s10255-013-0263-x.

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16

Zhu, Yuanguo. "Solutions to variational inequalities of parabolic type." Journal of Mathematical Analysis and Applications 321, no. 1 (September 2006): 24–31. http://dx.doi.org/10.1016/j.jmaa.2005.08.031.

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17

El-Dessouky, A. T. "A note on strongly nonlinear parabolic variational inequalities." Applied Mathematics and Nonlinear Sciences 2, no. 2 (October 23, 2017): 443–48. http://dx.doi.org/10.21042/amns.2017.2.00035.

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AbstractWe prove the existence of weak solutions of variational inequalities for general quasilinear parabolic operators of order m = 2 with strongly nonlinear perturbation term. The result is based on a priori bound for the time derivatives of the solutions.

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18

Maticiuc, Lucian, Etienne Pardoux, Aurel Răşcanu, and Adrian Zălinescu. "Viscosity solutions for systems of parabolic variational inequalities." Bernoulli 16, no. 1 (February 2010): 258–73. http://dx.doi.org/10.3150/09-bej204.

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19

Kubo, Masahiro, and Noriaki Yamazaki. "Elliptic-parabolic variational inequalities with time-dependent constraints." Discrete & Continuous Dynamical Systems - A 19, no. 2 (2007): 335–59. http://dx.doi.org/10.3934/dcds.2007.19.335.

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20

Solonukha, O. V. "Solvability of Unilateral Parabolic Problems and Variational Inequalities." Journal of Mathematical Sciences 127, no. 5 (June 2005): 2284–314. http://dx.doi.org/10.1007/s10958-005-0179-y.

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21

Jeong, Jin-Mun, Eun Young Ju, and Kyeong Yeon Lee. "Controllability for Nonlinear Variational Inequalities of Parabolic Type." Taiwanese Journal of Mathematics 15, no. 2 (April 2011): 857–73. http://dx.doi.org/10.11650/twjm/1500406238.

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22

Hintermüller, Michael, and Carlos N. Rautenberg. "Parabolic Quasi-variational Inequalities with Gradient-Type Constraints." SIAM Journal on Optimization 23, no. 4 (January 2013): 2090–123. http://dx.doi.org/10.1137/120874308.

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23

Kacur, Jozef, and Roger Van Keer. "Solution of degenerate parabolic variational inequalities with convection." ESAIM: Mathematical Modelling and Numerical Analysis 37, no. 3 (May 2003): 417–31. http://dx.doi.org/10.1051/m2an:2003035.

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24

Ziemer, William P. "Regularity of weak solutions of parabolic variational inequalities." Transactions of the American Mathematical Society 309, no. 2 (February 1, 1988): 763. http://dx.doi.org/10.1090/s0002-9947-1988-0961612-9.

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25

Kapustyan, Vladimir Y. "Optimal Control of Parabolic Singular Perturbated Variational Inequalities." IFAC Proceedings Volumes 30, no. 10 (July 1997): 31–35. http://dx.doi.org/10.1016/s1474-6670(17)43099-0.

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26

Chen, Qihong, Delin Chu, and Roger C. E. Tan. "Bilateral Obstacle Control Problem of Parabolic Variational Inequalities." SIAM Journal on Control and Optimization 46, no. 4 (January 2007): 1518–37. http://dx.doi.org/10.1137/050638047.

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27

Besenyei, Á. "On nonlinear parabolic variational inequalities containing nonlocal terms." Acta Mathematica Hungarica 116, no. 1-2 (July 2007): 145–62. http://dx.doi.org/10.1007/s10474-007-6024-7.

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28

Achdou, Yves, Frédéric Hecht, and David Pommier. "A Posteriori Error Estimates for Parabolic Variational Inequalities." Journal of Scientific Computing 37, no. 3 (July 15, 2008): 336–66. http://dx.doi.org/10.1007/s10915-008-9215-7.

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29

Carl, S., and Vy K. Le. "Sub-supersolution method for quasilinear parabolic variational inequalities." Journal of Mathematical Analysis and Applications 293, no. 1 (May 2004): 269–84. http://dx.doi.org/10.1016/j.jmaa.2004.01.005.

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30

Lavrenyuk, S. P. "Systems of parabolic variational inequalities without initial conditions." Ukrainian Mathematical Journal 49, no. 4 (April 1997): 595–603. http://dx.doi.org/10.1007/bf02487323.

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31

Ju, Eun-Young, and Jin-Mun Jeong. "Optimal Control Problems for Nonlinear Variational Evolution Inequalities." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/724190.

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We deal with optimal control problems governed by semilinear parabolic type equations and in particular described by variational inequalities. We will also characterize the optimal controls by giving necessary conditions for optimality by proving the Gâteaux differentiability of solution mapping on control variables.

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32

Kubo, M., K. Shirakawa, and N. Yamazaki. "Variational inequalities for a system of elliptic–parabolic equations." Journal of Mathematical Analysis and Applications 387, no. 2 (March 2012): 490–511. http://dx.doi.org/10.1016/j.jmaa.2011.09.008.

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33

Carl, Siegfried. "Existence and extremal solutions of parabolic variational–hemivariational inequalities." Monatshefte für Mathematik 172, no. 1 (May 1, 2013): 29–54. http://dx.doi.org/10.1007/s00605-013-0502-5.

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34

Bernardi, Marco Luigi, and Gianni Arrigo Pozzi. "On a class of singular nonlinear parabolic variational inequalities." Annali di Matematica Pura ed Applicata 159, no. 1 (December 1991): 117–31. http://dx.doi.org/10.1007/bf01766297.

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35

Maksimov, V. I. "Dynamic modelling of unknown perturbations in parabolic variational inequalities." Journal of Applied Mathematics and Mechanics 52, no. 5 (January 1988): 579–85. http://dx.doi.org/10.1016/0021-8928(88)90105-0.

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36

Xing-ming, Guo, and Zhou Shi-xing. "Optimal control of parabolic variational inequalities with state constraint." Applied Mathematics and Mechanics 24, no. 7 (July 2003): 756–62. http://dx.doi.org/10.1007/bf02437807.

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37

Vivaldi, Maria Agostina. "Existence of strong solutions for nonlinear parabolic variational inequalities." Nonlinear Analysis: Theory, Methods & Applications 11, no. 2 (January 1987): 285–95. http://dx.doi.org/10.1016/0362-546x(87)90105-2.

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38

Kenmochi, Nobuyuki, and Irena Pawlow. "Asymptotic behavior of solutions to parabolic-elliptic variational inequalities." Nonlinear Analysis: Theory, Methods & Applications 13, no. 10 (October 1989): 1191–213. http://dx.doi.org/10.1016/0362-546x(89)90007-2.

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39

Colombo, Maria, Luca Spolaor, and Bozhidar Velichkov. "On the asymptotic behavior of the solutions to parabolic variational inequalities." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 768 (November 1, 2020): 149–82. http://dx.doi.org/10.1515/crelle-2019-0041.

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AbstractWe consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Łojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon ([22]) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems.

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40

Lieberman, Gary M. "Boundary regularity for linear and quasilinear variational inequalities." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 112, no. 3-4 (1989): 319–26. http://dx.doi.org/10.1017/s0308210500018771.

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SynopsisA method of Jensen is extended to show that the second derivatives of the solutions of various linear obstacle problems are bounded under weaker regularity hypotheses on the dataof the problem than were allowed by Jensen. They are, in fact, weak enough that the linear results imply the boundedness of the second derivatives for quasilinear problems as well. Comparisons are made with previously known results, some of which are proved by similar methods. Both Dirichlet and oblique derivative boundary conditions are considered. Corresponding results for parabolic obstacle problems are proved.

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41

Kulieva, G., and K. Kuliev. "Rothe's method for nonlinear parabolic variational inequalities in noncylindrical domains." Differential Equations & Applications, no. 3 (2020): 227–42. http://dx.doi.org/10.7153/dea-2020-12-15.

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42

Kunze, Markus, and M. D. P. Monteiro Marques. "On parabolic quasi-variational inequalities and state-dependent sweeping processes." Topological Methods in Nonlinear Analysis 12, no. 1 (September 1, 1998): 179. http://dx.doi.org/10.12775/tmna.1998.036.

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43

Jeong, Jin-Mun, Eun Young Ju, and Kyeong Yeon Lee. "Controllability for Variational Inequalities of Parabolic Type with Nonlinear Perturbation." Journal of Inequalities and Applications 2010 (2010): 1–16. http://dx.doi.org/10.1155/2010/768469.

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44

Nguyen, Thi Van Anh. "Periodic Solutions to Differential Variational Inequalities of Parabolic-elliptic Type." Taiwanese Journal of Mathematics 24, no. 6 (December 2020): 1497–527. http://dx.doi.org/10.11650/tjm/200301.

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45

Le, Vy Khoi, and Klaus Schmitt. "On Parabolic Variational Inequalities with Multivalued Terms and Convex Functionals." Advanced Nonlinear Studies 18, no. 2 (April 1, 2018): 269–87. http://dx.doi.org/10.1515/ans-2018-0004.

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Abstract In this paper, we consider the following parabolic variational inequality containing a multivalued term and a convex functional: Find {u\in L^{p}(0,T;W^{1,p}_{0}(\Omega))} and {f\in F(\cdot,\cdot,u)} such that {u(\cdot,0)=u_{0}} and \langle u_{t}+Au,v-u\rangle+\Psi(v)-\Psi(u)\geq\int_{Q}f(v-u)\,dx\,dt\quad% \text{for all }v\in L^{p}(0,T;W^{1,p}_{0}(\Omega)), where A is the principal term; F is a multivalued lower-order term; {\Psi(u)=\int_{0}^{T}\psi(t,u)\,dt} is a convex functional. Moreover, we study the existence and other properties of solutions of this inequality assuming certain growth conditions on the lower-order term F.

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46

Carl, Siegfried, and Vy K. Le. "Quasilinear parabolic variational inequalities with multi-valued lower-order terms." Zeitschrift für angewandte Mathematik und Physik 65, no. 5 (August 22, 2013): 845–64. http://dx.doi.org/10.1007/s00033-013-0357-6.

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47

Gimperlein, Heiko, and Jakub Stocek. "Space–time adaptive finite elements for nonlocal parabolic variational inequalities." Computer Methods in Applied Mechanics and Engineering 352 (August 2019): 137–71. http://dx.doi.org/10.1016/j.cma.2019.04.019.

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48

Migórski, Stanisław, Van Thien Nguyen, and Shengda Zeng. "Solvability of parabolic variational-hemivariational inequalities involving space-fractional Laplacian." Applied Mathematics and Computation 364 (January 2020): 124668. http://dx.doi.org/10.1016/j.amc.2019.124668.

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49

Solonukha, O. V. "Existence of solutions of parabolic variational inequalities with one-sided restrictions." Mathematical Notes 77, no. 3-4 (March 2005): 424–39. http://dx.doi.org/10.1007/s11006-005-0041-z.

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50

Nagase, H. "On a few properties of solutions of nonlinear parabolic variational inequalities." Nonlinear Analysis: Theory, Methods & Applications 47, no. 3 (August 2001): 1659–69. http://dx.doi.org/10.1016/s0362-546x(01)00299-1.

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