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Journal articles on the topic 'Systems of Parabolic Equations'

Author: Grafiati

Published: 9 March 2023

Last updated: 31 July 2025

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1

Simon, László. "On some singular systems of parabolic functional equations." Mathematica Bohemica 135, no. 2 (2010): 123–32. http://dx.doi.org/10.21136/mb.2010.140689.

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2

Hrytchuk, M., and I. Klevchuk. "BIFURCATION OF TORI FOR PARABOLIC SYSTEMS OF DIFFERENTIAL EQUATIONS WITH WEAK DIFFUSION." Bukovinian Mathematical Journal 11, no. 2 (2023): 100–103. http://dx.doi.org/10.31861/bmj2023.02.10.

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The aim of the present article is to investigate of some properties of quasiperiodic solutions of nonlinear autonomous parabolic systems with the periodic condition. The research is devoted to the investigation of parabolic systems of differential equations with the help of integral manifolds method in the theory of nonlinear oscillations. We prove the existence of quasiperiodic solutions in autonomous parabolic system of differential equations with weak diffusion on the circle. We study existence and stability of an arbitrarily large finite number of tori for a parabolic system with weak diff

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3

Klevchuk, I. I. "Existence and stability of traveling waves in parabolic systems of differential equations with weak diffusion." Carpathian Mathematical Publications 14, no. 2 (2022): 493–503. http://dx.doi.org/10.15330/cmp.14.2.493-503.

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The aim of the present paper is to investigate of some properties of periodic solutions of a nonlinear autonomous parabolic systems with a periodic condition. We investigate parabolic systems of differential equations using an integral manifolds method of the theory of nonlinear oscillations. We prove the existence of periodic solutions in an autonomous parabolic system of differential equations with weak diffusion on the circle. We study the existence and stability of an arbitrarily large finite number of cycles for a parabolic system with weak diffusion. The periodic solution of parabolic eq

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4

Amann, Herbert. "Quasilinear evolution equations and parabolic systems." Transactions of the American Mathematical Society 293, no. 1 (1986): 191. http://dx.doi.org/10.1090/s0002-9947-1986-0814920-4.

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5

Kozhevnikov, A. "Multi–Weighted Parabolic Equations and Systems." Journal of Mathematical Sciences 193, no. 2 (2013): 267–82. http://dx.doi.org/10.1007/s10958-013-1452-0.

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6

Liu, Xingyu. "A System of Parabolic Laplacian Equations That Are Interrelated and Radial Symmetry of Solutions." Symmetry 17, no. 7 (2025): 1112. https://doi.org/10.3390/sym17071112.

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We utilize the moving planes technique to prove the radial symmetry along with the monotonic characteristics of solutions for a system of parabolic Laplacian equations. In this system, the solutions of the two equations are interdependent, with the solution of one equation depending on the function of the other. By use of the maximal regularity theory that has been established for fractional parabolic equations, we ensure the solvability of these systems. Our initial step is to formulate a narrow region principle within a parabolic cylinder. This principle serves as a theoretical basis for imp

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7

Al-Sultani, Mohamed Saleh Mehdi, and Igor Boglaev. "Block monotone iterations for solving coupled systems of nonlinear parabolic equations." ANZIAM Journal 61 (July 28, 2020): C166—C180. http://dx.doi.org/10.21914/anziamj.v61i0.15144.

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The article deals with numerical methods for solving a coupled system of nonlinear parabolic problems, where reaction functions are quasi-monotone nondecreasing. We employ block monotone iterative methods based on the Jacobi and Gauss–Seidel methods incorporated with the upper and lower solutions method. A convergence analysis and the theorem on uniqueness of a solution are discussed. Numerical experiments are presented. References Al-Sultani, M. and Boglaev, I. ''Numerical solution of nonlinear elliptic systems by block monotone iterations''. ANZIAM J. 60:C79–C94, 2019. doi:10.21914/anziamj.v

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8

Omuraliev, Asan, and Ella Abylaeva. "Regularized asymptotics of the solution of systems of parabolic differential equations." Filomat 36, no. 16 (2022): 5591–602. http://dx.doi.org/10.2298/fil2216591o.

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The regularization method for singularly perturbed problems of S. A. Lomov is generalized to constructing the asymptotics of the solution of the first boundary value problem for systems of differential equations of parabolic type with a small parameter at all derivatives.It is shown that the asymptotics of the solution of the problem contains n exponential, 2n parabolic and 2n angle boundary layer functions. The exponential boundary layer function describes the boundary layer along t = 0, the boundary layer along x = 0 and x = 1 is described by parabolic boundary layer functions.

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9

Kurima, Shunsuke. "Time discretization of an abstract problem from linearized equations of a coupled sound and heat flow." Electronic Journal of Differential Equations 2020, no. 01-132 (2020): 96. http://dx.doi.org/10.58997/ejde.2020.96.

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Recently, a time discretization of simultaneous abstract evolution equations applied to parabolic-hyperbolic phase-field systems has been studied. This article focuses on a time discretization of an abstract problem that has application to linearized equations of coupled sound and heat flow. As examples, we also study some parabolic-hyperbolic phase-field systems. For more information see https://ejde.math.txstate.edu/Volumes/2020/96/abstr.html

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10

Kavian, Otared, and Luz de Teresa. "Unique continuation principle for systems of parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 16, no. 2 (2009): 247–74. http://dx.doi.org/10.1051/cocv/2008077.

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11

Bensoussan, Alain, and Jens Frehse. "Smooth Solutions of systems of quasilinear parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 169–93. http://dx.doi.org/10.1051/cocv:2002059.

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12

Eden, A., B. Michaux, and J. M. Rakotoson. "Doubly nonlinear parabolic-type equations as dynamical systems." Journal of Dynamics and Differential Equations 3, no. 1 (1991): 87–131. http://dx.doi.org/10.1007/bf01049490.

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13

Rogovchenko, Yuri V. "Comparison principles for systems of impulsive parabolic equations." Annali di Matematica Pura ed Applicata 170, no. 1 (1996): 311–28. http://dx.doi.org/10.1007/bf01758993.

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14

Lyubanova, Anna Sh. "On nonlocal problems for systems of parabolic equations." Journal of Mathematical Analysis and Applications 421, no. 2 (2015): 1767–78. http://dx.doi.org/10.1016/j.jmaa.2014.08.027.

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15

Galaktionov, Victor A. "Dynamical systems of inequalities and nonlinear parabolic equations." Communications in Partial Differential Equations 24, no. 11-12 (1999): 2191–236. http://dx.doi.org/10.1080/03605309908821499.

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16

Nee, Janpou. "Almost periodic solutions to systems of parabolic equations." Journal of Applied Mathematics and Stochastic Analysis 7, no. 4 (1994): 581–86. http://dx.doi.org/10.1155/s1048953394000456.

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In this paper we show that the second-order differential solution is 𝕃2-almost periodic, provided it is 𝕃2-bounded, and the growth of the components of a non-linear function of a system of parabolic equation is bounded by any pair of con-secutive eigenvalues of the associated Dirichlet boundary value problems.

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17

Ревина, С. В. "Diffusion Instability Region for Systems of Parabolic Equations." Владикавказский математический журнал, no. 4 (December 14, 2022): 117–26. http://dx.doi.org/10.46698/d6373-9335-7338-n.

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Рассматривается система двух уравнений реакции-диффузии в ограниченной области $m$-мерного пространства с краевыми условиями Неймана на границе, для которой слагаемые реакции $f(u,v)$ и $g(u,v)$ зависят от двух параметров $a$ и $b$. Предполагается, что система имеет пространственно-однородное решение $(u_0,v_0)$, причем $f_u(u_0,v_0)>0$, а $-g_v(u_0,v_0)=F( \mathrm{Det (\mathrm{J})})$, где $\mathrm{J}$~--- матрица Якоби соответствующей линеаризованной системы в бездиффузионном приближении, $F$~--- гладкая монотонно возрастающая функция. Предложен способ аналитического описания области необх

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18

Revina, S. V. "Diffusion Instability Domains for Systems of Parabolic Equations." Siberian Mathematical Journal 65, no. 2 (2024): 487–94. http://dx.doi.org/10.1134/s0037446624020216.

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19

Avalishvili, Gia, and Mariam Avalishvili. "On nonclassical problems for first-order evolution equations." gmj 18, no. 3 (2011): 441–63. http://dx.doi.org/10.1515/gmj.2011.0028.

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Abstract The present paper deals with nonclassical initial-boundary value problems for parabolic equations and systems and their generalizations in abstract spaces. Nonclassical problems with nonlocal initial conditions for an abstract first-order evolution equation with time-dependent operator are considered, the existence and uniqueness results are proved and the algorithm of approximation of nonlocal problems by a sequence of classical problems is constructed. Applications of the obtained general results to initial-boundary value problems for parabolic equations and systems are considered.

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20

Ishida, Sachiko, and Tomomi Yokota. "Stabilization in degenerate parabolic equations in divergence form and application to chemotaxis systems." Archivum Mathematicum, no. 2 (2023): 181–89. http://dx.doi.org/10.5817/am2023-2-181.

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21

Biccari, Umberto, Dongnam Ko, and Enrique Zuazua. "Dynamics and control for multi-agent networked systems: A finite-difference approach." Mathematical Models and Methods in Applied Sciences 29, no. 04 (2019): 755–90. http://dx.doi.org/10.1142/s0218202519400050.

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We analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the nonlocal transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the nonlocal transport model can be obtained by a suitable averaging of the diffusive one. We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to

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22

Hong, Keum S., and Joseph Bentsman. "Stability Criterion For Linear Oscillatory Parabolic Systems." Journal of Dynamic Systems, Measurement, and Control 114, no. 1 (1992): 175–78. http://dx.doi.org/10.1115/1.2896501.

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This paper presents a stability criterion for a class of distributed parameter systems governed by linear oscillatory parabolic partial differential equations with Neumann boundary conditions. The results of numerical simulations that support the criterion are presented as well.

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23

Li, Wei Nian, Bao Tong Cui, and Lokenath Debnath. "Oscillation of systems of certain neutral delay parabolic differential equations." Journal of Applied Mathematics and Stochastic Analysis 16, no. 1 (2003): 81–94. http://dx.doi.org/10.1155/s1048953303000066.

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24

Boyer, Franck, and Víctor Hernández-Santamaría. "Carleman estimates for time-discrete parabolic equations and applications to controllability." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 12. http://dx.doi.org/10.1051/cocv/2019072.

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In this paper, we prove a Carleman estimate for a time-discrete parabolic operator under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate is then used to obtain relaxed observability estimates that yield, by duality, some controllability results for linear and semi-linear time-discrete parabolic equations. We also discuss the application of this Carleman estimate to the controllability of time-discrete coupled parabolic systems.

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25

Cywiak-Códova, D., G. Gutiérrez-Juárez, and And M. Cywiak-Garbarcewicz. "Spectral generalized function method for solving homogeneous partial differential equations with constant coefficients." Revista Mexicana de Física E 17, no. 1 Jan-Jun (2020): 11. http://dx.doi.org/10.31349/revmexfise.17.11.

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A method based on a generalized function in Fourier space gives analytical solutions to homogeneous partial differential equations with constant coefficients of any order in any number of dimensions. The method exploits well-known properties of the Dirac delta, reducing the differential mathematical problem into the factorization of an algebraic expression that finally has to be integrated. In particular, the method was utilized to solve the most general homogeneous second order partial differential equation in Cartesian coordinates, finding a general solution for non-parabolic partial differe

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26

Caussignac, Ph. "Incompletely Parabolic Systems from Friedrichs Theory Point of View." Mathematical Models and Methods in Applied Sciences 07, no. 08 (1997): 1141–52. http://dx.doi.org/10.1142/s0218202597000566.

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An incompletely parabolic system of partial differential equations consist of one parabolic subsystem coupled to a hyperbolic subsystem. For the initial-boundary value problem, it has been shown, by requiring that a solution remains bounded at any time by the data, that boundary conditions which make both subsystems well-posed render the global system well-posed too. In this paper, we establish the same type of result with the help of the notion of semi-admissible boundary conditions in the theory of Friedrichs positive systems of differential equations. Our aim is not to obtain the same resul

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27

Quittner, Pavol, and Philippe Souplet. "Admissible Lp norms for local existence and for continuation in semilinear parabolic systems are not the same." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 6 (2001): 1435–56. http://dx.doi.org/10.1017/s0308210500001475.

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We say that a Banach space E is a continuation space for a given parabolic problem if the E-norm of any non-global solution has to become unbounded. We will prove that for large classes of parabolic systems of two equations, the space E = Lr1 × Lr2 can be a continuation space even though the problem is not locally well posed in E. This stands in contrast with classical results for analogous scalar equations.

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28

Bensoussan, Alain, Jens Frehse, and Sheung Chi Phillip Yam. "Systems of quasilinear parabolic equations in Rn and systems of quadratic backward stochastic differential equations." Journal de Mathématiques Pures et Appliquées 149 (May 2021): 135–85. http://dx.doi.org/10.1016/j.matpur.2021.01.006.

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29

Melnig, Elena-Alexandra. "Stability in 𝐿𝑞-norm for inverse source parabolic problems". Journal of Inverse and Ill-posed Problems 28, № 6 (2020): 797–814. http://dx.doi.org/10.1515/jiip-2019-0081.

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AbstractWe consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in {L^{q}}-norms, {2\leq q\leq\infty}, for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.

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30

Sivergina, Irina F., Michael P. Polis, and Ilya Kolmanovsky. "Source Identification for Parabolic Equations." Mathematics of Control, Signals, and Systems (MCSS) 16, no. 2-3 (2003): 141–57. http://dx.doi.org/10.1007/s00498-003-0136-6.

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31

Heinrichs, Wilhelm. "Enclosure statements for systems of semilinear parabolic differential equations." Applications of Mathematics 36, no. 2 (1991): 96–122. http://dx.doi.org/10.21136/am.1991.104448.

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32

Bondarenko, Viktor, Anna Kravchenko, and Tetiana Sobko. "Generalization of the Trotter–Daletsky formula for systems of the "reaction–diffusion" type." System research and information technologies, no. 4 (December 22, 2021): 102–14. http://dx.doi.org/10.20535/srit.2308-8893.2021.4.08.

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An iterative method for constructing a solution to the Cauchy problem for a system of parabolic equations with a nonlinear potential has been proposed and substantiated. The method is based on the Trotter–Daletsky formula, generalized for a nonlinear perturbation of an elliptic operator. The idea of generalization is the construction of a composition of the semigroup generated by the Laplacian and the phase flow corresponding to a system of ordinary differential equations. A computational experiment performed for a two-dimensional system of semilinear parabolic equations of the “reaction–diffu

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33

Papageorgiou, N. "Optimality conditions for systems driven by nonlinear evolution equations." Mathematical Problems in Engineering 1, no. 1 (1995): 27–36. http://dx.doi.org/10.1155/s1024123x95000044.

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Using the Dubovitskii-Milyutin theory we derive necessary and sufficient conditions for optimality for a class of Lagrange optimal control problems monitored by a nonlinear evolution equation and involving initial and/or terminal constraints. An example of a parabolic control system is also included.

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34

Fattorini, H. O. "Robustness and convergence of suboptimal controls in distributed parameter systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 6 (1997): 1153–79. http://dx.doi.org/10.1017/s0308210500026998.

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A fundamental prerequisite for the numerical computation of optimal controls is to show that sequences of suboptimal (that is, close-to-optimal) controls converge. We show this in a version that applies to hyperbolic and parabolic distributed parameter systems, the latter including the Navier–Stokes equations. The optimal problems include control and state constraints; in the parabolic case, the constraints may be pointwise on the solution and the gradient.

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35

Pao, C. V. "Numerical Analysis of Coupled Systems of Nonlinear Parabolic Equations." SIAM Journal on Numerical Analysis 36, no. 2 (1999): 393–416. http://dx.doi.org/10.1137/s0036142996313166.

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36

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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37

Engler, Hans, Bernhard Kawohl, and Stephan Luckhaus. "Gradient estimates for solutions of parabolic equations and systems." Journal of Mathematical Analysis and Applications 147, no. 2 (1990): 309–29. http://dx.doi.org/10.1016/0022-247x(90)90350-o.

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38

Mierczyński, Janusz, and Wenxian Shen. "Persistence in Forward Nonautonomous Competitive Systems of Parabolic Equations." Journal of Dynamics and Differential Equations 23, no. 3 (2010): 551–71. http://dx.doi.org/10.1007/s10884-010-9181-2.

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39

Simon, L. "On systems of strongly nonlinear parabolic functional differential equations." Periodica Mathematica Hungarica 33, no. 2 (1996): 135–51. http://dx.doi.org/10.1007/bf02093511.

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40

Pao, C. V. "Systems of Parabolic Equations with Continuous and Discrete Delays." Journal of Mathematical Analysis and Applications 205, no. 1 (1997): 157–85. http://dx.doi.org/10.1006/jmaa.1996.5177.

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41

Cao, Yanzhao, Max Gunzburger, and James Turner. "The Controllability of Systems Governed by Parabolic Differential Equations." Journal of Mathematical Analysis and Applications 215, no. 1 (1997): 174–89. http://dx.doi.org/10.1006/jmaa.1997.5633.

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42

Belopolskaya, Yana. "Stochastic models for forward systems of nonlinear parabolic equations." Statistical Papers 59, no. 4 (2018): 1505–19. http://dx.doi.org/10.1007/s00362-018-1033-x.

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43

Bokalo, M. M., and A. M. Tsebenko. "Optimal control problem for systems governed by nonlinear parabolic equations without initial conditions." Carpathian Mathematical Publications 8, no. 1 (2016): 21–37. http://dx.doi.org/10.15330/cmp.8.1.21-37.

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An optimal control problem for systems described by Fourier problem for nonlinear parabolic equations is studied. Control functions occur in the coefficients of the state equations. The existence of the optimal control in the case of final observation is proved.

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44

Kashchenko, Sergey. "Infinite Turing Bifurcations in Chains of Van der Pol Systems." Mathematics 10, no. 20 (2022): 3769. http://dx.doi.org/10.3390/math10203769.

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A chain of coupled systems of Van der Pol equations is considered. We study the local dynamics of this chain in the vicinity of the zero equilibrium state. We make a transition to the system with a continuous spatial variable assuming that the number of elements in the chain is large enough. The critical cases corresponding to the Turing bifurcations are identified. It is shown that they have infinite dimension. Special nonlinear parabolic equations are proposed on the basis of the asymptotic algorithm. Their nonlocal dynamics describes the local behavior of solutions to the original system. I

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45

Mitra, Sourav. "Stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 66. http://dx.doi.org/10.1051/cocv/2019036.

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In this article, we study the local boundary stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel around Poiseuille flow which is a stationary solution for the system under consideration. The feedback control operator we construct has finite dimensional range. The homogeneous Navier–Stokes equations are of parabolic nature and the stabilization result for such system is well studied in the literature. In the present article we prove a stabilization result for non-homogeneous Navier–Stokes equations which involves coupled parabolic and hyperbolic dynamics by using only o

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46

Shakhmurov, Veli B. "Maximal regular boundary value problems in Banach-valued function spaces and applications." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–26. http://dx.doi.org/10.1155/ijmms/2006/92134.

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The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non-selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach-valuedLp-spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial boundary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value p

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47

Kahane, Charles S. "The Feynman-Kac formula for a system of parabolic equations." Czechoslovak Mathematical Journal 44, no. 4 (1994): 579–602. http://dx.doi.org/10.21136/cmj.1994.128499.

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48

Ling, Rina. "Unstable periodic wave solutions of Nerve Axion diffusion equations." International Journal of Mathematics and Mathematical Sciences 10, no. 4 (1987): 787–96. http://dx.doi.org/10.1155/s0161171287000875.

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49

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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50

Schneider, Guido, and Hannes Uecker. "Nonlinear coupled mode dynamics in hyperbolic and parabolic periodically structured spatially extended systems." Asymptotic Analysis 28, no. 2 (2001): 163–80. https://doi.org/10.3233/asy-2001-472.

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Nonlinear coupled mode equations occur as universal modulation equations in various circumstances. It is the purpose of this paper to prove exact estimates between the approximations obtained via the nonlinear coupled mode equations and solutions of the original parabolic or hyperbolic systems. The models which we consider contain all difficulties which have to be overcome in the general case.

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