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Relevant bibliographies by topics / Systems of Parabolic Equations

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Academic literature on the topic 'Systems of Parabolic Equations'

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Published: 9 March 2023

Last updated: 10 March 2023

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Journal articles on the topic "Systems of Parabolic Equations"

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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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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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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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 equation is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce a norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with retarded argument and weak diffusion. We use bifurcation theory for delay differential equations and quasilinear parabolic equations. The existence of periodic solutions in an autonomous parabolic system of differential equations on the circle with retarded argument and small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with retarded argument and weak diffusion are investigated.

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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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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.v60i0.13986 Al-Sultani, M. ''Numerical solution of nonlinear parabolic systems by block monotone iterations''. Tech. Report, 2019. https://arxiv.org/abs/1905.03599 Boglaev, I. ''Inexact block monotone methods for solving nonlinear elliptic problems'' J. Comput. Appl. Math. 269:109–117, 2014. doi:10.1016/j.cam.2014.03.029 Lui, S. H. ''On monotone iteration and Schwarz methods for nonlinear parabolic PDEs''. J. Comput. Appl. Math. 161:449–468, 2003. doi:doi.org/10.1016/j.cam.2003.06.001 Pao, C. V. Nonlinear parabolic and elliptic equations. Plenum Press, New York, 1992. doi:10.1007/s002110050168 Pao C. V. ''Numerical analysis of coupled systems of nonlinear parabolic equations''. SIAM J. Numer. Anal. 36:393–416, 1999. doi:10.1137/S0036142996313166 Varga, R. S. Matrix iterative analysis. Springer, Berlin, 2000. 10.1007/978-3-642-05156-2 Zhao, Y. Numerical solutions of nonlinear parabolic problems using combined-block iterative methods. Masters Thesis, University of North Carolina, 2003. http://dl.uncw.edu/Etd/2003/zhaoy/yaxizhao.pdf

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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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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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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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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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Dissertations / Theses on the topic "Systems of Parabolic Equations"

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Crooks, Elaine Craig Mackay. "Travelling-wave solutions for parabolic systems." Thesis, University of Bath, 1996. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.319218.

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Yolcu, Türkay. "Parabolic systems and an underlying Lagrangian." Diss., Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29760.

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In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L¹.

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Yolcu, Türkay. "Parabolic systems and an underlying Lagrangian." Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29760.

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Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2010.<br>Committee Chair: Gangbo, Wilfrid; Committee Member: Chow, Shui-Nee; Committee Member: Harrell, Evans; Committee Member: Swiech, Andrzej; Committee Member: Yezzi, Anthony Joseph. Part of the SMARTech Electronic Thesis and Dissertation Collection.

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Reichelt, Sina. "Two-scale homogenization of systems of nonlinear parabolic equations." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2015. http://dx.doi.org/10.18452/17385.

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Ziel dieser Arbeit ist es zwei verschiedene Klassen von Systemen nichtlinearer parabolischer Gleichungen zu homogenisieren, und zwar Reaktions-Diffusions-Systeme mit verschiedenen Diffusionslängenskalen und Gleichungen vom Typ Cahn-Hilliard. Wir betrachten parabolische Gleichungen mit periodischen Koeffizienten, wobei die Periode dem Verhältnis der charakteristischen mikroskopischen zu der makroskopische Längenskala entspricht. Unser Ziel ist es, effektive Gleichungen rigoros herzuleiten, um die betrachteten Systeme besser zu verstehen und den Simulationsaufwand zu minimieren. Wir suchen also einen Konvergenzbegriff, mit dem die Lösung des Ausgangsmodells im Limes der Periode gegen Null gegen die Lösung des effektiven Modells konvergiert. Um die periodische Mikrostruktur und die verschiedenen Diffusivitäten zu erfassen, verwenden wir die Zwei-Skalen Konvergenz mittels periodischer Auffaltung. Der erste Teil der Arbeit handelt von Reaktions-Diffusions-Systemen, in denen einige Spezies mit der charakteristischen Diffusionslänge der makroskopischen Skala und andere mit der mikroskopischen diffundieren. Die verschiedenen Diffusivitäten führen zu einem Verlust der Kompaktheit, sodass wir nicht direkt den Grenzwert der nichtlinearen Terme bestimmen können. Wir beweisen mittels starker Zwei-Skalen Konvergenz, dass das effektive Modell ein zwei-skaliges Modell ist, welches von der makroskopischen und der mikroskopischen Skale abhängt. Unsere Methode erlaubt es uns, explizite Raten für die Konvergenz der Lösungen zu bestimmen. Im zweiten Teil betrachten wir Gleichungen vom Typ Cahn-Hilliard, welche ortsabhängige Mobilitätskoeffizienten und allgemeine Potentiale beinhalten. Wir beweisen evolutionäre Gamma-Konvergenz der zugehörigen Gradientensysteme basierend auf der Gamma-Konvergenz der Energien und der Dissipationspotentiale.<br>The aim of this thesis is to derive homogenization results for two different types of systems of nonlinear parabolic equations, namely reaction-diffusion systems involving different diffusion length scales and Cahn-Hilliard-type equations. The coefficient functions of the considered parabolic equations are periodically oscillating with a period which is proportional to the ratio between the charactersitic microscopic and macroscopic length scales. In view of greater structural insight and less computational effort, it is our aim to rigorously derive effective equations as the period tends to zero such that solutions of the original model converge to solutions of the effective model. To account for the periodic microstructure as well as for the different diffusion length scales, we employ the method of two-scale convergence via periodic unfolding. In the first part of the thesis, we consider reaction-diffusion systems, where for some species the diffusion length scale is of order of the macroscopic length scale and for other species it is of order of the microscopic one. Based on the notion of strong two-scale convergence, we prove that the effective model is a two-scale reaction-diffusion system depending on the macroscopic and the microscopic scale. Our approach supplies explicit rates for the convergence of the solution. In the second part, we consider Cahn-Hilliard-type equations with position-dependent mobilities and general potentials. It is well-known that the classical Cahn-Hilliard equation admits a gradient structure. Based on the Gamma-convergence of the energies and the dissipation potentials, we prove evolutionary Gamma-convergence, for the associated gradient system such that we obtain in the limit of vanishing periods a Cahn-Hilliard equation with homogenized coefficients.

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Liu, Weian, Yin Yang, and Gang Lu. "Viscosity solutions of fully nonlinear parabolic systems." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2621/.

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In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].

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Floater, Michael S. "Blow-up of solutions to nonlinear parabolic equations and systems." Thesis, University of Oxford, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.235037.

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Chen, Mingxiang. "Structural stability of periodic systems." Diss., Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/29341.

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Floridia, Giuseppe. "Approximate multiplicative controllability for degenerate parabolic problems and regularity properties of elliptic and parabolic systems." Doctoral thesis, Università di Catania, 2012. http://hdl.handle.net/10761/1051.

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This thesis consists of two parts, both related to the theory of parabolic equations and systems. The first part is devoted to control theory which studies the possibility of influencing the evolution of a given system by an external action called control. Here we address approximate controllability problems via multiplicative controls, motivated by our interest in some differential models for the study of climatology. In the second part of the thesis we address regularity issues on the local differentiability and H\"older regularity for weak solutions of nonlinear systems in divergence form. In order to improve readability, the two parts have been organized as completely independent chapters, with two separate introductions and bibliographies. All the new results of this thesis have been presented at conferences and workshops, and most of them appeared or are to appear as research articles in international journals. Related directions for future research are also outlined in body of the work.

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Al, Refai Mohammed. "Sequential eigenfunction expansion for certain non-linear parabolic systems and wave type equations." Thesis, McGill University, 2000. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=36747.

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In trying to solve nonlinear partial differential equations with time dependence using the Galerkin method, one ends up with solving nonlinear systems of ordinary differential equations, which are not easily solved. In this thesis we introduce a new iterative method based on eigenfunction expansion to deal with the finite non-linear systems sequentially.<br>We apply the new method to integrate the semi-linear parabolic equation ut=12u+fu ,x∈D with homogeneous Dirichlet or Robin boundary conditions. We prove the convergence of the new iterative method, and use it to find the multiple solutions of the system, which are difficult to obtain using the Galerkin method.<br>We next apply the new method to solve a parabolic system of two semi-linear equations<br>ut=12u+f u,q qt=12 q+gu,q ,x∈D with homogeneous boundary conditions Au = 0 and Bv = 0. We prove the convergence of the new method for the case when A = B. If A ≠ B no analytical statements are obtained. However, the proof of convergence is a sufficient, but not a necessary condition, and numerical calculations indicate that the solution obtained by the new method still converges to that obtained by the Galerkin method for the case when A ≠ B. We apply the new method to integrate a system in combustion theory, and we are able to find critical (as defined in Chapter 4) solutions for the system, which are not easily found using the Galerkin method.<br>To see that the new method can be applied to more general systems, we use it to integrate the Kuramoto-Sivashinsky equation<br>6tu+41+e 264xu+e6 2xu+12 6xu2=0 ,x∈0,&ell; We prove the convergence of the iterative method and use it to find the first term of the eigenfunction expansion analytically, and from that we notice that the equation has two solutions, one stable and the other unstable. This kind of observation can not be obtained using the Galerkin method.<br>Finally, we apply the new method to solve a wave type equation governing the motion of a fluid in a conveying pipe,<br>EI64w 6x4+&parl0;MU2 t+M6U6t L-x&parr0;6 2w6x2+2MU6 2w6x6t+M +m62w 6t2=0. In all of the above systems, numerical calculations indicate that the solutions obtained by the new method and the Galerkin method coincide.

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Zhao, Yaxi. "Numerical solutions of nonlinear parabolic problems using combined-block iterative methods /." Electronic version (PDF), 2003. http://dl.uncw.edu/etd/2003/zhaoy/yaxizhao.pdf.

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Books on the topic "Systems of Parabolic Equations"

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Zheng, Songmu. Nonlinear parabolic equations and hyperbolic-parabolic coupled systems. Longman, 1995.

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Zheng, S. Nonlinear parabolic equations and hyperbolic-parabolic coupled systems. Longman, 1995.

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I, Volʹpert A. Traveling wave solutions of parabolic systems. American Mathematical Society, 1994.

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Volpert, Vitaly A., 1958- author and Volpert, Vladimir A., 1954- author, eds. Traveling wave solutions of parabolic systems. American Mathematical Society, 2004.

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Miroslav, Krstić, ed. Adaptive control of parabolic PDEs. Princeton University Press, 2010.

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I, Koshelev A. Regularity problem for quasilinear elliptic and parabolic systems. Springer, 1995.

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Nonlinear parabolic-hyperbolic coupled systems and their attractors. Birkhäuser, 2008.

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1972-, Mingione Giuseppe, and Steffen Klaus 1945-, eds. Parabolic systems with polynomial growth and regularity. American Mathematical Society, 2011.

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Smyshlyaev, Andrey. Adaptive control of parabolic PDEs. Princeton University Press, 2010.

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Smyshlyaev, Andrey. Adaptive control of parabolic PDEs. Princeton University Press, 2010.

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Book chapters on the topic "Systems of Parabolic Equations"

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Banks, H. T., and K. Kunisch. "Parabolic Equations." In Estimation Techniques for Distributed Parameter Systems. Birkhäuser Boston, 1989. http://dx.doi.org/10.1007/978-1-4612-3700-6_5.

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Kavdia, Mahendra. "Parabolic Differential Equations, Diffusion Equation." In Encyclopedia of Systems Biology. Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4419-9863-7_273.

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Wang, Mingxin. "Weakly Coupled Parabolic Systems." In Nonlinear Second Order Parabolic Equations. CRC Press, 2021. http://dx.doi.org/10.1201/9781003150169-4.

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Rozovskii, B. L. "Ito’s Second Order Parabolic Equations." In Stochastic Evolution Systems. Springer Netherlands, 1990. http://dx.doi.org/10.1007/978-94-011-3830-7_4.

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Rozovsky, Boris L., and Sergey V. Lototsky. "Itô’s Second-Order Parabolic Equations." In Stochastic Evolution Systems. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-94893-5_4.

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Smith, Hal. "Quasimonotone systems of parabolic equations." In Mathematical Surveys and Monographs. American Mathematical Society, 2008. http://dx.doi.org/10.1090/surv/041/07.

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Pao, C. V. "Systems with Nonlinear Boundary Conditions." In Nonlinear Parabolic and Elliptic Equations. Springer US, 1992. http://dx.doi.org/10.1007/978-1-4615-3034-3_9.

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Pao, C. V. "Coupled Systems of Reaction Diffusion Equations." In Nonlinear Parabolic and Elliptic Equations. Springer US, 1992. http://dx.doi.org/10.1007/978-1-4615-3034-3_8.

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Kochubei, Anatoly N. "Fractional-parabolic equations and systems. Cauchy problem." In Fractional Differential Equations, edited by Anatoly Kochubei and Yuri Luchko. De Gruyter, 2019. http://dx.doi.org/10.1515/9783110571660-007.

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Pao, C. V. "Applications of Coupled Systems to Model Problems." In Nonlinear Parabolic and Elliptic Equations. Springer US, 1992. http://dx.doi.org/10.1007/978-1-4615-3034-3_12.

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Conference papers on the topic "Systems of Parabolic Equations"

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Arkhipova, Arina. "New a priori estimates for nondiagonal strongly nonlinear parabolic systems." In Parabolic and Navier–Stokes equations. Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc81-0-1.

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Zadrzyńska, Ewa, and Wojciech M. Zajączkowski. "Some linear parabolic system in Besov spaces." In Parabolic and Navier–Stokes equations. Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc81-0-36.

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Picard, Rainer. "The Stokes system in the incompressible case–revisited." In Parabolic and Navier–Stokes equations. Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc81-0-23.

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Cieślak, Tomasz, Philippe Laurençot, and Cristian Morales-Rodrigo. "Global existence and convergence to steady states in a chemorepulsion system." In Parabolic and Navier–Stokes equations. Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc81-0-7.

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Pawłow, Irena, and Wojciech M. Zajączkowski. "Global existence and uniqueness of weak solutions to Cahn-Hilliard-Gurtin system in elastic solids." In Parabolic and Navier–Stokes equations. Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc81-0-22.

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ARCEO, CARLENE P., JOSE MA L. ESCANER, MITSUHARU ÔTANI, and POLLY W. SY. "PARABOLIC EQUATIONS WITH SINGULARITY ON THE BOUNDARY." In Proceedings of Modelling and Control of Mechanical Systems. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812776594_0002.

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Stańczy, Robert. "On radially symmetric solutions of some chemotaxis system." In Nonlocal and Abstract Parabolic Equations and their Applications. Institute of Mathematics Polish Academy of Sciences, 2009. http://dx.doi.org/10.4064/bc86-0-19.

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Viglialoro, Giuseppe, Stella Vernier Piro, and Monica Marras. "Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system." In The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Madrid, Spain). American Institute of Mathematical Sciences, 2015. http://dx.doi.org/10.3934/proc.2015.0809.

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Abels, Helmut. "Longtime behavior of solutions of a Navier-Stokes/Cahn-Hilliard system." In Nonlocal and Abstract Parabolic Equations and their Applications. Institute of Mathematics Polish Academy of Sciences, 2009. http://dx.doi.org/10.4064/bc86-0-1.

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Salmani, Abdelhafid, Youssef Akdim, and Mounir Mekkour. "Renormalized solutions for nonlinear anisotropic parabolic equations." In 2019 International Conference on Intelligent Systems and Advanced Computing Sciences (ISACS). IEEE, 2019. http://dx.doi.org/10.1109/isacs48493.2019.9068873.

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Reports on the topic "Systems of Parabolic Equations"

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Dalang, Robert C., and N. Frangos. Stochastic Hyperbolic and Parabolic Partial Differential Equations. Defense Technical Information Center, 1994. http://dx.doi.org/10.21236/ada290372.

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Rundell, William, and Michael S. Pilant. Undetermined Coefficient Problems for Quasi-Linear Parabolic Equations. Defense Technical Information Center, 1992. http://dx.doi.org/10.21236/ada256012.

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Nohel, John A. A Class of One-Dimensional Degenerate Parabolic Equations. Defense Technical Information Center, 1985. http://dx.doi.org/10.21236/ada160962.

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Dresner, L. On some general properties of parabolic conservation equations. Office of Scientific and Technical Information (OSTI), 1993. http://dx.doi.org/10.2172/10119060.

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Pilant, Michael S., and William Rundell. Undetermined Coefficient Problems for Quasi-Linear Parabolic Equations. Defense Technical Information Center, 1989. http://dx.doi.org/10.21236/ada218462.

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Ostashev, Vladimir, Michael Muhlestein, and D. Wilson. Extra-wide-angle parabolic equations in motionless and moving media. Engineer Research and Development Center (U.S.), 2021. http://dx.doi.org/10.21079/11681/42043.

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Wide-angle parabolic equations (WAPEs) play an important role in physics. They are derived by an expansion of a square-root pseudo-differential operator in one-way wave equations, and then solved by finite-difference techniques. In the present paper, a different approach is suggested. The starting point is an extra-wide-angle parabolic equation (EWAPE) valid for small variations of the refractive index of a medium. This equation is written in an integral form, solved by a perturbation technique, and transformed to the spectral domain. The resulting split-step spectral algorithm for the EWAPE accounts for the propagation angles up to 90° with respect to the nominal direction. This EWAPE is also generalized to large variations in the refractive index. It is shown that WAPEs known in the literature are particular cases of the two EWAPEs. This provides an alternative derivation of the WAPEs, enables a better understanding of the underlying physics and ranges of their applicability, and opens an opportunity for innovative algorithms. Sound propagation in both motionless and moving media is considered. The split-step spectral algorithm is particularly useful in the latter case since complicated partial derivatives of the sound pressure and medium velocity reduce to wave vectors (essentially, propagation angles) in the spectral domain.

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Carasso, Alfred S. Compensating Operators and Stable Backward in Time Marching in Nonlinear Parabolic Equations. National Institute of Standards and Technology, 2013. http://dx.doi.org/10.6028/nist.ir.7967.

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Hale, Jack K. Large Diffusivity and Asymptotic Behavior in Parabolic Systems. Defense Technical Information Center, 1985. http://dx.doi.org/10.21236/ada166197.

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Angenent, Sigurd. Parabolic Equations for Curves on Surfaces. 2. Intersections, Blow Up and Generalized Solutions. Defense Technical Information Center, 1989. http://dx.doi.org/10.21236/ada212890.

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Babuska, Ivo, and Tadeusz Janik. The p-Version of the Finite Element Method for Parabolic Equations. Part 1. Defense Technical Information Center, 1988. http://dx.doi.org/10.21236/ada197786.

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