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Artykuły w czasopismach na temat „Numerical blow-up”

Autor: Grafiati
Data publikacji: 3 czerwca 2025
Data aktualizacji: 31 lipca 2025

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1

Cho, Chien-Hong, and Ying-Jung Lu. "On the numerical solutions for a parabolic system with blow-up." AIMS Mathematics 6, no. 11 (2021): 11749–77. http://dx.doi.org/10.3934/math.2021683.

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<abstract><p>We study the finite difference approximation for axisymmetric solutions of a parabolic system with blow-up. A scheme with adaptive temporal increments is commonly used to compute an approximate blow-up time. There are, however, some limitations to reproduce the blow-up behaviors for such schemes. We thus use an algorithm, in which uniform temporal grids are used, for the computation of the blow-up time and blow-up behaviors. In addition to the convergence of the numerical blow-up time, we also study various blow-up behaviors numerically, including the blow-up set, blow
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2

Maan, A. Rasheed, and Farhan A.G. "On Blow-up Time and Rate Of The Numerical Solutions of The Semilinear Heat Equation with Reaction Term." Journal of Progressive Research in Mathematics 9, no. 1 (2016): 1333–40. https://doi.org/10.5281/zenodo.3976761.

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In this paper, we study the numerical blow-up solutions and times of the semilinear heat equation with reaction term. We compute the blow-up growth rate in the numerical solutions of two numerical experiments, depending on the blow-up solutions and times, those have been computed using a finite difference method.
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3

Fernández Bonder, Julián, Pablo Groisman, and Julio D. Rossi. "On numerical blow-up sets." Proceedings of the American Mathematical Society 130, no. 7 (2002): 2049–55. http://dx.doi.org/10.1090/s0002-9939-02-06350-5.

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4

Cho, Chien-Hong. "Numerical detection of blow-up: a new sufficient condition for blow-up." Japan Journal of Industrial and Applied Mathematics 33, no. 1 (2015): 81–98. http://dx.doi.org/10.1007/s13160-015-0198-0.

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5

FERREIRA, RAÚL, PABLO GROISMAN, and JULIO D. ROSSI. "NUMERICAL BLOW-UP FOR A NONLINEAR PROBLEM WITH A NONLINEAR BOUNDARY CONDITION." Mathematical Models and Methods in Applied Sciences 12, no. 04 (2002): 461–83. http://dx.doi.org/10.1142/s021820250200174x.

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In this paper we study numerical approximations for positive solutions of a nonlinear heat equation with a nonlinear boundary condition. We describe in terms of the nonlinearities when solutions of a semidiscretization in space exist globally in time and when they blow up in finite time. We also find the blow-up rates and the blow-up sets. In particular we prove that regional blow-up is not reproduced by the numerical scheme. However, in the appropriate variables we can reproduce the correct blow-up set when the mesh parameter goes to zero.
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6

Khalil, Manar, Ishak Hashim, Maan Rasheed, Faieza Samat, and Shaher Momani. "Numerical Finite-Difference Approximations of a Coupled Reaction-Diffusion System with Gradient Terms." European Journal of Pure and Applied Mathematics 17, no. 3 (2024): 1516–38. http://dx.doi.org/10.29020/nybg.ejpam.v17i3.5246.

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This study focuses on the derivation of explicit and implicit finite difference formulas.The objective of this study is to derive an estimation of the blow-up time for a coupled reaction-diffusion system incorporating gradient terms, employing numerical finite difference approximations. Furthermore, an examination is conducted on the consistency, stability, and convergence of the proposed schemes. Additionally, the study presents two numerical experiments. In each instance, the numerical blow-up time is calculated benefit the suggested methodologies, employing varying space steps and non-fixed
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7

Lekporo, K. Z., K. B. Edja,, N. Koffi, and K. A. Touré. "SIMULTANEOUS NUMERICAL BLOW-UP IN A FOUR-COMPONENT SYSTEM OF HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS." Advances in Mathematics: Scientific Journal 14, no. 2 (2025): 157–86. https://doi.org/10.37418/amsj.14.2.3.

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This paper investigates the numerical approximation of a system of heat equations with nonlinear boundary conditions. We prove that the solution of a semidiscrete form of above problem blows up in a finite time. We also establish certain conditions under which simultaneous blow-up occurs for the solution of the semidiscrete problem. After showing that the numerical blow-up time converges to the theoretical blow-up time as the mesh size tends to zero, we finally present some numerical results to illustrate key points of our work.
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8

Chung, Soon-Yeong, and Jae-Hwang Lee. "Blow-up for discrete reaction-diffusion equations on networks." Applicable Analysis and Discrete Mathematics 9, no. 1 (2015): 103–19. http://dx.doi.org/10.2298/aadm150210005c.

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In this paper, we discuss the conditions under which blow-up occurs for the solutions of reaction-diffusion equations on networks. The analysis of this class of problems includes the existence of blow-up in finite time and the determination of the blow-up time and the corresponding blow-up rate. In addition, when the solution blows up, we give estimates for the blow-up time and also provide the blow-up rate. Finally, we show some numerical illustrations which describe the main results.
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9

Ferreira, Raŭl, and Arturo De Pablo. "Numerical Blow-up for the p-Aplacian Equation with a Source." Computational Methods in Applied Mathematics 5, no. 2 (2005): 137–54. http://dx.doi.org/10.2478/cmam-2005-0007.

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AbstractWe study numerical approximations of nonnegative solutions of the p-Laplacian equation with a nonlinear source. We describe when solutions of a semidiscretization in space exist globally in time and when they blow up in a finite time. We also find the blow-up rates and the blow-up sets by means of the discrete self-similar profiles.
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10

Stuart, A. M., and M. S. Floater. "On the computation of blow-up." European Journal of Applied Mathematics 1, no. 1 (1990): 47–71. http://dx.doi.org/10.1017/s095679250000005x.

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Numerical methods for initial-value problems which develop singularities in finite time are analyzed. The objective is to determine simple strategies which produce the correct asymptotic behaviour and give an accurate approximation of the blow-up time. Fixed step methods for scalar ordinary differential equations are studied first and it is shown that there is a natural embedding of the discrete process in a continuous one. This shows clearly how and why the fixed-step strategy fails. A class of time-stepping strategies that correspond to a time- continuous re-scaling of the underlying differe
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11

Achille, Adou Koffi, Diop Fatou N., N’Guessan Koffi, and Touré Kidjégbo Augustin. "NUMERICAL BLOW-UP TIME FOR NONLINEAR PARABOLIC PROBLEMS." Advances in Differential Equations and Control Processes 28 (August 10, 2022): 135–52. http://dx.doi.org/10.17654/0974324322028.

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12

Cho, C. H. "A numerical algorithm for blow-up problems revisited." Numerical Algorithms 75, no. 3 (2016): 675–97. http://dx.doi.org/10.1007/s11075-016-0216-6.

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13

N’Gohisse, Firmin K., and Théodore K. Boni. "Numerical blow-up for a nonlinear heat equation." Acta Mathematica Sinica, English Series 27, no. 5 (2011): 845–62. http://dx.doi.org/10.1007/s10114-011-8464-9.

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14

Hamadneh, Tareq, Zainouba Chebana, Ibraheem Abu Falahah, et al. "On Finite-Time Blow-Up Problem for Nonlinear Fractional Reaction Diffusion Equation: Analytical Results and Numerical Simulations." Fractal and Fractional 7, no. 8 (2023): 589. http://dx.doi.org/10.3390/fractalfract7080589.

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The study of the blow-up phenomenon for fractional reaction–diffusion problems is generally deemed of great importance in dealing with several situations that impact our daily lives, and it is applied in many areas such as finance and economics. In this article, we expand on some previous blow-up results for the explicit values and numerical simulation of finite-time blow-up solutions for a semilinear fractional partial differential problem involving a positive power of the solution. We show the behavior solution of the fractional problem, and the numerical solution of the finite-time blow-up
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15

Kauranen, Aapo, and Pekka Koskela. "Boundary blow-up under Sobolev mappings." Analysis & PDE 7, no. 8 (2014): 1839–50. http://dx.doi.org/10.2140/apde.2014.7.1839.

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16

N'Guessan, Koffi, Nabongo Diabate, and Kidjegbo Augustin Toure. "Blow-up for Semidiscretizations of some Semilinear Parabolic Equations with a Convection Term." Journal of Progressive Research in Mathematics 5, no. 2 (2015): 499–518. https://doi.org/10.5281/zenodo.3979619.

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This paper concerns the study of the numerical approximation for the following parabolic equations with a convection term  where p >1. We obtain some conditions under which the solution of the semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove that the semidiscrete blowup time converges to the real one, when the mesh size goes to zero. Finally, we give some numerical experiments to illustrate ours analysis.
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17

Edja, Kouame Beranger, Kidjegbo Augustin Toure, and Brou Jean-Claude Koua. "Numerical Blow-up for A Heat Equation with Nonlinear Boundary Conditions." Journal of Mathematics Research 10, no. 5 (2018): 119. http://dx.doi.org/10.5539/jmr.v10n5p119.

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We study numerical approximations of solutions of a heat equation with nonlinear boundary conditions which produce blow-up of the solutions. By a semidiscretization using a finite difference scheme in the space variable we get a system of ordinary differential equations which is an approximation of the original problem. We obtain sufficient conditions which guarantee the blow-up solution of this system in a finite time. We also show that this blow-up time converges to the theoretical one when the mesh size goes to zero. We present some numerical results to illustrate certain point of our work.
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18

Klein, Christian, Christof Sparber, and Peter Markowich. "Numerical study of fractional nonlinear Schrödinger equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2172 (2014): 20140364. http://dx.doi.org/10.1098/rspa.2014.0364.

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Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically co
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19

Olmstead, W. E., C. A. Roberts, and K. Deng. "Coupled Volterra Equations with Blow-Up Solutions." Journal of Integral Equations and Applications 7, no. 4 (1995): 499–516. http://dx.doi.org/10.1216/jiea/1181075901.

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20

Assalé, Louis A., Théodore K. Boni, and Diabate Nabongo. "Numerical Blow-Up Time for a Semilinear Parabolic Equation with Nonlinear Boundary Conditions." Journal of Applied Mathematics 2008 (2008): 1–29. http://dx.doi.org/10.1155/2008/753518.

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We obtain some conditions under which the positive solution for semidiscretizations of the semilinear equationut=uxx−a(x,t)f(u), 0<x<1, t∈(0,T), with boundary conditionsux(0,t)=0,ux(1,t)=b(t)g(u(1,t)), blows up in a finite time and estimate its semidiscrete blow-up time. We also establish the convergence of the semidiscrete blow-up time and obtain some results about numerical blow-up rate and set. Finally, we get an analogous result taking a discrete form of the above problem and give some computational results to illustrate some points of our analysis.
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21

Cho, Chien-Hong. "On the computation of the numerical blow-up time." Japan Journal of Industrial and Applied Mathematics 30, no. 2 (2013): 331–49. http://dx.doi.org/10.1007/s13160-013-0101-9.

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22

Barro, Geneviève, Benjamin Mampassi, Longin Some, Jean Ntaganda, and Ousséni So. "Full discretization of some reaction diffusion equation with blow up." Open Mathematics 4, no. 2 (2006): 260–69. http://dx.doi.org/10.1007/s11533-006-0002-0.

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AbstractThis paper aims at the development of numerical schemes for nonlinear reaction diffusion problems with a convection that blows up in a finite time. A full discretization of this problem that preserves the blow — up property is presented as well as a numerical simulation. Efficiency of the method is derived via a numerical comparison with a classical scheme based on the Runge Kutta scheme.
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23

BRÄNDLE, CRISTINA, PABLO GROISMAN, and JULIO D. ROSSI. "FULLY DISCRETE ADAPTIVE METHODS FOR A BLOW-UP PROBLEM." Mathematical Models and Methods in Applied Sciences 14, no. 10 (2004): 1425–50. http://dx.doi.org/10.1142/s0218202504003751.

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We present adaptive procedures in space and time for the numerical study of positive solutions to the following problem: [Formula: see text] with p,m>0. We describe how to perform adaptive methods in order to reproduce the exact asymptotic behavior (the blow-up rate and the blow-up set) of the continuous problem.
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24

FRAUENDIENER, JÖRG, and RALF PETER. "BLOW-UP OF THE NONEQUIVARIANT ()-DIMENSIONAL WAVE MAP." ANZIAM Journal 55, no. 2 (2013): 151–61. http://dx.doi.org/10.1017/s1446181113000400.

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AbstractIt has been known for a long time that the equivariant $2+1$ wave map into the $2$-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit violations of equivariance.
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25

Groisman, Pablo, and Julio D. Rossi. "Dependence of the blow‐up time with respect to parameters and numerical approximations for a parabolic problem." Asymptotic Analysis 37, no. 1 (2004): 79–91. https://doi.org/10.3233/asy-2004-597.

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We find a bound for the modulus of continuity of the blow‐up time for the problem ut=λΔu+up, with initial datum u(x,0)=ϕ(x)+hf(x) respect to the parameters λ, p and h. We also find an estimate for the rate of convergence of the blow‐up times for a semi‐discrete numerical scheme.
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26

Hou, Thomas Y. "Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier–Stokes equations." Acta Numerica 18 (May 2009): 277–346. http://dx.doi.org/10.1017/s0962492906420018.

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Whether the 3D incompressible Euler and Navier–Stokes equations can develop a finite-time singularity from smooth initial data with finite energy has been one of the most long-standing open questions. We review some recent theoretical and computational studies which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. We also investigate the dynamic stability of the 3D Navier–Stokes equations and the stabilizing effect of convection. A unique feature of our approach is the interplay between computation and analysis.
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27

Ramiandrisoa, Arthur. "Blow‐up profile for radial solutions of the nonlinear heat equation." Asymptotic Analysis 21, no. 3-4 (1999): 221–38. https://doi.org/10.3233/asy-1999-361.

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Assuming that the maximal classical solution $u$ of $u_t-\Delta u=u^p$ on $B(0,R)$ has, at time $T_{\max}$ , a blow‐up set $\mathbf{B}(u)$ that is not the singleton $\left\{0\right\}$ , we prove that ${\|u\|}_{L^q (\varOmega)}$ blows up for all $q>(p-1)/2$ , the energy blows up and $u$ blows up completely after $T_{\max}$ in the sense of Baras–Cohen (equivalent to the fact that there is no weak solution extending $u$ beyond $T_{\max}$ ). We also study this type of blow‐up through numerical experiments as well as degenerate blow‐up.
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28

Cho, Chien-Hong, and Chun-Yi Liu. "Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-Up Problem." East Asian Journal on Applied Mathematics 7, no. 4 (2017): 679–96. http://dx.doi.org/10.4208/eajam.220816.300517a.

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AbstractWe consider the second order nonlinear ordinary differential equation u″ (t) = u1+α (α > 0) with positive initial data u(0) = a0, u′(0) = a1, whose solution becomes unbounded in a finite time T. The finite time T is called the blow-up time. Since finite difference schemes with uniform meshes can not reproduce such a phenomenon well, adaptively-defined grids are applied. Convergence with mesh sizes of certain smallness has been considered before. However, more iterations are required to obtain an approximate blow-up time if smaller meshes are applied. As a consequence, we consider in
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29

ROTTSCHÄFER, V., J. C. TZOU, and M. J. WARD. "Transition to blow-up in a reaction–diffusion model with localized spike solutions." European Journal of Applied Mathematics 28, no. 6 (2017): 1015–55. http://dx.doi.org/10.1017/s0956792517000043.

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For certain singularly perturbed two-component reaction–diffusion systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behaviour in terms of some parameter in the system. For some such systems, such as the Gray–Scott model, a spike self-replication behaviour is observed as the parameter varies across the saddle-node point. We demonstrate and analyse a qualitatively new type of transition as a parameter is slowly decreased below the saddle node value, which is characterized by a finite-time blow-up of the spike solution. More specifically, we use a
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30

Eremin, Alexey, Emiko Ishiwata, Tetsuya Ishiwata, and Yukihiko Nakata. "Delay-induced blow-up in a planar oscillation model." Japan Journal of Industrial and Applied Mathematics 38, no. 3 (2021): 1037–61. http://dx.doi.org/10.1007/s13160-021-00475-x.

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AbstractIn this paper we study a system of delay differential equations from the viewpoint of a finite time blow-up of the solution. We prove that the system admits blow-up solutions, no matter how small the length of the delay is. In the non-delay system every solution approaches to a stable unit circle in the plane, thus time delay induces blow-up of solutions, which we call “delay-induced blow-up” phenomenon. Furthermore, it is shown that the system has a family of infinitely many periodic solutions, while the non-delay system has only one stable limit cycle. The system studied in this pape
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31

Ganon, Ardjouma, Manin Mathurin Taha, and Kidjégbo Augustin Touré. "Blow-up for Semidiscretization of Semilinear Parabolic Equation With Nonlinear Boundary Condition." Journal of Mathematics Research 11, no. 5 (2019): 1. http://dx.doi.org/10.5539/jmr.v11n5p1.

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This paper deals with the study of the numerical approximation for the following semilinear equation with a nonlinear absorption term ut = uxx− λup, 0 < x < 1, t > 0, and a nonlinear flux boundary condition ux(0,t) = 0, ux(1,t) = uq(1,t), t > 0. We give conditions under which the positive semidiscrete solution blows up in a finite time. Convergence of the numerical blow-up time to the theoretical one when the mesh size goes to zero is established. Finally, we use an efficient algorithm to estimate the blow-up time.
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32

Klein, Christian, Benson Muite, and Kristelle Roidot. "Numerical study of blow-up in the Davey-Stewartson system." Discrete & Continuous Dynamical Systems - B 18, no. 5 (2013): 1361–87. http://dx.doi.org/10.3934/dcdsb.2013.18.1361.

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33

Ferreira, R. "Adaptive numerical schemes for a parabolic problem with blow-up." IMA Journal of Numerical Analysis 23, no. 3 (2003): 439–63. http://dx.doi.org/10.1093/imanum/23.3.439.

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34

Al’shin, A. B., and E. A. Al’shina. "Numerical diagnosis of blow-up of solutions of pseudoparabolic equations." Journal of Mathematical Sciences 148, no. 1 (2008): 143–62. http://dx.doi.org/10.1007/s10958-007-0542-2.

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35

Matsue, Kaname, and Akitoshi Takayasu. "Numerical validation of blow-up solutions with quasi-homogeneous compactifications." Numerische Mathematik 145, no. 3 (2020): 605–54. http://dx.doi.org/10.1007/s00211-020-01125-z.

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36

Takayasu, Akitoshi, Kaname Matsue, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi, and Shin’ichi Oishi. "Numerical validation of blow-up solutions of ordinary differential equations." Journal of Computational and Applied Mathematics 314 (April 2017): 10–29. http://dx.doi.org/10.1016/j.cam.2016.10.013.

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37

Brunner, H., and Z. W. Yang. "Blow-up behavior of Hammerstein-type Volterra integral equations." Journal of Integral Equations and Applications 24, no. 4 (2012): 487–512. http://dx.doi.org/10.1216/jie-2012-24-4-487.

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38

Fino, A. Z., I. Dannawi, and M. Kirane. "Blow-up of solutions for semilinear fractional Schrödinger equations." Journal of Integral Equations and Applications 30, no. 1 (2018): 67–80. http://dx.doi.org/10.1216/jie-2018-30-1-67.

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39

PELZ, RICHARD B. "Symmetry and the hydrodynamic blow-up problem." Journal of Fluid Mechanics 444 (September 25, 2001): 299–320. http://dx.doi.org/10.1017/s0022112001005298.

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The problem of whether a spontaneous singularity can occur in finite time in an incompressible inviscid fluid flow is addressed. As suggested by previous numerical simulations, candidate flows are restricted to be invariant under the octahedral group of symmetries and to have a compact vortex tube in the fundamental domain. It is shown that in such a flow the image vorticity contributes strongly to the axial strain rate on the fundamental in a way which is only weakly proportional to the curvature of the vortex lines. Analysis of a model flow shows that axial strain rate scales as the inverse
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40

Arora, Anudeep Kumar, Svetlana Roudenko, and Kai Yang. "On the focusing generalized Hartree equation." Mathematics in Applied Sciences and Engineering 9999, no. 9999 (2020): 1–20. http://dx.doi.org/10.5206/mase/10855.

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In this paper we give a review of the recent progress on the focusing generalized Hartree equation, which is a nonlinear Schrodinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the Riesz potential. We describe the local well-posedness in H1 and Hs settings, discuss the extension to the global existence and scattering, or finite time blow-up. We point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the L2 -critical setting. We finish by showing known analytical results about the stab
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41

Ma, Wenyuan, and Baoqiang Yan. "Global Existence and Uniform Blow-Up to a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising in a Thermal Explosion Theory." Mathematics 11, no. 9 (2023): 1993. http://dx.doi.org/10.3390/math11091993.

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This paper deals with a nonlinear nonlocal parabolic system with nonlinear heat-loss boundary conditions, which arise in the thermal explosion model. Firstly, we prove a comparison principle for some kinds of parabolic systems under nonlinear boundary conditions. Using this, we improve a new theorem of the sub-and-super solution. Secondly, based on the new sub-and-super solution theorem, the sufficient conditions that the solution exists and blows up uniformly in finite time are presented. Then, we generalize some of the lemmas related to uniform blow-up solutions, which are used to introduce
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42

Cole, Justin T., Abdullah M. Aurko, and Ziad H. Musslimani. "Collapse dynamics for two-dimensional space-time nonlocal nonlinear Schrödinger equations." Nonlinearity 37, no. 4 (2024): 045001. http://dx.doi.org/10.1088/1361-6544/ad1efa.

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Abstract The question of collapse (blow-up) in finite time is investigated for the two-dimensional (non-integrable) space-time nonlocal nonlinear Schrödinger equations. Starting from the two-dimensional extension of the well known AKNS q , r system, three different cases are considered: (i) partial and full parity-time (PT) symmetric, (ii) reverse-time (RT) symmetric, and (iii) general q , r system. Through extensive numerical experiments, it is shown that collapse of Gaussian initial conditions depends on the value of its quasi-power. The collapse dynamics (or lack thereof) strongly depends o
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Li, Shaoyong, and Zhengrong Liu. "The Traveling Wave Solutions and Their Bifurcations for the BBM-LikeB(m,n)Equations." Journal of Applied Mathematics 2013 (2013): 1–17. http://dx.doi.org/10.1155/2013/490341.

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We investigate the traveling wave solutions and their bifurcations for the BBM-likeB(m,n)equationsut+αux+β(um)x−γ(un)xxt=0by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-likeB(3,2)equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-likeB(4,2)equation, we construct two periodic wave solut
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44

Chen, Xueyong, Fuxing Hu, Jianhua Zhang, and Jianwei Shen. "GLOBAL EXISTENCE AND BLOW-UP FOR A CHEMOTAXIS SYSTEM." Mathematical Modelling and Analysis 22, no. 2 (2017): 237–51. http://dx.doi.org/10.3846/13926292.2017.1292323.

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In this paper we consider a Keller-Segel-type chemotaxis model with reaction term under no-flux boundary conditions, where the kinetics term of the system is power function. Assuming some growth conditions, the existence of bounded global strong solution to the parabolic-parabolic system is given. We also give the numerical test and find out that there exists a threshold. When the power frequency greater than the threshold, both global solution and blow-up solution exist.
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45

Budd, C. J., V. A. Galaktionov, and Jianping Chen. "Focusing blow-up for quasilinear parabolic equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 5 (1998): 965–92. http://dx.doi.org/10.1017/s0308210500030018.

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We study the behaviour of the non-negative blowing up solutions to the quasilinear parabolic equation with a typical reaction–diffusion right-hand side and with a singularity in the space variable which takes the formwhere m ≧ 1, p > 1 are arbitrary constants, in the critical exponent case q = (p–1)/m > 0. We impose zero Dirichlet boundary conditions at the singular point x = 0 and at x = 1, and take large initial data. For a class of ‘concave’ initial functions, we prove focusing at the origin of the solutions as t approaches the blow-up time T in the sense that x = 0 belongs to the blo
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Brunner, Hermann, Hongwei Li, and Xiaonan Wu. "Numerical Solution of Blow-Up Problems for Nonlinear Wave Equations on Unbounded Domains." Communications in Computational Physics 14, no. 3 (2013): 574–98. http://dx.doi.org/10.4208/cicp.160412.111012a.

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AbstractThe numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered. Applying the unified approach, which is based on the operator splitting method, we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation, and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain. Then the finite difference method is used to solve the reduced problem on the bounded computational domain. Finally, a broad range of numerical examples are given to d
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Diabaté, Paterne A. T. "NUMERICAL BLOW-UP FOR A SEMILINEAR PARABOLIC EQUATION WITH A POTENTIAL." International Journal of Numerical Methods and Applications 20, no. 2 (2021): 115–33. http://dx.doi.org/10.17654/nm020020115.

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Ganon, Ardjouma, Manin Mathurin Taha, N'guessan Koffi, and Augustin Kidjegbo Toure. "Numerical blow-up for nonlinear diffusion equation with neumann boundary conditions." Journal of Nonlinear Sciences and Applications 14, no. 02 (2020): 80–88. http://dx.doi.org/10.22436/jnsa.014.02.03.

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Klein, Christian, and Nikola Stoilov. "Numerical Study of Blow-Up Mechanisms for Davey-Stewartson II Systems." Studies in Applied Mathematics 141, no. 1 (2018): 89–112. http://dx.doi.org/10.1111/sapm.12214.

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Korpusov, M. O., R. S. Shafir, and A. K. Matveeva. "Numerical Diagnostics of Solution Blow-Up in a Thermoelectric Semiconductor Model." Computational Mathematics and Mathematical Physics 64, no. 7 (2024): 1595–602. http://dx.doi.org/10.1134/s0965542524700647.

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