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Page 3355 of Mathematical Reviews Vol. , Issue 92f [page]

1992 Mathematical Reviews  
For fixed parameter values and arbitrarily fine discretizations we construct solutions which blow up in finite time for two semi- discrete schemes.  ...  This connection between the blow-up phenome- non and spurious steady states is also explored for Galerkin and nonlinear Galerkin semi-discrete approximations.  ... 

Page 4063 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews  
An initial-boundary value problem for a semilinear parabolic equa- tion with parameter ¢~? in front of the nonlinear reaction term is considered.  ...  (RA-UBAS; Buenos Aires) Adaptive numerical schemes for a parabolic problem with blow-up. (English summary) IMA J. Numer. Anal. 23 (2003), no. 3, 439-463.  ... 

Page 6126 of Mathematical Reviews Vol. , Issue 97J [page]

1997 Mathematical Reviews  
In this paper the authors consider a semi-implicit discretization of a scalar one-dimensional parabolic equation of the form u, = ux. + f(u) with homogeneous Dirichlet boundary conditions in [0, 1].  ...  Carvalho (BR-SPL3-IC; Sado Carlos) 97j:35080 35K60 35840 Yang, Shi Xin (PRC-XIAM; Xiamen) Extinction of solutions of semilinear heat equations with a gradient term. (Chinese.  ... 

Page 5483 of Mathematical Reviews Vol. , Issue 2001H [page]

2001 Mathematical Reviews  
The paper deals with the large-time behaviour of sign-changing solutions of inhomogeneous parabolic equations and systems.  ...  The well-posedness of this problem was already studied by the authors. Let up be a solution of the initial-boundary value problem without source term.  ... 

Page 1049 of Mathematical Reviews Vol. , Issue 2004b [page]

2004 Mathematical Reviews  
1049 35K Parabolic equations and systems priate hypotheses, we establish the local existence and uniqueness of a positive classical solution and obtain that the solution either exists globally or blows  ...  A weakly coupled system of parabolic equations 8,u'?)  ... 

Page 6123 of Mathematical Reviews Vol. , Issue 99i [page]

1999 Mathematical Reviews  
Qing Fang (J-EHIMSS; Matsuyama) 991:35066 35K55 35B05 Zaag, Hatem (F-ENS-MI; Paris) Blow-up results for vector-valued nonlinear heat equations with no gradient structure. (English summary) Ann. Inst.  ...  The reaction-diffusion equation u, = Au + (i+ id)|u\?~'u, ue, is considered. The author proves existence of a blow-up solution for small 6 and suitable initial data up.  ... 

Page 3348 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews  
A 8 (1993), no. 4, 351-359. Summary: “The blow-up of the solutions to initial-boundary value problems for a class of nonlinear pseudohyperbolic equations u,, — Uxx = au?  ...  Nguyén Hong Thai (PL-SZCZ; Szczecin) 95:35030 35B40 35B05 35L70 Yang, Zhi Jian Blow-up of solutions to initial-boundary value problems for a class of nonlinear evolution equations. (Chinese.  ... 

Parabolic Monge–Ampère methods for blow-up problems in several spatial dimensions

C J Budd, J F Williams
2006 Journal of Physics A: Mathematical and General  
The scheme is based on computing a Legendre transformation from a regular to a spatially non-uniform mesh via the solution of a relaxed form of the Monge-Ampere equation.  ...  Numerical examples are presented in one and two dimensions which demonstrate the effectiveness of this approach.  ...  The map from computational to physical coordinates is determined in terms of the gradient of a mesh potential Q which satisfies a parabolic partial differential equation.  ... 
doi:10.1088/0305-4470/39/19/s06 fatcat:6ykzsga5bnhwfdrymgebjrwdgy

Refined blow-up asymptotics for a perturbed nonlinear heat equation with a gradient and a non-local term [article]

Bouthaina Abdelhedi, Hatem Zaag
2021 arXiv   pre-print
We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term.  ...  In some earlier works [1, 2], we constructed a solution u for that equation such that u and ∇ u both blow up at the origin and only there. We also gave the final blow-up profile.  ...  Zaag, Construction of a blow-up solution for a perturbed nonlinear heat equation with a gradient term, J. Differential Equations, 272 (2021) 1-45. [2] B. Abdelhedi and H.  ... 
arXiv:2112.03039v1 fatcat:zhew4vze5bdchotmb23iu3kxnu

Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations [article]

Jian-Guo Liu and Li Wang and Zhennan Zhou
2016 arXiv   pre-print
We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver.  ...  With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.  ...  The initial condition is taken the same as in Blow up In this subsection, we focus on the cases when ρ blows up, and show that our schemes, (4.8) In Fig. 5 on the left, we plot a slice of solution  ... 
arXiv:1610.03016v2 fatcat:g3mvrzk6qze4hlbvifecwukrsq

Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

José A. Carrillo, Helene Ranetbauer, Marie-Therese Wolfram
2016 Journal of Computational Physics  
Positivity of solutions as well as energy decrease of the semi-discrete scheme are guaranteed by its construction. We illustrate this properties with various examples in spatial dimension one and two.  ...  In this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance  ...  Acknowledgments JAC was partially supported by the Royal Society via a Wolfson Research Merit Award.  ... 
doi:10.1016/j.jcp.2016.09.040 fatcat:y7hymheapjdh7jl73gk6chbiii

author index volume

2007 Journal of Computational and Applied Mathematics  
Vulkov, Blow-up of continuous and semidiscrete solutions to elliptic equations with semilinear dynamical boundary conditions of parabolic type 414-434 Koulaei, M.H. and F.  ...  matrices 258-265 Du, L., Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources 237-247 Gegenhasi see Wang, H.  ... 
doi:10.1016/s0377-0427(07)00064-7 fatcat:32gvtji2gvf5lfvlq74xjnkpsi

Convergence of a Semi-Discrete Numerical Method for a Class of Nonlocal Nonlinear Wave Equations [article]

H. A. Erbay, S. Erbay, A. Erkip
2018 arXiv   pre-print
In this article, we prove the convergence of a semi-discrete numerical method applied to a general class of nonlocal nonlinear wave equations where the nonlocality is introduced through the convolution  ...  Finally, we present some numerical experiments that confirm numerically both the expected convergence rate of the semi-discrete scheme and the ability of the method to capture finite-time blow-up of solutions  ...  blow-up of solutions.  ... 
arXiv:1805.07264v1 fatcat:ccutdizbybadpjongbxyrb4fiu

Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations

H.A. Erbay, S. Erbay, A. Erkip
2018 Mathematical Modelling and Numerical Analysis  
In this article, we prove the convergence of a semi-discrete numerical method applied to a general class of nonlocal nonlinear wave equations where the nonlocality is introduced through the convolution  ...  Finally, we present some numerical experiments that confirm numerically both the expected convergence rate of the semi-discrete scheme and the ability of the method to capture finite-time blow-up of solutions  ...  blow-up of solutions.  ... 
doi:10.1051/m2an/2018035 fatcat:2qrfatubljbxhnpupu4vvrcyeq

Page 1134 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
Blowing-up of solutions for a class of higher-order reaction-diffusion equations. (Chinese. English and Chinese summaries) Math. Practice Theory 28 (1998), no. 3, 220-225.  ...  Sufficient conditions for blow-up of solutions for the two systems are obtained.  ... 
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