This paper considers the asymptotic behavior of solutions of equations of evolutions, and concentrates on the analysis of the critical blow-up solutions for a class of evolutions for nonlinear Schrödinger equations in a bounded domain. More precisely, the numerical approximation  of the blow-up rate below the one of the known explicit explosive solutions is studied, which has strictly positive energy for the following initial-boundary value problem:

where  u is a complex-valued function of the variable Δ is the Laplace operator in and the time t ≥ 0.

The paper proposes a general setting to study and understand the behavior of the blow-up solutions in a finite time as a function of the parameters α, β, with initial condition u(0, x) = u₀, in the energy space  also in the case where  is large enough and its size d is taken as parameter. Some assumptions are found under which the solution of the above problem blows-up in a finite time, study the dynamics of blow-up solutions and estimate its blow-up time. Finally, some numerical experiments to illustrate the analysis have been provided.

discretizations, finite difference method, blow-up, finite element method, numerical blow-up time, nonlinear Schrödinger equation.

How to cite this article: N’takpe Jean-Jacques, L. Boua Sobo Blin, Nachid Halima and Kambire D. Gnowille, The limit of blow-up dynamics solutions for a class of nonlinear critical Schrödinger equations, Advances in Differential Equations and Control Processes 31(2) (2024), 207-238. https://doi.org/10.17654/0974324324011

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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