THE LIMIT OF BLOW-UP DYNAMICS SOLUTIONS FOR A CLASS OF NONLINEAR CRITICAL SCHRÖDINGER EQUATIONS

N’takpe Jean-Jacques; L.  Boua Sobo Blin; Nachid Halima; Kambire D.  Gnowille

N’takpe Jean-Jacques Department of Mathematics Nangui Abrogoua University, Ivory Cost
L. Boua Sobo Blin Laboratoire des Sciences et Technologies de 1environnement (LSTE), Jean Lorougon GUEDE University, Daloa, Ivory Cost
Nachid Halima (Department of Mathematics Nangui Abrogoua University, Ivory Cost)
Kambire D. Gnowille (Department of Mathematics Nangui Abrogoua University, Ivory Cost)

Article ID: 2434

Keywords: discretizations; finite difference method; blow-up; finite element method; numerical blow-up time; nonlinear Schrödinger equation

Abstract

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: $$ (P) \begin{cases}u_t(t, x)-i \alpha \Delta u(t, x)-i \beta f(t, x)=0, & x \in \mathbb{R}^d, t \in(0, T), \\ u(x, 0)=u_0(x), & x \in \mathbb{R}^d,\end{cases} $$ where $i=\sqrt{-1}, \quad \alpha \in \mathbb{R}, \quad \beta \in \mathbb{R}, \quad d \geq 1, \quad u$ is a complex-valued function of the variable $x \in \mathbb{R}^d, \Delta$ is the Laplace operator in $\mathbb{R}^d$ and the time $t \geq 0$. 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 $\alpha, \beta$, with initial condition $u(0, x)=u_0$, in the energy space $H^1 \in \mathbb{R}^d$, also in the case where $\mathbb{R}^d$ 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.

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