Electron. J. Differential Equations, Vol. 2017 (2017), No. 157, pp. 1-16.

Asymptotic behavior for Dirichlet problems of nonlinear Schrodinger equations with Landau damping on a half line

Liliana Esquivel

Abstract:
This article is a continuation of the study in [5], where we proved the existence of solutions, global in time, for the initial-boundary value problem
$$\displaylines{ u_{t}+iu_{xx}+i|u|^{2}u+|\partial _x|^{1/2}u=0,\quad t\geq 0,\; x\geq 0; \cr u(x,0)=u_{0}(x),\quad x>0 \cr u_x(0,t)=h(t),\quad t>0, }$$
where $|\partial _x|^{1/2}$ is the module-fractional derivative operator defined by the modified Riesz Potential
$$ |\partial _x|^{1/2}=\frac{1}{\sqrt{2\pi }}\int_{0}^{\infty } \frac{\hbox{sign}(x-y) }{\sqrt{|x-y|}}u_y(y)dy. $$
Here, we study the asymptotic behavior of the solution.

Submitted March 6, 2017. Published June 28, 2017.
Math Subject Classifications: 35Q55, 35B40.
Key Words: Fractional Schrodinger equation; boundary value; Landau damping.

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Liliana Esquivel
Universidad de Pamplona
Santander, Colombia
email: lesquivel@unipamplona.edu.co

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