Asymptotic behavior for Dirichlet problems of nonlinear Schrodinger equations with Landau damping on a half line
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 $$\displaylines{ u_{t}+iu_{xx}+i|u|^{2}u+|\partial _x|^{1/2}u=0,\quad t\geq 0,\; x\geq 0; \cr u(x,0)=u_{0}(x),\quad x>0 \cr u_x(0,t)=h(t),\...
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2017-06-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2017/157/abstr.html |
| Summary: | This article is a continuation of the study in [5],
where we proved the existence of solutions, global in time,
for the initial-boundary value problem
$$\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. |
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| ISSN: | 1072-6691 |