Nonexistence of non-trivial global weak solutions for higher-order nonlinear Schrodinger equations
We study the initial-value problem for the higher-order nonlinear Schrodinger equation $$ i\partial_{t}u-(-\Delta)^{m}u=\lambda| u|^{p}, $$ subject to the initial data $$ u(x,0)=f(x), $$ where $u=u(x,t)\in\mathbb{C}$ is a complex-valued function, $(x,t)\in\mathbb{R}^{N}\times[0,+\infty)$,...
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-12-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/312/abstr.html |
| Summary: | We study the initial-value problem for the higher-order nonlinear
Schrodinger equation
$$
i\partial_{t}u-(-\Delta)^{m}u=\lambda| u|^{p},
$$
subject to the initial data
$$
u(x,0)=f(x),
$$
where $u=u(x,t)\in\mathbb{C}$ is a complex-valued function,
$(x,t)\in\mathbb{R}^{N}\times[0,+\infty)$, $p>1$, $m\geq 1$,
$\lambda\in\mathbb{C}\backslash\{0\},$ and $f(x)$
is a given complex-valued function. We prove nonexistence of
a nontrivial global weak solution. Furthermore, we prove that
the $L^2$-norm of the local in time $L^2$-solution blows
up at a finite time. |
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| ISSN: | 1072-6691 |