TY - JOUR
T1 - Asymptotics of solutions to the fourth order Schrödinger type equation with a dissipative nonlinearity
AU - Segata, Junichi
AU - Shimomura, Akihiro
PY - 2006/1/1
Y1 - 2006/1/1
N2 - In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrödinger type equation with a cubic dissipative nonlinearity λ|u|2u, where λ is a complex constant satisfying Im λ < 0, is studied. This nonlinearity is a long-range interaction. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as t → +∞ in a suitable function space. Our modified free dynamics decays like (t log t)-1/2 as t → ∞.
AB - In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrödinger type equation with a cubic dissipative nonlinearity λ|u|2u, where λ is a complex constant satisfying Im λ < 0, is studied. This nonlinearity is a long-range interaction. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as t → +∞ in a suitable function space. Our modified free dynamics decays like (t log t)-1/2 as t → ∞.
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U2 - 10.1215/kjm/1250281786
DO - 10.1215/kjm/1250281786
M3 - Article
AN - SCOPUS:33846315496
VL - 46
SP - 439
EP - 456
JO - Kyoto Journal of Mathematics
JF - Kyoto Journal of Mathematics
SN - 0023-608X
IS - 2
ER -