TY - JOUR
T1 - Local well-posedness of the complex Ginzburg–Landau equation in bounded domains
AU - Kuroda, Takanori
AU - Ôtani, Mitsuharu
N1 - Funding Information:
The second author’s research was partly supported by the Grant-in-Aid for Scientific Research [Grant Number 1 5K13451 ], the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan .
Publisher Copyright:
© 2018 Elsevier Ltd
PY - 2019/2
Y1 - 2019/2
N2 - In this paper, we are concerned with the local well-posedness of the initial–boundary value problem for complex Ginzburg–Landau (CGL) equations in bounded domains. There are many studies for the case where the real part of its nonlinear term plays as dissipation. This dissipative case is intensively studied and it is shown that (CGL) admits a global solution when parameters appearing in (CGL) belong to the so-called CGL-region. This paper deals with the non-dissipative case. We regard (CGL) as a parabolic equation perturbed by monotone and non-monotone perturbations and follows the basic strategy developed in Ôtani (1982) to show the local well-posedness of (CGL) and the existence of small global solutions provided that the nonlinearity is the Sobolev subcritical.
AB - In this paper, we are concerned with the local well-posedness of the initial–boundary value problem for complex Ginzburg–Landau (CGL) equations in bounded domains. There are many studies for the case where the real part of its nonlinear term plays as dissipation. This dissipative case is intensively studied and it is shown that (CGL) admits a global solution when parameters appearing in (CGL) belong to the so-called CGL-region. This paper deals with the non-dissipative case. We regard (CGL) as a parabolic equation perturbed by monotone and non-monotone perturbations and follows the basic strategy developed in Ôtani (1982) to show the local well-posedness of (CGL) and the existence of small global solutions provided that the nonlinearity is the Sobolev subcritical.
KW - Complex Ginzburg–Landau equation
KW - Initial–boundary value problem
KW - Local well-posedness
KW - Subdifferential operator
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U2 - 10.1016/j.nonrwa.2018.08.006
DO - 10.1016/j.nonrwa.2018.08.006
M3 - Article
AN - SCOPUS:85052870335
SN - 1468-1218
VL - 45
SP - 877
EP - 894
JO - Nonlinear Analysis: Real World Applications
JF - Nonlinear Analysis: Real World Applications
ER -