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Well-posedness for the complex Ginzburg-Landau equations. (English) Zbl 1208.35143
Aiki, T. (ed.) et al., Current advances in nonlinear analysis and related topics. Collected papers of the conference on nonlinear evolution equations and related topics, Tokyo, Japan, October 10–12, 2009, the 4th Polish-Japanese days on current advances in applied nonlinear analysis and mathematical modelling issues, Warsaw, Poland, May 18–21, 2009 and the RIMS conference on nonlinear evolution equations and mathematical modeling, Kyoto, Japan, October 20–24, 2009. Tokyo: Gakkōtosho (ISBN 978-4-7625-0457-0/hbk). GAKUTO International Series. Mathematical Sciences and Applications 32, 429-442 (2010).
Summary: The $$L^p$$ well-posedness for the complex Ginzburg-Landau equation $$\partial u/\partial t- (\lambda+i\mu)\Delta u+ (\kappa+i\nu)|u|^{q-2}u-\gamma u=0$$ in $$\Omega\times(0,\infty)$$ with Dirichlet boundary condition $$u=0$$ on $$\partial\Omega\times(0,\infty)$$ is studied under the assumption that $$|\mu|/\lambda< 2\sqrt{p-1}/|p-2|$$ and $$2\leq q\leq 2+2p/N$$. The result due to Ginibre and Velo is improved in that the obtained solution is of class $$C^1$$, the Lipschitz continuous dependence of solutions on their initial data is obtained and their result remains true even if $$1<p<2$$ and $$\Omega$$ is a general domain in $$\mathbb R^N$$ with smooth boundary.
For the entire collection see [Zbl 1192.35004].

##### MSC:
 35Q56 Ginzburg-Landau equations 47H20 Semigroups of nonlinear operators 35B65 Smoothness and regularity of solutions to PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs