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Stable solutions and their spatial structure of the Ginzburg-Landau equation. (English) Zbl 0877.35049
Summary: In recent papers [{\it S. Jimbo} and {\it Y. Morita}, Nonlinear Anal., Theory Methods Appl. 22, No. 6, 753-770 (1994; Zbl 0798.35019), SIAM Math. Anal. 27, 1360-1385 (1996)] and [{\it S. Jimbo}, {\it Y. Morita} and {\it J. Zhai}, Commun. Partial Differ. Equations 20, No. 11-12, 2093-2112 (1995; Zbl 0841.35041)], we studied the Ginzburg-Landau equation $\Delta\Phi+\lambda(1-|\Phi|^2) \Phi=0$, $\Phi=u_1+ iu_2$ in a bounded domain $\Omega\subset\bbfR^n$ with homogeneous Neumann boundary conditions. Those works revealed the instability of non-constant solutions in any convex domain and the existence of stable non-constant solutions in topologically non-trivial domains. This report surveys these studies together with an introduction of a new result.
35J65Nonlinear boundary value problems for linear elliptic equations
35B35Stability of solutions of PDE
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