On the Initial-Value Problem for the Generalized Two-Dimensional Ginzburg–Landau Equation☆

Abstract The Ginzburg–Landau-type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics, physics, and chemistry. Most work so far concentrates on Ginzburg–Landau-type equations with one spatial dimension (1D). In this paper, the authors study a complex generalized Ginzburg–Landau equation with two spatial dimensions (2D). Sufficient conditions for the existence and uniqueness of global solutions for the initial-value problem of the generalized 2D Ginzburg–Landau equation are obtained. This rigorously establishes the foundation for further investigation of this type of model.

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