Asymptotic Expansions with Optimal Convergent Rates of Global Solutions to Semilinear Parabolic Equations

Asymptotic Expansions with Optimal Convergent Rates of Global Solutions to Semilinear Parabolic Equations Asymptotic expansion, Complex Ginzburg-Landau equation, Convection-diffusion equation This doctoral dissertation is devoted to the higher-order asymptotic expansions of global solutions to the complex Ginzburg-Landau type equation or the convection-diffusion equation. For the former equation, the Taylor expansion of the complex Gaussian kernel with respect to both the spatial and time variables plays a crucial role. For the latter equation, the asymptotic self-similarity of the asymptotic profiles enables us to determine whether the decay rates of the remainders are optimal. This dissertation also investigates the weighted estimates of the global solutions, which serve as the basis for the higher-order asymptotic expansions. In this context, a new approach based on the commutation relations between the complex Ginzburg-Landau semigroup and weights is developed.