Rapid Boundary Stabilization of One-Dimensional Nonlinear Parabolic Equations

We investigate the rapid boundary exponential stabilization of one-dimensional nonlinear parabolic equations by combining the framework of Coron and Trélat [ SIAM J. Control Optim., 43 (2004), pp. 549–569] with the frequency Lyapunov method introduced by Xiang [ ESAIM Control Optim. Calc. Var., 30 (2024), 40]. We construct a boundary feedback control that achieves the local rapid exponential stabilization under minimal low-frequency structural requirements. This design, when coupled with the intrinsic global dissipative dynamics of the equations, yields global stabilization results for both dissipative superlinear heat equations and viscous Burgers equations.