Well-posedness for a Third-Order PDE with Dissipation - Santiago Ayala, Yolanda Silvia (2025)
In this work, we prove that the Cauchy problem associated with a third-order equation with dissipation in periodic Sobolev spaces admits a unique solution. We also show that the solution depends continuously on the initial data. Our approach combines both an intuitive method, based on Fourier theory, and a more abstract framework using semigroup theory. Furthermore, by employing an alternative method, we demonstrate the uniqueness of the solution through its dissipative nature, drawing inspiration from the contributions of Iorio [1] and Santiago [2]. To deepen and enrich our study, we investigate the infinite dimensional space in which differentiability occurs and its connection to the initial data. Finally, we extend our results to equations of arbitrary nth order.