Cauchy problem for the time-fractional generalized Kuramoto-Sivashinsky equation

TL;DR

This paper establishes the global solvability of a generalized time-fractional Kuramoto-Sivashinsky equation in Schwartz space using Fourier analysis and successive approximations.

Contribution

It provides a rigorous analytical framework for fractional Kuramoto-Sivashinsky equations in topological function spaces, combining linear Fourier analysis with nonlinear approximation methods.

Findings

Proved global solvability in Schwartz space.

Developed uniform estimates for solutions.

Established convergence of approximation sequences.

Abstract

This paper studies global solvability of the Cauchy problem for a generalized time-fractional Kuramoto-Sivashinsky equation in the Shwartz space, which is a complete topological space generated by a family of semi-norms. The main approach is based on separating the linear and nonlinear parts of the equation and applying appropriate analytical methods to each of them. The linear part of the equation is analyzed using the Fourier transform. The nonlinear equation is treated by the method of successive approximations, and uniform estimates for the constructed sequence are derived. Furthermore, taking into account the topological structure of the Schwartz space, the convergence of the sequence in the sense of semi-norms is rigorously established. The results provide a rigorous analytical framework for fractional Kuramoto-Sivashinsky type equations in topological function spaces.