A generalized time-fractional Kuramoto-Sivashinsky equation in the Schwartz space: global solvability and stability
We study the Cauchy problem for a generalized time-fractional Kuramoto--Sivashinsky equation on \(\mathbb R\) with the regularized Caputo derivative in the Schwartz space. The linear problem is solved by the Fourier transform and a Mittag--Leffler representation, with preservation of the Schwartz class. For the nonlinear problem, local solvability is established by successive approximations with convergence in the full Schwartz topology. A continuation principle accounting for the complete fractional memory is developed: the history term is estimated uniformly with respect to the continuation point, which yields global solvability on every finite time interval. An \(L^2(\mathbb R)\) stability estimate is also obtained, implying uniqueness.
math.AP↗