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arXiv · 2604.08041

A generalized time-fractional Kuramoto-Sivashinsky equation in the Schwartz space: global solvability and stability

R. R. Ashurov·Z. A. Sobirov·R. B. Norkulova
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Abstract

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.

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R. R. Ashurov, Z. A. Sobirov, R. B. Norkulova. 2026-09-22. A generalized time-fractional Kuramoto-Sivashinsky equation in the Schwartz space: global solvability and stability. https://arxiv.org/abs/2604.08041

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