Evolution of the radius of analyticity for mKdV-type equations

TL;DR

This paper establishes new lower bounds for the evolution of the radius of analyticity in solutions to mKdV-type equations, showing algebraic decay for standard mKdV and constant bounds for generalized mKdV with damping.

Contribution

It provides the first global-in-time lower bounds for the radius of analyticity for mKdV equations with damping, improving previous results for the standard mKdV.

Findings

Algebraic lower bound $oldsymbol{cT^{-rac{1}{2}}}$ for mKdV solutions.

Constant lower bounds for the radius of analyticity in mKdV with damping.

Global well-posedness with preserved analyticity radius for mKdV with damping.

Abstract

In this paper, we obtain new lower bounds for the evolution of the radius of analyticity of solutions to two initial value problems (IVPs) with initial data belonging to the class of analytic functions $H^{\sigma,s}(\mathbb{R})$ defined via a hyperbolic cosine weight. First, we consider the IVP for the modified Korteweg-de Vries (mKdV) equation. For this problem, we prove that the evolution of the radius of analyticity $\sigma(T)$ of the solution admits an algebraic lower bound $cT^{-\frac 12}$ for some $c>0$ and given arbitrarily large $T>0$. Next, we analyze the IVP for the mKdV equation with generalized dispersion (mKdVm) and a damping term. For this problem, we guarantee the local well-posedness in $H^{\sigma,s}(\mathbb{R})$ and demonstrate that the local solution can be extended globally in time and admits constant lower bounds for the radius of analyticity $\sigma(t)$ as time goes to infinity. The outcome of this paper concerning the mKdV equation represents an improvement on that achieved by the authors' previous work in [R. O. Figueira and M. Panthee, New lower bounds for the radius of analyticity for the mKdV equation and a system of mKdV-type equations, J. Evol. Equ. 24 No. 42 (2024)]. As far as we know, the results for the mKdVm with damping are new.