Evolution of the radius of analyticity for mKdV-type equations

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 $H^{σ,s}(\mathbb{R})$ 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 $σ(T)$ 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 $H^{σ,s}(\mathbb{R})$ and demonstrate that the local solution can be extended globally in time and admits constant lower bounds for the radius of analyticity $σ(t)$ 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.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Lemma 2.1
• Lemma 2.2