Evolution of the radius of analyticity for the generalized Benjamin equation

Renata O. Figueira; Mahendra Panthee

doi:10.3934/dcds.2023039

Article Contents

2023, Volume 43, Issue 8: 3043-3059. Doi: 10.3934/dcds.2023039

This issue Previous Article Qualitative properties for solutions to conformally invariant fourth order critical systems Next Article Asymptotic properties of the Boussinesq equations with Dirichlet boundary conditions

Evolution of the radius of analyticity for the generalized Benjamin equation

Renata O. Figueira^1, and
Mahendra Panthee^1, ,

1.
Department of Mathematics, University of Campinas, 13083-859, SP, Brazil

^*Corresponding author: Mahendra Panthee
^*Corresponding author: Mahendra Panthee

Received: January 2023

Revised: April 2023

Early access: April 2023

Published: August 2023

The first author is supported by FAPESP, Brazil (#2021/04999-9).
The second author is supported by FAPESP, Brazil (#2020/14833-8) and CNPq, Brazil (#307790/2020- 7).

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Abstract

In this work we consider the initial value problem (IVP) for the generalized Benjamin equation

$ \begin{equation} \begin{cases} \partial_t u-l\mathcal{H} \partial_x^2u-\partial_x^3u+u^p\partial_xu = 0, \quad x,\; t\in \mathbb{R},\; p\geq 1, \\ u(x,0) = u_0(x), \end{cases} \;\;\;\;\;\;\;\;\;(1)\end{equation} $

where $ u = u(x,t) $ is a real valued function, $ 0<l<1 $ and $ \mathcal{H} $ is the Hilbert transform. This model was introduced by Benjamin [3] and describes unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity.

We prove that the local solution to the IVP (1) for given data in the spaces of functions analytic on a strip around the real axis continue to be analytic without shrinking the width of the strip in time. We also study the evolution in time of the radius of spatial analyticity and show that it can decrease as the time advances. Finally, we present an algebraic lower bound on the possible rate of decrease in time of the uniform radius of spatial analyticity.

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Mathematics Subject Classification: Primary: 35A20, 35Q53, 35Q35; Secondary: 35Q55, 35A20, 35Q53, 35B40, 35Q35.

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