A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

Ruy Coimbra Charão; Alessandra Piske; Ryo Ikehata

doi:10.3934/dcds.2021189

Article Contents

2022, Volume 42, Issue 5: 2215-2255. Doi: 10.3934/dcds.2021189

This issue Previous Article Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains Next Article On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersion

A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

1.
Department of Mathematics, Graduate Program in Pure and Applied Mathematics, Federal University of Santa Catarina, 88040-900, Florianopolis, Brazil
2.
Department of Mathematics, Division of Educational Sciences, Graduate School of Humanities and Social Sciences, Hiroshima University, Higashi-Hiroshima 739-8524, Japan

^*Corresponding author: Ruy Coimbra Charão
^*Corresponding author: Ruy Coimbra Charão

Received: April 30, 2021

Revised: September 30, 2021

Published: May 2022

The work of the first author (R. C. CHARÃO) was partially supported by PRINT/CAPES - Process 88881.310536/2018-00, the work of the second author (A. PISKE) was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and the work of the third author (R. IKEHATA) was supported in part by Grant-in-Aid for Scientific Research (C)20K03682 of JSPS

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Abstract

We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in $ {{\bf R}}^{n} $, and study the asymptotic profile and optimal decay rates of solutions as $ t \to \infty $ in $ L^{2} $-sense. The operator $ L $ considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [7]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.

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Mathematics Subject Classification: Primary: 35G10, 35L05; Secondary: 35B40, 35C20, 35S05, 33C05.

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