Uniform decay rate estimates for the 2D wave equation posed in an inhomogeneous medium with exponential growth source term, locally distributed nonlinear dissipation, and dynamic Cauchy–Ventcel–type boundary conditions

José G. Simion Antunes, Marcelo M. Cavalcanti and Valéria N. Domingos Cavalcanti

Mathematische Nachrichten, 2024, vol. 297, issue 3, 962-997

Abstract: We study the wellposedness and stabilization for a Cauchy–Ventcel problem in an inhomogeneous medium Ω⊂R2$\Omega \subset \mathbb {R}^2$ with dynamic boundary conditions subject to a exponential growth source term and a nonlinear damping distributed around a neighborhood ω$\omega$ of the boundary according to the geometric control condition. We, in particular, generalize substantially the work due to Almeida et al. (Commun. Contemp. Math. 23 (2021), no. 03, 1950072), in what concerns an exponential growth for source term instead of a polynomial one. We give a proof based on the truncation of a equivalent problem and passage to the limit in order to obtain in one shot, the energy identity as well as the observability inequality, which are the essential ingredients to obtain uniform decay rates of the energy. We show that the energy of the equivalent problem goes uniformly to zero, for all initial data of finite energy taken in bounded sets of finite energy phase space. One advantage of our proof is that the decay rate is independent of the nonlinearity.

Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.202200288

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:297:y:2024:i:3:p:962-997

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().