Existence, uniqueness, and exponential stability for the Kirchhoff equation in whole hyperbolic space

In this paper, we prove the existence, uniqueness, and exponential stability for a damped nonlinear wave equation of Kirchhoff type which is defined in whole hyperbolic space B N . Our strategy consists of changing the problem into a singular problem defined in the unitary ball of R N endowed with t...

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Bibliographic Details
Published in:Journal of evolution equations 2022-09, Vol.22 (3), Article 67
Main Authors: Carrião, Paulo Cesar, Vicente, André
Format: Article
Language:English
Subjects:
Analysis
Galerkin method
Hyperbolic coordinates
Mathematics
Mathematics and Statistics
Stability
Uniqueness
Wave equations
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Summary:In this paper, we prove the existence, uniqueness, and exponential stability for a damped nonlinear wave equation of Kirchhoff type which is defined in whole hyperbolic space B N . Our strategy consists of changing the problem into a singular problem defined in the unitary ball of R N endowed with the Euclidean metric. One difficulty is to prove the existence of solution and the Faedo–Galerkin method was our main tool. It is well known that when we deal with the Kirchhoff model defined in whole space R N , the exponential stability is not expected. In this work, we prove that, in the hyperbolic space, the problem is exponentially stable. The main tool to reach the result is to combine the classical Nakao’s techniques with the use of Hardy inequality.
ISSN:1424-3199
1424-3202
DOI:10.1007/s00028-022-00821-7