Exponential decay for semilinear wave equation with localized damping in the hyperbolic space

Author(s)

Carriao, Paulo Cesar ; Miyagaki, Olimpio Hiroshi ; Vicente, Andre

• Document type: Journal article
• Source: Mathematische Nachrichten; v. 296, n. 1, p. 22-pg., 2022-11-01.

Abstract

In this paper, we study the exponential decay of the energy associated to an initial value problem involving the wave equation on the hyperbolic space B-N. The space B-N is the unit disc {x is an element of R-N : |x| < 1} of R-N endowed with the Riemannian metric g given by g(ij) = p(2)delta(ij), where p(x)=2/1-|x|(2) and delta(ij) = 0, if i not equal j. Making an appropriate change, the problem can be seen as a singular problem on the boundary of the open ball B-1 = { x is an element of R-N ; |x| < 1} endowed with the euclidean metric. The proof is based on the multiplier techniques combined with the use of Hardy's inequality, in a version due to the Brezis-Marcus, which allows us to overcome the difficulty involving the singularities. (AU)

• FAPESP's process: 19/24901-3 - Critical nonlocal quasilinear problem: existence, multiplicity and properties of the solutions
• Grantee: Olimpio Hiroshi Miyagaki
• Support Opportunities: Regular Research Grants