Asymptotic profiles for a wave equation with parameter-dependent logarithmic damping. (English) Zbl 1479.35089

Charão, Ruy Coimbra; D’Abbicco, Marcello; Ikehata, Ryo

Math. Methods Appl. Sci. 44, No. 18, 14003-14024 (2021).

Summary: We study a nonlocal wave equation with logarithmic damping, which is rather weak in the low-frequency zone as compared with frequently studied strong damping case. We consider the Cauchy problem for this model in \(\mathbb{R}^{n}\), and we study the asymptotic profile and optimal estimates of the solutions and the total energy as \(t \rightarrow \infty\) in \(L^2\) sense. In that case, some results on hypergeometric functions are useful.

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B45	A priori estimates in context of PDEs
35C20	Asymptotic expansions of solutions to PDEs
35L15	Initial value problems for second-order hyperbolic equations
35R09	Integro-partial differential equations
35S05	Pseudodifferential operators as generalizations of partial differential operators

Keywords:

asymptotic profiles; logarithmic \(\theta\) damping; optimal \(L^2\) and energy decay; nonlocal wave equation