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

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
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