## 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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