## Double diffusion structure of logarithmically damped wave equations with a small parameter.(English)Zbl 1481.35067

Summary: We consider a wave equation with a nonlocal logarithmic damping depending on a small parameter $$0 < \theta < \frac{ 1}{ 2}$$. This research is a counter part of that was initiated by Charão-D’Abbicco-Ikehata considered in [R. C. Charão et al., Math. Methods Appl. Sci. 44, No. 18, 14003–14024 (2021; Zbl 1479.35089)] for the large parameter case $$\theta > \frac{ 1}{ 2}$$. We study the Cauchy problem for this model in $$\mathbb{R}^n$$ to the case $$\theta \in(0, \frac{ 1}{ 2})$$, and we obtain an asymptotic profile and optimal estimates in time of solutions as $$t \to \infty$$ in $$L^2$$-sense. An important discovery in this research is that in the case when $$n = 1$$, we can present a threshold $$\theta^\ast = \frac{ 1}{ 4}$$ of the parameter $$\theta \in(0, \frac{ 1}{ 2})$$ such that the solution of the Cauchy problem decays with some optimal rate for $$\theta \in(0, \theta^\ast)$$ as $$t \to \infty$$, while the $$L^2$$-norm of the corresponding solution never decays for $$\theta \in [ \theta^\ast, \frac{ 1}{ 2})$$ and, in particular, in the case $$\theta \in [ \theta^\ast, \frac{ 1}{ 2})$$ it shows an infinite time $$L^2$$ blow-up of the corresponding solutions. The former (i.e., $$\theta \in(0, \theta^\ast)$$ case) indicates an usual diffusion phenomenon, while the latter (i.e., $$\theta \in [ \theta^\ast, \frac{ 1}{ 2})$$ case) implies, so to speak, a singular diffusion phenomenon. Such a singular diffusion in the one dimensional case is a quite novel phenomenon discovered through our new model produced by logarithmic damping with a small parameter $$\theta$$. It might be already prepared in the usual structural damping case such as $$( - \Delta)^\theta u_t$$ with $$\theta \in(0, 1 / 2)$$, however unfortunately nobody has ever just pointed out even in the structural damping case.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35B20 Perturbations in context of PDEs 35L15 Initial value problems for second-order hyperbolic equations 35S05 Pseudodifferential operators as generalizations of partial differential operators

Zbl 1479.35089
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### References:

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