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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

Citations:

Zbl 1479.35089
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References:

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