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.


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
Full Text: DOI arXiv


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