Double diffusion structure of logarithmically damped wave equations with a small parameter

Abstract

We consider a wave equation with a nonlocal logarithmic damping depending on a small parameter 0<θ<12. This research is a counter part of that was initiated by Charão-D'Abbicco-Ikehata considered in [5] for the large parameter case θ>12. We study the Cauchy problem for this model in Rn to the case θ∈(0,12), and we obtain an asymptotic profile and optimal estimates in time of solutions as t→∞ in L2-sense. An important discovery in this research is that in the case when n=1, we can present a threshold θ⁎=14 of the parameter θ∈(0,12) such that the solution of the Cauchy problem decays with some optimal rate for θ∈(0,θ⁎) as t→∞, while the L2-norm of the corresponding solution never decays for θ∈[θ⁎,12) and, in particular, in the case θ∈[θ⁎,12) it shows an infinite time L2 blow-up of the corresponding solutions. The former (i.e., θ∈(0,θ⁎) case) indicates an usual diffusion phenomenon, while the latter (i.e., θ∈[θ⁎,12) 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 θ. It might be already prepared in the usual structural damping case such as (−Δ)θut with θ∈(0,1/2), however unfortunately nobody has ever just pointed out even in the structural damping case.