2021/09/21 by Alessandra Piske, Piske, Alessandra, Ruy Coimbra Charão +3
Engineering · Mathematics · #35B20 #35B40 #35L05 #35R12 #35S05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2109.09944
openalex publication_date 2021/09/21 · openalex created_date 2022/11/27 · openalex updated_date 2026/07/28
We consider a wave equation with a nonlocal logarithmic damping depending on a small parameter θ∈ (0,1/2). This research is a counter part of that was initiated by Charao-D'Abbicco-Ikehata considered in [5] for the large parameter case θ∈ (1/2,1). We study the Cauchy problem for this model in the whole space for the small parameter case, and we obtain an asymptotic profile and optimal estimates in time of solutions as time goes to infinity in L2-sense. An important discovery in this research is that in the one dimensional case, we can present a threshold θ* = 1/4 of the parameter θ such that the solution of the Cauchy problem decays with some optimal rate for θ∈ (0,θ*), while the L2-norm of the corresponding solution blows up in infinite time for θ∈ [θ*,1/2). The former (i.e., θ∈ (0,θ*) case) indicates an usual diffusion phenomenon, while the latter (i.e., θ∈ [θ*,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 θ.