Strongly damped wave equations with mass-like terms of the logarithmic-Laplacian

Abstract

We consider strongly damped wave equations with logarithmic mass-like terms with a parameter θ∈(0,1]. This research is a part of a series of wave equations that was initiated by Charão-Ikehata [6], Charão-D'Abbicco-Ikehata considered in [5] depending on a parameter θ∈(1/2,1) and Piske-Charão-Ikehata [26] for small parameter θ∈(0,1/2). We derive a leading term as t→∞ of the solution, and by using it, a growth and a decay property of the solution itself can be precisely studied in terms of L2-norm. An interesting aspect appears in the case of n=1, roughly speaking, a small θ produces a diffusive property, and a large θ gives a kind of singularity, expressed by growth rates.