Asymptotic behavior for a class of logarithmic wave equations with Balakrishnan-Taylor damping, nonlinear weak damping and strong linear damping

Abstract

In spaces with any dimension, a class of logarithmic wave equations with Balakrishnan-Taylor damping, nonlinear weak damping and strong linear damping was considered $ {u_{tt}} - M(t)\Delta u + \left( {g * \Delta u} \right)(t) + h({u_t}){u_t} - \Delta {u_t} + f(u) + u = u\ln {\left| u \right|^k} $ with Dirichlet boundary condition. Under a set of specified assumptions, we established the existence of global weak solutions and elucidated the decay rate of the energy function for particular initial data. This contribution extended and surpassed prior investigations, as documented by A. Peyravi (2020), which omitted the consideration of Balakrishnan-Taylor damping and strong linear damping while being confined to three spatial dimensions. Our findings underscored the pivotal role of these overlooked damping factors. Furthermore, our demonstration underscored that Balakrishnan-Taylor damping, weak damping and strong damping collectively induced an exponential decay, although the precise nature of this decay was contingent upon the differentiable function associated with the memory damping term $ {\zeta (t)} $. Consequently, the absence of the damping term in reference by A. Peyravi (2020) was unequivocally shown not to augment the decay rate. This insight enhanced our understanding of the nuanced dynamics involved and contributed to the refinement of existing models.

Keywords

• global weak solutions
• energy decay
• logarithmic wave equation
• balakrishnan-taylor damping