Asymptotic behavior and finite-time blow-up of solutions to a nonlinear viscoelastic love equation

In this paper, we undertake a detailed investigation of the long-time dynamics of the Love equation with a viscoelastic term. First, by applying Banach's fixed point theorem, we establish the local existence and uniqueness of weak solutions. Next, under mild and general assumptions on the asymptotic behavior of the memory kernel, we derive sharp and explicit energy decay estimates, thereby recovering both optimal exponential and polynomial decay rates. Finally, employing the concavity method, we prove that weak solutions blow up in finite time under suitable conditions on the nonlinear source and the memory kernel.