Large time behavior for the classical wave equation with different regular data and its applications

In this paper, we mainly consider large time behavior for the classical free wave equation $u_{tt}-\mathrm{\Delta }u=0$ in ${\mathbb{R}}^n$. We derive some large time optimal estimates for the quantity of solution $\mathrm{\parallel }u(t,\cdot ){\mathrm{\parallel }}_{L^2}$ with initial data belonging to $L^2$ or with additional weighted $L^1$ integrabilities. Particularly, some thresholds are discovered for the (local or global in time) stabilization of this quantity. We also apply these results to the wave equation with scale-invariant terms, the undamped $\sigma$-evolution equation, the critical Moore-Gibson-Thompson equation, and the linearized compressible Euler system.