In this paper, we mainly consider large time behavior for the classical free wave equation $u_{ tt}-\Delta u=0$ in $\mathbb{ R}^n$. We derive some large time optimal estimates for the quantity of solution $\|u(t,\cdot)\|_{ L^2}$ 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.
Categories: math.AP