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For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation.

My question is...why is the above PDE so named? That is, how does $a$ represent a damping mechanism?

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The reason is that one expects solutions to decay to $0$ (under the right additional assumptions for $a$), just as in the case of a damped harmonic oscillator.

In the special case where $a > 0$ is a constant and $u(\cdot,0), \, u_t(\cdot, 0)$ are both multiples of an eigenfunction $\tilde u$ of $-\Delta_g$ with eigenvalue $\lambda > 0$, the solution is in fact $u(x,t) = \psi(t)\tilde u(x)$ where $\psi'' + a\psi' + \lambda \psi = 0$, by separation of variables. So the damped oscillator appears naturally.

In the general case where only $a \in C^\infty, \, a \ge 0$ is assumed, the name "damped wave equation" mainly expresses a hope and a program. Without additional geometric assumptions, it can't be expected that solutions will in fact decay to $0$ (think of $M$ having two connected components, with $a = 0$ on one of them).

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