I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial data, in a non-trapping medium without obstacles, in $2$ or $3$ spatial dimensions.

In particular, if $u$ is a solution of

$$ \begin{cases} u_{tt} - \nabla\left(c^2(x)\nabla \cdot u\right) = 0 \textrm{ in } \mathbb R \times \mathbb R^n \\ u(0,x) = 0, \quad u_t(0,x) = g(x), \end{cases} $$ with $g$ compactly supported, (and more generally, $u(0,x) = f$ compactly supported, though I don't presently need this), I'm looking for a reference to an estimate of the form

Given $B \subset \mathbb{R}^n$, there exists $T_0$ (depending on $B$) such that the solution of the above IVP satisfies $$\left|u_t(t,x) \right| \leq C\,\eta(t)\,\|g\|_{L^2}, \quad t > T_0, x \in B$$ where $C$ depends on $B$, and $\eta(t) = t^{-2}$ if $n=2$, and $\eta(t) = e^{-\delta t}$ if $n=3$, where $\delta$ depends on $c(x)$.

More generally, the left-hand-side could be a higher-order derivative, and $C$ and $\eta$ (in $n=2$) will depend on the order.

For my purposes, it is also fine if the left-hand-side is the $L^2(B)$-norm of $u_t$ (rather than the sup norm above).

My search started with Hristova Time reversal in thermoacoustic tomography - an error estimate, where a similar result is stated (Theorem 2), but for an operator in "principal" form $\partial_{tt} - c^2(x) \Delta$.

Following the other references therein, Egorov and Shubin (Linear partial differential equations: Foundations of the classical theory) contains the same result, and Vainberg On the short-wave asymptotic behavior... seems to be concerned with the exterior problem, i.e. in $\mathbb R^n \backslash \Omega_0 $ where $\Omega_0$ is a bounded domain, and either Dirichlet or Neumann values are specified on $\partial \Omega_0$. (though admittedly, I am not sure I understand all the results in this paper after only a quick scan, it is somewhat dense).

While I think such obstacles usually worsen the results, I'm not comfortable citing these unless they specifically allow $\Omega_0 = \emptyset$.

I have also looked at Morawetz Exponential decay of solutions of the wave equation, but this seems to assume a constant-speed medium.

Searching through the more modern literature produces a LARGE variety of results, on variations of this result (for damping media, for non-smooth coefficients, etc.) which I have sort of found myself lost among. Can anyone point me in the right direction? Much thanks.

  • $\begingroup$ Interesting question. Could you find an answer in the meantime? $\endgroup$ – tomglabst Dec 4 '15 at 7:39

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