In this paper, we prove new results on the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients, not necessarily in divergence form. Under suitable assumptions on the coefficients and on the source term, we establish the LAP for space dimensions 2 and 3. This result is extended to one space dimension with an appropriate modification. We also quantify the LAP and thus provide estimates for the convergence of the time-domain solution to the frequency-domain solution. Our proofs are based on time-decay results of solutions of some auxiliary problems. The obtained results are illustrated numerically on radially symmetric problems in dimensions 1,2 and 3.
翻译:在本文中,我们用非恒定系数的波形方程式限制振幅原则(LAP)的有效性证明有新的结果,不一定以差异形式出现,根据对系数和源术语的适当假设,我们为空间2和3确定LAP,这一结果经适当修改后扩大到一个空间层面,我们还对LAP进行量化,从而提供时间-域解决方案与频率-域解决方案趋同的估计值。我们的证据以某些辅助问题解决方案的时间-下降结果为基础。所获得的结果用数字方式用参数1-2和3的对称问题来说明。