# American Institute of Mathematical Sciences

October  2022, 21(10): 3389-3405. doi: 10.3934/cpaa.2022105

## On the exponential time-decay for the one-dimensional wave equation with variable coefficients

 1 Institute of Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria 2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria 3 St. Petersburg Department of V. A. Steklov Mathematical Institute, RAS, Fontanka 27, 191023 St. Petersburg, Russia

*Corresponding author

Received  January 2022 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: A. Arnold, S. Geevers, and I. Perugia have been funded by the Austrian Science Fund (FWF) through the project F 65 "Taming Complexity in Partial Differential Systems". I. Perugia has also been funded by the FWF through the project P 29197-N32. A. Arnold and D. Ponomarev were supported by the bi-national FWF-project I3538-N32

We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates.

Citation: Anton Arnold, Sjoerd Geevers, Ilaria Perugia, Dmitry Ponomarev. On the exponential time-decay for the one-dimensional wave equation with variable coefficients. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3389-3405. doi: 10.3934/cpaa.2022105
##### References:
 [1] A. Arnold, J. Carrillo and M. Tidriri, Large-time behavior of discrete kinetic equations with non-symmetric interactions, Math. Models Methods Appl. Sci., 12 (2002), 1555-1564.  doi: 10.1142/S0218202502002239. [2] A. Arnold, S. Geevers, I. Perugia and D. Ponomarev, On the limiting amplitude principle for the wave equation with variable coefficients, preprint, 2022, arXiv: 2202.10105. [3] J. M. Bouclet and N. Burq, Sharp resolvent and time-decay estimates for dispersive equations on asymptotically Euclidean backgrounds, Duke Mathematical Journal, 170 (2021), 2575-2629.  doi: 10.1215/00127094-2020-0080. [4] R. Charao and R. Ikehata, A note on decay rates of the local energy for wave equations with Lipschitz wavespeeds, J. Math. Anal. Appl., 483 (2020), 1-14.  doi: 10.1016/j.jmaa.2019.123636. [5] L. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, 2010. doi: 10.1090/gsm/019. [6] J. Lewis, The heterogeneous string: coupled helices in Hilbert space, Quart. Appl. Math., 38 (1981), 461-467.  doi: 10.1090/qam/614553. [7] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [8] M. Reed, Abstract Non-linear Wave Equations, Springer-Verlag, New York, 1976. [9] M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 2004. [10] J. Shapiro, Local energy decay for Lipschitz wavespeeds, Comm. Partial Differ. Equ., 43 (2018), 839-858.  doi: 10.1080/03605302.2018.1475491. [11] W. A. Strauss, Nonlinear Wave Equations, American Mathematical Society, Providence, 1989. [12] G. Teschl, Mathematical Methods in Quantum Mechanics: with Applications to Schrödinger operators, 2$^{nd}$ edition, American Mathematical Society, Providence, 2009. doi: 10.1090/gsm/099.

show all references

##### References:
 [1] A. Arnold, J. Carrillo and M. Tidriri, Large-time behavior of discrete kinetic equations with non-symmetric interactions, Math. Models Methods Appl. Sci., 12 (2002), 1555-1564.  doi: 10.1142/S0218202502002239. [2] A. Arnold, S. Geevers, I. Perugia and D. Ponomarev, On the limiting amplitude principle for the wave equation with variable coefficients, preprint, 2022, arXiv: 2202.10105. [3] J. M. Bouclet and N. Burq, Sharp resolvent and time-decay estimates for dispersive equations on asymptotically Euclidean backgrounds, Duke Mathematical Journal, 170 (2021), 2575-2629.  doi: 10.1215/00127094-2020-0080. [4] R. Charao and R. Ikehata, A note on decay rates of the local energy for wave equations with Lipschitz wavespeeds, J. Math. Anal. Appl., 483 (2020), 1-14.  doi: 10.1016/j.jmaa.2019.123636. [5] L. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, 2010. doi: 10.1090/gsm/019. [6] J. Lewis, The heterogeneous string: coupled helices in Hilbert space, Quart. Appl. Math., 38 (1981), 461-467.  doi: 10.1090/qam/614553. [7] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [8] M. Reed, Abstract Non-linear Wave Equations, Springer-Verlag, New York, 1976. [9] M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 2004. [10] J. Shapiro, Local energy decay for Lipschitz wavespeeds, Comm. Partial Differ. Equ., 43 (2018), 839-858.  doi: 10.1080/03605302.2018.1475491. [11] W. A. Strauss, Nonlinear Wave Equations, American Mathematical Society, Providence, 1989. [12] G. Teschl, Mathematical Methods in Quantum Mechanics: with Applications to Schrödinger operators, 2$^{nd}$ edition, American Mathematical Society, Providence, 2009. doi: 10.1090/gsm/099.
Blue: function $\lambda_*(\gamma_0)$, implicitly defined by (3.21). Red: function $\lambda_0(\gamma_0)$, defined in (3.22)
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