TY - JOUR

T1 - ON THE EXPONENTIAL TIME-DECAY FOR THE ONE-DIMENSIONAL WAVE EQUATION WITH VARIABLE COEFFICIENTS

AU - Arnold, Anton

AU - Geevers, Sjoerd

AU - Perugia, Ilaria

AU - Ponomarev, Dmitry

N1 - Funding Information:
2020 Mathematics Subject Classification. Primary: 35L05, 35L10; Secondary: 35B40. Key words and phrases. Wave equation with variable coefficients, local energy decay, long-time asymptotics, decay rate estimates, one-dimensional dynamics in heterogeneous media. 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. ∗Corresponding author.
Publisher Copyright:
© 2022 American Institute of Mathematical Sciences. All rights reserved.

PY - 2022/10

Y1 - 2022/10

N2 - 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 de- cay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The sec- ond 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.

AB - 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 de- cay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The sec- ond 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.

KW - decay rate estimates

KW - local energy decay

KW - long-time asymptotics

KW - one-dimensional dynamics in heterogeneous media

KW - Wave equation with variable coefficients

UR - http://www.scopus.com/inward/record.url?scp=85138544338&partnerID=8YFLogxK

U2 - 10.3934/cpaa.2022105

DO - 10.3934/cpaa.2022105

M3 - Article

AN - SCOPUS:85138544338

VL - 21

SP - 3389

EP - 3405

JO - Communications in Pure and Applied Analysis (CPAA)

JF - Communications in Pure and Applied Analysis (CPAA)

SN - 1534-0392

IS - 10

ER -