## Optimal energy decay for the wave-heat system on a rectangular domain

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**Optimal energy decay for the wave-heat system on a rectangular domain.** / Batty, Charles; Paunonen, Lassi; Seifert, David.

Research output: Contribution to journal › Article › Scientific › peer-review

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*SIAM JOURNAL ON MATHEMATICAL ANALYSIS*, vol. 51, no. 2, pp. 808-819. https://doi.org/10.1137/18M1195796

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*SIAM JOURNAL ON MATHEMATICAL ANALYSIS*,

*51*(2), 808-819. https://doi.org/10.1137/18M1195796

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TY - JOUR

T1 - Optimal energy decay for the wave-heat system on a rectangular domain

AU - Batty, Charles

AU - Paunonen, Lassi

AU - Seifert, David

PY - 2019

Y1 - 2019

N2 - We study the rate of energy decay for solutions of a coupled wave-heat system on a rectangular domain. Using techniques from the theory of C 0 -semigroups, and in particular a well-known result due to Borichev and Tomilov, we prove that the energy of classical solutions decays like t - 2/ 3 as t \rightarrow \infty . This rate is moreover shown to be sharp. Our result implies in particular that a general estimate in the literature, which predicts at least logarithmic decay and is known to be best possible in general, is suboptimal in the special case under consideration here. Our strategy of proof involves direct estimates based on separation of variables and a refined version of the technique developed in our earlier paper for a one-dimensional wave-heat system.

AB - We study the rate of energy decay for solutions of a coupled wave-heat system on a rectangular domain. Using techniques from the theory of C 0 -semigroups, and in particular a well-known result due to Borichev and Tomilov, we prove that the energy of classical solutions decays like t - 2/ 3 as t \rightarrow \infty . This rate is moreover shown to be sharp. Our result implies in particular that a general estimate in the literature, which predicts at least logarithmic decay and is known to be best possible in general, is suboptimal in the special case under consideration here. Our strategy of proof involves direct estimates based on separation of variables and a refined version of the technique developed in our earlier paper for a one-dimensional wave-heat system.

KW - C -semigroups

KW - Coupled

KW - Energy

KW - Heat equation

KW - Rates of decay

KW - Rectangular domain

KW - Resolvent estimates

KW - Wave equation

U2 - 10.1137/18M1195796

DO - 10.1137/18M1195796

M3 - Article

VL - 51

SP - 808

EP - 819

JO - SIAM JOURNAL ON MATHEMATICAL ANALYSIS

JF - SIAM JOURNAL ON MATHEMATICAL ANALYSIS

SN - 0036-1410

IS - 2

ER -