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# Forum Mathematicum

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Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 30, Issue 5

# Space-time L2 estimates, regularity and almost global existence for elastic waves

Kunio Hidano
• Department of Mathematics, Faculty of Education, Mie University, 1577 Kurima-machiya-cho, Tsu, Mie 514-8507, Japan
• Email
• Other articles by this author:
/ Dongbing Zha
Published Online: 2018-06-20 | DOI: https://doi.org/10.1515/forum-2018-0050

## Abstract

In this paper, we first establish a kind of weighted space-time ${L}^{2}$ estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time ${L}^{2}$ estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in ${H}^{3}×{H}^{2}$.

MSC 2010: 35L52; 35Q74

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Revised: 2018-05-21

Published Online: 2018-06-20

Published in Print: 2018-09-01

Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: JP15K04955

Award identifier / Grant number: JP18K03365

The first author was supported in part by JSPS KAKENHI Grants JP15K04955 and JP18K03365. The second author was supported by the Shanghai Sailing Program (no. 17YF1400700) and Fundamental Research Funds for the Central Universities (no. 17D110913).

Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1291–1307, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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