Almost Global Existence for 3-D Quasilinear Wave Equations in Exterior Domains with Neumann Boundary Conditions

Meng Yuan

doi:10.3934/cpaa.2022120

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2022, Volume 21, Issue 11: 3721-3753. Doi: 10.3934/cpaa.2022120

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Almost Global Existence for 3-D Quasilinear Wave Equations in Exterior Domains with Neumann Boundary Conditions

Meng Yuan

School of Computer Science and Artificial Intelligence, Aliyun School of Big Data, School of Software, Changzhou University, Changzhou 213164, China

Received: April 2022

Revised: June 2022

Early access: August 2022

Published: November 2022

The author is supported by Jiangsu Shuangchuang Program (No. JSSCBS20210901).

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Abstract

We are concerned with the initial-boundary value problems of 3-D quasilinear wave equations outside compact convex obstacles with Neumann boundary conditions. When the surfaces of 3-D compact convex obstacles are smooth and the quadratic nonlinearities in the quasilinear wave equations do not fulfill the null condition, we establish the almost global existence of smooth small-amplitude solutions to the above initial-boundary value problems. The lower bound of the lifespan is proved to be $ e^{\kappa/\varepsilon} $ with some positive $ \kappa $ and the small positive parameter $ \varepsilon $ as the size of the initial data. This lower bound is optimal as shown in the 3-D boundaryless case.

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Mathematics Subject Classification: Primary: 35L05, 35L10, 35L20, 35L72.

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