Lifespan estimates of solutions to quasilinear wave equations with damping and negative mass term

Jie Yang; Sen Ming; Wei Han; Xiongmei Fan

doi:10.3934/mcrf.2022022

Article Contents

2023, Volume 13, Issue 3: 946-966. Doi: 10.3934/mcrf.2022022

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Lifespan estimates of solutions to quasilinear wave equations with damping and negative mass term

1.
Data Science and Technology, North University of China, Taiyuan 030051, China
2.
Department of Mathematics, North University of China, Taiyuan 030051, China

^*Corresponding author
^*Corresponding author

Received: November 2021

Early access: May 2022

Published: September 2023

The project is supported by Fundamental Research Program of Shanxi Province(No. 20210302123021, No. 20210302123045), Innovative Research Team of North University of China (No. TD201901), Program for the Innovative Talents of Higher Education Institutions of Shanxi Province

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Abstract

The main goal of this work is to investigate formation of singularities for solutions to the quasilinear wave equations with damping terms, negative mass terms and divergence form nonlinearities in the critical and sub-critical cases. Upper bound lifespan estimates of solutions are derived by applying the rescaled test function method and iteration technique. The results are the same as corresponding wave equation without damping term and mass term. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent. To the best of our knowledge, the results in Theorems $ 1.1-1.4 $ are new. Moreover, the changing trends of semilinear wave equations are illustrated through numerical simulation.

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Mathematics Subject Classification: Primary: 35L70, 58J45.

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Figure 1. wave equation with power nonlinearity

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Figure 2. wave equation with derivative nonlinearity

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Figure 3. wave equation with combined nonlinearities

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