Blow-up and lifespan estimate for wave equations with critical damping term of space-dependent type related to Glassey conjecture

Ahmad Z. Fino; Mohamed Ali Hamza

doi:10.3934/dcdss.2022192

Article Contents

2023, Volume 16, Issue 6: 1383-1400. Doi: 10.3934/dcdss.2022192

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Blow-up and lifespan estimate for wave equations with critical damping term of space-dependent type related to Glassey conjecture

Ahmad Z. Fino^1, and
Mohamed Ali Hamza^2, ,

1.
College of Engineering and Technology, American University of the Middle East, Egaila 54200, Kuwait
2.
Department of Basic Sciences, Deanship of Preparatory Year and Supporting Studies, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saoudi Arabia

^*Corresponding author: Mohamed Ali Hamza
^*Corresponding author: Mohamed Ali Hamza

Received: April 2022

Revised: October 2022

Early access: November 22, 2022

Published online: November 22, 2022

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Abstract

The main purpose of the present paper is to study the blow-up problem of the wave equation with space-dependent damping in the scale-invariant case and time derivative nonlinearity with small initial data. Under appropriate initial data which are compactly supported, by using a test function method and taking into account the effect of the damping term ($ \frac{\mu}{\sqrt{1+|x|^2}}u_t $), we show that in higher dimensions the blow-up region is given by $ p \in (1, p_G(N+\mu)] $ where $ p_G(N) $ is the Glassey exponent. Furthermore, we shall establish a blow-up region, independent of $ \mu $ given by $ p\in (1, 1+\frac{2}{N}), $ for appropriate initial data in the energy space with noncompact support.

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Mathematics Subject Classification: Primary: 35L71, 35B44, 35L15; Secondary: 35L71, 35L05.

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