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Volume 36, Issue 4
The Lifespan of Smooth Solutions to Semilinear Wave Equations in Schwarzschild Space-Time

Qiong Lou & Shaoying Luo

DOI: 10.4208/jpde.v36.n4.6

J. Part. Diff. Eq., 36 (2023), pp. 404-413.

Published online: 2023-11

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  • Abstract

This paper considers the Cauchy problem of the semilinear wave equations with small initial data in the Schwarzschild space-time, $□_gu = |u_t | ^p ,$ where $g$ denotes the Schwarzschild metric. When $1< p<2$ and the initial data are supported far away from the black hole, we can prove that the lifespan of the spherically symmetric solution obtains the same order as the semilinear wave equation evolving in the Minkowski space-time by introducing an auxiliary function.

  • Keywords

Semilinear wave equations, Schwarzschild spacetime, blow-up, lifespan.

  • AMS Subject Headings

35L05, 35L15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-36-404, author = {Lou , Qiong and Luo , Shaoying}, title = {The Lifespan of Smooth Solutions to Semilinear Wave Equations in Schwarzschild Space-Time}, journal = {Journal of Partial Differential Equations}, year = {2023}, volume = {36}, number = {4}, pages = {404--413}, abstract = {

This paper considers the Cauchy problem of the semilinear wave equations with small initial data in the Schwarzschild space-time, $□_gu = |u_t | ^p ,$ where $g$ denotes the Schwarzschild metric. When $1< p<2$ and the initial data are supported far away from the black hole, we can prove that the lifespan of the spherically symmetric solution obtains the same order as the semilinear wave equation evolving in the Minkowski space-time by introducing an auxiliary function.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v36.n4.6}, url = {http://global-sci.org/intro/article_detail/jpde/22137.html} }
TY - JOUR T1 - The Lifespan of Smooth Solutions to Semilinear Wave Equations in Schwarzschild Space-Time AU - Lou , Qiong AU - Luo , Shaoying JO - Journal of Partial Differential Equations VL - 4 SP - 404 EP - 413 PY - 2023 DA - 2023/11 SN - 36 DO - http://doi.org/10.4208/jpde.v36.n4.6 UR - https://global-sci.org/intro/article_detail/jpde/22137.html KW - Semilinear wave equations, Schwarzschild spacetime, blow-up, lifespan. AB -

This paper considers the Cauchy problem of the semilinear wave equations with small initial data in the Schwarzschild space-time, $□_gu = |u_t | ^p ,$ where $g$ denotes the Schwarzschild metric. When $1< p<2$ and the initial data are supported far away from the black hole, we can prove that the lifespan of the spherically symmetric solution obtains the same order as the semilinear wave equation evolving in the Minkowski space-time by introducing an auxiliary function.

Qiong Lou & Shaoying Luo. (2023). The Lifespan of Smooth Solutions to Semilinear Wave Equations in Schwarzschild Space-Time. Journal of Partial Differential Equations. 36 (4). 404-413. doi:10.4208/jpde.v36.n4.6
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