Remark on a semirelativistic equation in the energy space

Kazumasa Fujiwara; Shuji Machihara; Tohru  Ozawa

doi:10.3934/proc.2015.0473

Article Contents

2015, Volume 2015: 473-478. Doi: 10.3934/proc.2015.0473 open access

This issue Previous Article Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$ Next Article Estimates for solutions of nonautonomous semilinear ill-posed problems

Remark on a semirelativistic equation in the energy space

1.
Department of Pure and Applied Physics, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555
2.
Faculty of Science, Saitama University, 255 Shimo-Okubo, Saitama 338-8570
3.
Department of Applied Physics, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555

Received: August 31, 2014

Revised: January 31, 2015

Published: October 2015

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Abstract

Well-posedness of the Cauchy problem for a semirelativistic equation with cubic nonlinearity is shown in the energy space $H^{1/2}$. Solutions are constructed as a limit of approximation solutions, where the argument on the convergence depends on the completeness of $L^2$ and is independent of compactness. The Yudovitch type argument plays an important role for the convergence arguments.

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Mathematics Subject Classification: Primary: 35Q40; Secondary: 35Q55.

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References

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