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<p>We consider the Cauchy problem for systems of semilinear hyperbolic equations of first order with multiple propagation speeds. We obtain a sufficient condition for the global existence of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-solutions with small initial data, and also the “almost” optimal lifespan of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-solutions under weaker conditions than this sufficient condition. Application to systems of semilinear wave equations in one space dimension with multiple propagation speeds is discussed, and we obtain the “almost” optimal lifespan estimate under some condition weaker than the null condition.</p>

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Lifespan of small solutions to semilinear hyperbolic systems in one space dimension

  • 期刊论文

  • 发表日期：

    2026年1月23日

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  • Soichiro Katayama,
  • Kyouhei Wakasa

摘要

We consider the Cauchy problem for systems of semilinear hyperbolic equations of first order with multiple propagation speeds. We obtain a sufficient condition for the global existence of \(C^1\) C 1 -solutions with small initial data, and also the “almost” optimal lifespan of \(C^1\) C 1 -solutions under weaker conditions than this sufficient condition. Application to systems of semilinear wave equations in one space dimension with multiple propagation speeds is discussed, and we obtain the “almost” optimal lifespan estimate under some condition weaker than the null condition.