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<p>The Kawahara equation models the deformation waves in shape memory alloys, the evolution of magneto-acoustic waves in plasmas and the propagation of nonlinear water-waves in the long-wave length region. In this paper, we prove the well-posedness of the classical solutions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^\infty _{\textrm{loc}}(0,\infty ;H^4(\mathbb {R}))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mtext>loc</mtext> <mi>∞</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo>;</mo> <msup> <mi>H</mi> <mn>4</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to the Cauchy problem, associated with this equation. The existence argument is based on a vanishing viscosity type approximation of the problem and the Aubin--Lions lemma. The main tool for the uniqueness and the stability with respect to the initial data is energy estimates. We improve the existing literature considering a larger class of fluxes.</p>

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A note on a Kawahara type equation

  • 期刊论文

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  • 发表日期：

    2026年3月18日

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  • Giuseppe Maria Coclite,
  • Lorenzo di Ruvo

摘要

The Kawahara equation models the deformation waves in shape memory alloys, the evolution of magneto-acoustic waves in plasmas and the propagation of nonlinear water-waves in the long-wave length region. In this paper, we prove the well-posedness of the classical solutions in \(L^\infty _{\textrm{loc}}(0,\infty ;H^4(\mathbb {R}))\) L loc ∞ ( 0 , ∞ ; H 4 ( R ) ) to the Cauchy problem, associated with this equation. The existence argument is based on a vanishing viscosity type approximation of the problem and the Aubin--Lions lemma. The main tool for the uniqueness and the stability with respect to the initial data is energy estimates. We improve the existing literature considering a larger class of fluxes.