This note proves well-posedness of classical solutions for the Cauchy problem associated with the Kawahara equation, indicating improved understanding of its dynamics.
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^∞ loc(0,∞ ;H⁴(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.
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Coclite et al. (2026) studied this question.
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