A note on a Kawahara type equation

Source abstract

Abstract 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.