Oscillatory solutions and smoothing of a higher-order p-Laplacian operator

The goal of this paper was to provide a general analysis of the solutions to a higher-order p-Laplacian operator with nonlinear advection. Generally speaking, it is well known that any solution to a higher-order operator exhibits oscillations. In the present study, an advection term is introduced. This will allow us to analyze smoothing conditions in the solutions. The study of existence and uniqueness is based on a variational approach. Solutions are analyzed with an energy formulation initially discussed by Saint-Venant and extended in the works by Tikhonov and Täklind. This variational principle is supported by the definition of generalized norms under Hilbert-Sobolev spaces, enabling focus on the oscillating properties of solutions. Afterward, the paper introduces an analysis to characterize the traveling wave kind of solutions together with their characterization to understand the oscillations. Finally, a numerical exploration focuses on the smoothing conditions by the action of the nonlinear advection term. As a main finding to report: There exist a traveling wave speed ( $\lambda$) and an advection coefficient ( $c^{\ast }$) for which the profile's first minimum is almost positive, and such positivity holds beyond the first minimum.