Instability of energy solutions, travelling waves, and scaling invariance for a fourth-order p-Laplacian operator with superlinear reaction

Abstract

This analysis explores the oscillatory behavior of traveling wave solutions for a higher-order p-Laplacian operator with a superlinear reaction term. The study employs an energy-based approach, incorporating generalized Sobolev spaces to examine relevant properties of the solutions, including oscillations, diffusive mollification, and compact support. Based on this energy framework, the regularity of the involved operator is established. The problem is then reformulated using a traveling wave approach, revealing the oscillatory nature of solutions near the null solution. Numerical simulations are conducted for each wave speed to validate the analytical results, yielding the corresponding traveling profiles. Notably, one of the most significant f indings is the attraction towards the null critical point, which helps prevent blow-up formation. Finally, the study delves into the equation’s scale-invariant properties, leading to the derivation of self-similar solutions.