Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 68, pp. 1-22.

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

Author: Jose Luis Diaz Palencia (Univ. a Distancia de Madrid, Spain)

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 findings 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.


Submitted June 18, 2024. Published November 07, 2024.
Math Subject Classifications: 35K92, 35K91, 35K55.
Key Words: Higher order p-Laplacian operator; travelling waves; homotopy; superlinear reaction.