Linearized stability implies asymptotic stability for radially symmetric equilibria of p-Laplacian boundary value problems in the unit ball in ℝN

Research output: Contribution to journalArticle

4 Downloads (Pure)

Abstract

We consider the parabolic initial-boundary value problem (Forumala presented). where Ω = B 1 is the unit ball centered at the origin in R N, with N ≥ 2, p > 1, and ∆ p denotes the p-Laplacian on Ω. The function f: [0, 1] × R → R is continuous, and the partial derivative f v exists and is continuous and bounded on [0, 1] × R. It will be shown that (under certain additional hypotheses) the ‘principle of linearized stability’ holds for radially symmetric equilibrium solutions u 0 of the equation. That is, the asymptotic stability, or instability, of u 0 is determined by the sign of the principal eigenvalue of a linearization of the problem at u 0. It is well-known that this principle holds for the semilinear case p = 2 (∆ 2 is the linear Laplacian), but has not been shown to hold when p ≠ 2. We also consider a bifurcation type problem similar to the one above, having a line of trivial solutions and a curve of non-trivial solutions bifurcating from the line of trivial solutions at the principal eigenvalue of the p-Laplacian. We characterize the stability, or instability, of both the trivial solutions and the non-trivial bifurcating solutions, in a neighbourhood of the bifurcation point, and we obtain a result on ‘exchange of stability’ at the bifurcation point, analogous to the well-known result when p = 2.

Original languageEnglish
Pages (from-to)1-17
Number of pages17
JournalElectronic Journal of Differential Equations
Volume2019
Issue number94
Publication statusPublished - 29 Jul 2019

Keywords

  • Bifurcation
  • P-Laplacian
  • Parabolic boundary value problem
  • Stability

ASJC Scopus subject areas

  • Analysis

Fingerprint Dive into the research topics of 'Linearized stability implies asymptotic stability for radially symmetric equilibria of <i>p</i>-Laplacian boundary value problems in the unit ball in ℝ<sup>N</sup>'. Together they form a unique fingerprint.

Cite this