### 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 language | English |
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Pages (from-to) | 1-17 |

Number of pages | 17 |

Journal | Electronic Journal of Differential Equations |

Volume | 2019 |

Issue number | 94 |

Publication status | Published - 29 Jul 2019 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis