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

2019-07-29
Rynne, Bryan
##### Publisher
Texas State University, Department of Mathematics
##### Abstract
We consider the parabolic initial-boundary value problem ∂v / ∂t = ∆p(v) + ƒ(|x|, v), in Ω x (0, ∞), v = 0, in ∂Ω x [0, ∞), v = v0 ∈ C00(Ω̅), in Ω̅ x {0}, where Ω = B1 is the unit ball centered at the origin in ℝN, with N ≥ 2, p > 1, and ∆p denotes the p-Laplacian on Ω. The function ƒ : [0, 1] x ℝ → ℝ is continuous, and the partial derivative ƒv exists and is continuous and bounded on [0, 1] x ℝ. It will be shown that (under certain additional hypotheses) the 'principle of linearized stability' holds for radially symmetric equilibrium solutions u0 of the equation. That is, the asymptotic stability, or instability, of u0 is determined by the sign of the principal eigenvalue of a linearization of the problem at u0. 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 and a curve of non-trivial solutions bifurcating from the 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 neighborhood 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.
##### Keywords
p-Laplacian, Parabolic boundary value problem, Stability, Bifurcation
##### Citation
Rynne, B. P. (2019). Linearized stability implies asymptotic stability for radially symmetric equilibria of p-Laplacian boundary value problems in the unit ball in ℝN. <i>Electronic Journal of Differential Equations, 2019</i>(94), pp. 1-17.