Electron. J. Differential Equations, Vol. 2019 (2019), No. 94, pp. 1-17.

Linearized stability implies asymptotic stability for radially symmetric equilibria of p-Laplacian boundary value problems in the unit ball in $\mathbb R^N$

Bryan P. Rynne

Abstract:
We consider the parabolic initial-boundary value problem
$$\displaylines{ \frac{\partial v}{\partial t} = \Delta_p(v) + f(|x|,v) , \quad \text{in } \Omega \times (0,\infty),\cr v = 0 , \quad \text{in } \partial \Omega \times [0,\infty),\cr v = v_0 \in C_0^0(\overline\Omega) , \quad \text{in } \overline\Omega \times \{0\}, }$$
where $\Omega = B_1$ is the unit ball centered at the origin in $\mathbb R^N$, with $ N \ge 2$, $ p > 1 $, and $\Delta_p$ denotes the p-Laplacian on $\Omega$. The function $f:[0,1]\times{\mathbb R}\to{\mathbb R}$ is continuous, and the partial derivative $f_v$ exists and is continuous and bounded on $[0,1]\times{\mathbb R}$. 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 ($\Delta_2$ is the linear Laplacian), but has not been shown to hold when $p \ne 2$. 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.

Submitted November 27, 2018. Published July 29, 2019.
Math Subject Classifications: 35B35, 35B32, 35K55.
Key Words: p-Laplacian; parabolic boundary value problem; stability; bifurcation.

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Bryan P. Rynne
Department of Mathematics and the Maxwell Institute for Mathematical Sciences
Heriot-Watt University
Edinburgh EH14 4AS, Scotland, UK
email: B.P.Rynne@hw.ac.uk

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