Electron. J. Differential Equations,
Vol. 2019 (2019), No. 94, pp. 117.
Linearized stability implies asymptotic stability for radially symmetric
equilibria of pLaplacian boundary value problems in the unit ball in
Bryan P. Rynne
Abstract:
We consider the parabolic initialboundary value problem
where
is the unit ball centered at the origin in
,
with
,
, and
denotes the pLaplacian on
.
The function
is continuous, and the
partial derivative
exists and is continuous and bounded on
.
It will be shown that (under certain additional hypotheses) the
"principle of linearized stability" holds for radially symmetric equilibrium
solutions
of the equation.
That is, the asymptotic stability, or instability, of
is determined by
the sign of the principal eigenvalue of a linearization of the problem at
.
It is wellknown that this principle holds for the semilinear case p=2
(
is the linear Laplacian), but has not been shown to hold when
.
We also consider a bifurcation type problem similar to the one above, having a
line of trivial solutions and a curve of nontrivial solutions bifurcating
from the line of trivial solutions at the principal eigenvalue of the pLaplacian.
We characterize the stability, or instability, of both the trivial solutions and
the nontrivial 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 wellknown result when p=2.
Submitted November 27, 2018. Published July 29, 2019.
Math Subject Classifications: 35B35, 35B32, 35K55.
Key Words: pLaplacian; parabolic boundary value problem; stability; bifurcation.
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Bryan P. Rynne
Department of Mathematics and the Maxwell Institute for Mathematical Sciences
HeriotWatt University
Edinburgh EH14 4AS, Scotland, UK
email: B.P.Rynne@hw.ac.uk

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