We consider the parabolic, initial-boundary value problem
1
$$\matrix{ {\displaystyle{{\partial v} \over {\partial t}} = \Delta _p(v) + f(x,v),} & {{\rm in}({\rm - 1},{\rm 1}) \times ({\rm 0},\infty ),} \cr {v( \pm 1,t) = 0,} \hfill \hfill \hfill & {{\rm t}\in [{\rm 0},\infty ),} \hfill \hfill \cr {v = v_0\in C_0^0 ([-1,1]),} & {{\rm in}[{\rm - 1},{\rm 1}] \times \{ {\rm 0}\} ,} \cr } $$
where Δ
p denotes the
p-Laplacian on ( − 1, 1), with
p > 1, and the function
f:[ − 1, 1] × ℝ → ℝ is continuous, and the partial derivative
fv exists and is continuous and bounded on [ − 1, 1] × ℝ. It will be shown that (under certain additional hypotheses) the ‘principle of linearized stability’ holds for equilibrium solutions
u0 of (1). That is, the asymptotic stability, or instability, of
u0 is determined by the sign of the principal eigenvalue of a suitable linearization of the problem (1) at
u0. It is well-known that this principle holds for the semilinear case
p = 2 (Δ
2 is the linear Laplacian), but has not been shown to hold when
p ≠ 2.
We also consider a bifurcation type problem similar to (1), having a line of trivial solutions. We characterize the stability or instability of the trivial solutions, and the bifurcating, non-trivial solutions, and show that there is an ‘exchange of stability’ at the bifurcation point, analogous to the well-known result when p = 2.