Wayne, C. E.
Invariant Manifolds for Parabolic Partial Differential
Equations on Unbounded Domains
(64K, Plain TeX)
ABSTRACT. In this paper finite dimensional invariant manifolds
for nonlinear parabolic partial differential equations of the
form
$$
{{\partial u}\over{\partial \tau}} = \Delta u + F(u)
~;~ u = u(\xi,\tau)~,\quad \xi \in \real^d~, \tau \ge 1~~,
$$
are constructed.
Such results are somewhat surprising because of the continuous
spectrum of the linearized equation.
These manifolds control the long time behavior of solutions
of these equations and can be
used to construct systematic expansions of the long-time asymptotics
in inverse powers of $\tau$. They also give a new perspective on
the change in the long-time asymptotics of the equation with
nonlinear term $F(u) = |u|^{p-1} u$, when $p$ passes through
the critical value $p_c = 1 + 2/d$.