 94202 Wayne, C. E.
 Invariant Manifolds for Parabolic Partial Differential
Equations on Unbounded Domains
(64K, Plain TeX)
Jun 17, 94

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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 longtime asymptotics
in inverse powers of $\tau$. They also give a new perspective on
the change in the longtime asymptotics of the equation with
nonlinear term $F(u) = u^{p1} u$, when $p$ passes through
the critical value $p_c = 1 + 2/d$.
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