94-202 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 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 $p_c = 1 + 2/d$.

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