The center manifold technique and complex dynamics of parabolic equations

Abstract

Consider the following scalar parabolic equation

EquationSourceu_t - Lu = f\left( {x,u,\nabla u} \right), t > 0,x \in \Omega$$

((1))

with Dirichlet boundary condition

EquationSource u\left( {u,t} \right) = 0, t > 0,x \in \partial \Omega $$

((2))

or Neumann boundary condition

EquationSource \frac{\partial } {{\partial v}}u\left( {x,t} \right) = 0, t > 0,x \in \partial \Omega $$

((3))

Here Ω ⊂ ℝ^N is a smooth bounded domain (most frequently just the open unit ball), L is a second order selfadjoint uniformly elliptic differential operator with smooth coefficients (most often Lu = Δu + a(x)u where a is a smooth function on EquationSource \bar \Omega $$ ) and v is the outer normal to the boundary ∂Ω of Ω. Finally,

$$ f:\left( {x,s,w} \right) \in \bar \Omega \times \mathbb{R} \times \mathbb{R}^N \mapsto f\left( {x,s,w} \right) \in \mathbb{R} $$

is some nonlinearity.

In these lectures we describe some recent realization results for vector fields or jets of vector fields on invariant manifolds of Eq. (1)–(2) and (1)–(3) and give an example of Eq. (1)–(2) with gradient independent nonlinearity and admitting a nonconvergent bounded trajectory. Some necessary mathematical background (center manifold theorem, Nash-Moser inverse mapping theorem) is also provided.