An L^p theory of invariant manifolds of parabolic partial differential equations of ℝ^d

We study the problem about the existence of finite-dimensional invariant manifolds for nonlinear heat equations of the form ∂u/∂τ = △u + F(u, ∇u) on ℝ^d × [1, ∞). We show that in spite of the fact that the linearized equation has continuous spectrum extending from negative infinity to zero, there exist finite dimensional invariant manifolds which control the long time asymptotics of solutions. We consider the problem for these equations in the framework of weighted Sobolev spaces of L^p type. The L^p theory of this problem gives the L^∞ estimate of the long-time asymptotics of solutions under natural assumptions on the nonlinear term F and their initial data.